defeo.bib

@article{koepf1999chebyshev,
  title={Efficient Computation of Chebyshev Polynomials in Computer Algebra},
  author={Koepf, Wolfram},
  journal={Computer Algebra Systems: A Practical Guide.},
  publisher={Wiley},
  pages={79--99},
  year={1999}
}

@book{silverman2007arithmetic,
  title={The arithmetic of dynamical systems},
  author={Silverman, Joseph H},
  series={Graduate Texts in Mathematics},
  volume={241},
  year={2007},
  publisher={Springer}
}

@article{hart2010flint,
  title={Fast library for number theory: an introduction},
  author={Hart, William},
  journal={Mathematical Software--ICMS 2010},
  pages={88--91},
  year={2010},
  publisher={Springer},
  url={http://www.flintlib.org},
}

@manual{Sage,
    author = {Stein, William A. and Others},
    citeulike-article-id = {10862504},
    comment = {{\url{http://www.sagemath.org}}},
    key = {Sage},
    organization = {The Sage Development Team},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    title = {{S}age {M}athematics {S}oftware ({V}ersion 5.5)},
    year = {2013}
}

@article{lenstra02-pell,
    author = {Lenstra, Hendrick W.},
    citeulike-article-id = {11897159},
    citeulike-linkout-0 = {http://www.ams.org/notices/200202/fea-lenstra.pdf},
    journal = {Notices of the {AMS}},
    keywords = {number\_theory},
    number = {2},
    pages = {182--192},
    posted-at = {2013-01-16 12:37:54},
    priority = {4},
    title = {Solving the {P}ell equation},
    url = {http://www.ams.org/notices/200202/fea-lenstra.pdf},
    volume = {49},
    year = {2002}
}

@misc{lemmermeyer03,
    abstract = {{The aim of this article is to show that the arithmetic of Pell conics admits
a description which is completely analogous to that of elliptic curves: there
is a theory of 2-descent with associated Selmer and Tate-Shafarevich groups,
and there should be an analog of the conjecture of Birch and Swinnerton-Dyer.}},
    archivePrefix = {arXiv},
    author = {Lemmermeyer, Franz},
    citeulike-article-id = {11890240},
    citeulike-linkout-0 = {http://arxiv.org/abs/math/0311306},
    citeulike-linkout-1 = {http://arxiv.org/pdf/math/0311306},
    day = {18},
    eprint = {math/0311306},
    keywords = {algebraic\_curves},
    posted-at = {2013-01-10 23:26:12},
    priority = {2},
    title = {{Conics - a Poor Man's Elliptic Curves}},
    url = {http://arxiv.org/abs/math/0311306},
    year = {2003}
}

@article{hambleton12,
    author = {Hambleton, Samuel A.},
    citeulike-article-id = {11890238},
    citeulike-linkout-0 = {http://www.ams.org/journals/proc/2012-140-08/S0002-9939-2011-11196-1/},
    journal = {Proceedings of the American Mathematical Society},
    key = {hambleton12},
    keywords = {algebraic\_curves, primality},
    pages = {2653--2661},
    posted-at = {2013-01-10 23:25:00},
    priority = {1},
    title = {Generalized {L}ucas-{L}ehmer tests using {P}ell conics},
    url = {http://www.ams.org/journals/proc/2012-140-08/S0002-9939-2011-11196-1/},
    volume = {140},
    year = {2012}
}

@article{LiRo01,
title = "Sylvester–Habicht Sequences and Fast Cauchy Index Computation",
journal = "J. Symbolic Comput.",
volume = "31",
number = "3",
pages = "315 - 341",
year = "2001",
note = "",
issn = "0747-7171",
doi = "10.1006/jsco.2000.0427",
url = "http://www.sciencedirect.com/science/article/pii/S0747717100904279",
author = "T. Lickteig and {M.-F.} Roy"
}

@article{KeUm11,
        Author = {Kedlaya, K. S. and Umans, C.},
        Journal = {SIAM J. Computing},
        Number = {6},
        Pages = {1767-1802},
        Title = {Fast Polynomial Factorization and Modular Composition},
        Volume = {40},
        Year = {2011}
}

@Article{GaSc12,
  author = 	 {P. Gaudry and {\'E}. Schost},
  title = 	 {Point-counting in genus 2 over prime fields},
  journal = {J. Symbolic Comput.},
  volume = 47,
 number = 4,
 pages = {368–400},
year = 2012
}

@inproceedings{LeSc12,
  author    = {R. Lebreton and {\'E}. Schost},
  title     = {Algorithms for the universal decomposition algebra},
  booktitle = {ISSAC'12},
 publisher = {ACM},
pages = {234-241},
  year      = {2012},
}

@Misc{DoSc12,
  author = 	 {Doliskani, J. and Schost, {\'E}.},
  title = 	 {A Note on Computations in Degree $2^k$-Extensions of Finite Fields},
  year = 	 2012,
  note = 	 {Manuscript}}

@InProceedings{Reischert97,
  author =       {Reischert, D.},
  title =        {Asymptotically Fast Computation of Subresultants},
  booktitle = {ISSAC},
  pages =        {233--240},
  year =         {1997},
  publisher = {ACM},
}

@Article{Shoup90,
  author = 	 {Shoup, Victor},
  title = 	 {New algorithms for finding irreducible polynomials over finite fields},
  journal = 	 {Math. Comp.},
  year = 	 1990,
  volume = 	 54,
  pages = 	 {435--447}}

@book{ireland1990classical,
  title={A Classical Introduction to Modern Number Theory},
  author={Ireland, K. and Rosen, M.},
  series={Graduate Texts in Mathematics},
volume = 84,
    edition = {Second},
  url={http://books.google.ca/books?id=jhAXHuP2y04C},
  year={1990},
  publisher={Springer}
}

@inproceedings{benjamin10,
    author = {Benjamin, Arthur T.},
    booktitle = {Congressus Numerantium},
    citeulike-article-id = {11696584},
    citeulike-linkout-0 = {http://www.math.hmc.edu/~benjamin/papers/LucasTriangle.pdf},
    keywords = {fibonacci, lucas},
    pages = {169--177},
    posted-at = {2012-11-15 22:11:00},
    priority = {0},
    title = {The {L}ucas Triangle Recounted},
    url = {http://www.math.hmc.edu/~benjamin/papers/LucasTriangle.pdf},
    volume = {200},
    year = {2010}
}

@article{eftekhari12,
    abstract = {We consider a key exchange procedure whose security is based on the
difficulty of computing discrete logarithms in a group, and where
exponentiation is hidden by a conjugation. We give a platform-dependent
cryptanalysis of this protocol. Finally, to take full advantage of this
procedure, we propose a group of matrices over a noncommutative ring as
platform group},
    archivePrefix = {arXiv},
    author = {Eftekhari, Mohammad},
    citeulike-article-id = {11353354},
    citeulike-linkout-0 = {http://arxiv.org/abs/1209.6144},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1209.6144},
    citeulike-linkout-2 = {http://dx.doi.org/10.1515/gcc-2012-0001},
    day = {27},
    doi = {10.1515/gcc-2012-0001},
    eprint = {1209.6144},
    issn = {1867-1144},
    journal = {Groups – Complexity – Cryptology},
    keywords = {affine\_group, braids, cryptography, finite\_field},
    month = sep,
    number = {3},
    pages = {185--203},
    posted-at = {2012-10-01 13:55:13},
    priority = {0},
    title = {A {D}iffie-{H}ellman Key Exchange Using Matrices Over Non Commutative Rings},
    url = {http://dx.doi.org/10.1515/gcc-2012-0001},
    volume = {3},
    year = {2012}
}

@electronic{li+wu+zhang12-jacobi-pairing,
    abstract = {In this paper, we propose an elaborate geometric approach to explain the group law on Jacobi quartic curves which are seen as the intersection of two quadratic surfaces in space. Using the geometry
 interpretation we construct the Miller function. Then we present explicit formulae for the addition and doubling steps in Miller's algorithm to compute Tate pairing on Jacobi quartic curves. Both the addition step and doubling step of our formulae for Tate pairing computation on Jacobi curves are faster than previously proposed ones.
 Finally, we present efficient formulas for Jacobi quartic curves with twists of degree 4 or 6. For twists of degree 4, both the addition steps and doubling steps in our formulas are faster than the fastest result on Weierstrass curves. For twists of degree 6, the addition steps of our formulae are faster than the fastest result on Weierstrass curves.},
    author = {Li, Liangze and Wu, Hongfeng and Zhang, Fan},
    citeulike-article-id = {11353303},
    citeulike-linkout-0 = {http://eprint.iacr.org/2012/551},
    day = {21},
    howpublished = {Cryptology ePrint Archive, Report 2012/551},
    keywords = {elliptic\_curve, pairing},
    month = sep,
    posted-at = {2012-10-01 13:28:13},
    priority = {0},
    title = {Faster Pairing Computation on {J}acobi quartic Curves with High-Degree Twists},
    url = {http://eprint.iacr.org/2012/551},
    year = {2012}
}

@misc{regev04,
    abstract = {In a recent paper, Kuperberg described the first subexponential time
algorithm for solving the dihedral hidden subgroup problem. The space
requirement of his algorithm is super-polynomial. We describe a modified
algorithm whose running time is still subexponential and whose space
requirement is only polynomial.},
    archivePrefix = {arXiv},
    author = {Regev, Oded},
    citeulike-article-id = {11333453},
    citeulike-linkout-0 = {http://arxiv.org/abs/quant-ph/0406151},
    citeulike-linkout-1 = {http://arxiv.org/pdf/quant-ph/0406151},
    day = {21},
    eprint = {quant-ph/0406151},
    keywords = {discrete\_log, quantum},
    month = jun,
    posted-at = {2012-09-27 11:15:26},
    priority = {2},
    title = {A Subexponential Time Algorithm for the Dihedral Hidden Subgroup Problem with Polynomial Space},
    url = {http://arxiv.org/abs/quant-ph/0406151},
    year = {2004}
}

@article{bisson+sutherland11-rho,
    abstract = {We describe a space-efficient algorithm for solving a generalization of the subset sum problem in a finite group G, using a Pollard-ρ approach. Given an element z and a sequence of elements S, our algorithm attempts to find a subsequence of S whose product in G is equal to z. For a random sequence S of length d log2 n, where n = \#G and d ≥ 2 is a constant, we find that its expected running time is \$\${O(\sqrt{n}\,{\rm log}\,n)}\$\$ group operations (we give a rigorous proof for d > 4), and it only needs to store O(1) group elements. We consider applications to class groups of imaginary quadratic fields, and to finding isogenies between elliptic curves over a finite field.},
    author = {Bisson, Gaetan and Sutherland, Andrew V.},
    citeulike-article-id = {9441544},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s10623-011-9527-8},
    citeulike-linkout-1 = {http://www.springerlink.com/content/4293k3621h3316j7},
    day = {9},
    doi = {10.1007/s10623-011-9527-8},
    issn = {0925-1022},
    journal = {Designs, Codes and Cryptography},
    keywords = {discrete\_log, isogenies},
    month = jun,
    number = {1},
    pages = {1--13},
    posted-at = {2012-09-27 10:44:26},
    priority = {4},
    publisher = {Springer Netherlands},
    title = {A low-memory algorithm for finding short product representations in finite groups},
    url = {http://dx.doi.org/10.1007/s10623-011-9527-8},
    volume = {63},
    year = {2011}
}

@electronic{arana+etal12-koblitz_curves,
    abstract = {We design a state-of-the-art software implementation of field and elliptic curve arithmetic in standard Koblitz curves at the 128-bit security level. Field arithmetic is carefully crafted by using the best formulae and implementation strategies available, and the increasingly common native support to binary field arithmetic in modern desktop computing platforms. The i-th power of the Frobenius automorphism on Koblitz curves is exploited to obtain new and faster interleaved versions of the well-known \$\tau\$NAF scalar multiplication algorithm. The usage of the \$\tau^{\lfloor m/3 \rfloor}\$ and
 \$\tau^{\lfloor m/4 \rfloor}\$ maps are employed to create analogues of the 3-and 4-dimensional GLV decompositions and in general, the \$\lfloor m/s \rfloor\$-th power of the Frobenius automorphism is applied as an analogue of an \$s\$-dimensional GLV decomposition. The effectiveness of these techniques is illustrated by timing the scalar multiplication operation for fixed, random and multiple points. To our knowledge, our library was the first to compute a random point scalar multiplication in less than 10^5 clock cycles among all curves with or without endomorphisms defined over binary or prime fields. The results of our optimized implementation suggest a trade-off between speed, compliance with the published standards and side-channel protection. Finally, we estimate the performance of curve-based cryptographic protocols instantiated using the proposed techniques and compare our results to related work.},
    author = {Aranha, Diego F. and Faz-Hern\'{a}ndez, Armando and L\'{o}pez, Julio and Rodr\'{\i}guez-Henr\'{\i}quez, Francisco},
    citeulike-article-id = {11192498},
    citeulike-linkout-0 = {http://eprint.iacr.org/2012/519},
    day = {4},
    howpublished = {Cryptology ePrint Archive, Report 2012/519},
    keywords = {elliptic\_curve, finite\_field, naf},
    month = sep,
    posted-at = {2012-09-06 12:44:38},
    priority = {3},
    title = {Faster implementation of scalar multiplication on Koblitz curves},
    url = {http://eprint.iacr.org/2012/519},
    year = {2012}
}

@book{davenport2000multiplicative,
    author = {Davenport, Harold and Montgomery, Hugh L.},
    citeulike-article-id = {11162843},
    keywords = {cyclotomic\_polynomial},
    posted-at = {2012-08-31 15:27:28},
    priority = {2},
    publisher = {Springer Verlag},
    title = {Multiplicative number theory},
    volume = {74},
    year = {2000}
}

@book{washington1997introduction,
    author = {Washington, Lawrence C.},
    citeulike-article-id = {11162842},
    keywords = {cyclotomic\_polynomial},
    posted-at = {2012-08-31 15:27:27},
    priority = {2},
    publisher = {Springer Verlag},
    title = {Introduction to cyclotomic fields},
    volume = {83},
    year = {1997}
}

@book{lang1990cyclotomic,
    author = {Lang, Serge},
    citeulike-article-id = {11162841},
    keywords = {cyclotomic\_polynomial},
    posted-at = {2012-08-31 15:27:27},
    priority = {2},
    publisher = {Springer},
    title = {Cyclotomic fields I and {II}},
    year = {1990}
}

@electronic{lenstra+desmit08-stdmodels,
    author = {Lenstra, Hendrick W. and De Smit, Bart},
    citeulike-article-id = {11159208},
    citeulike-linkout-0 = {http://www.math.leidenuniv.nl/~desmit/papers/standard_models.pdf},
    keywords = {cyclotomic\_polynomial, irreducible\_polynomials},
    pages = {1--4},
    posted-at = {2012-08-30 23:45:53},
    priority = {0},
    title = {Standard models for finite fields: the definition},
    url = {http://www.math.leidenuniv.nl/~desmit/papers/standard_models.pdf},
    year = {2008}
}

@article{thaine01,
    author = {Thaine, F.},
    citeulike-article-id = {11159205},
    journal = {Mathematics of computation},
    keywords = {cyclotomic\_polynomial},
    number = {236},
    posted-at = {2012-08-30 23:42:14},
    priority = {2},
    publisher = {American Mathematical Society},
    title = {Jacobi sums and new families of irreducible polynomials of {G}aussian periods},
    volume = {70},
    year = {2001}
}

@article{thaine00,
    author = {Thaine, F.},
    citeulike-article-id = {11159204},
    journal = {Mathematics of computation},
    keywords = {cyclotomic\_polynomial},
    number = {232},
    posted-at = {2012-08-30 23:42:13},
    priority = {2},
    publisher = {American Mathematical Society},
    title = {Families of irreducible polynomials of {G}aussian periods and matrices of cyclotomic numbers},
    volume = {69},
    year = {2000}
}

@article{gurak06powers,
    author = {Gurak, Stanley},
    citeulike-article-id = {11159183},
    journal = {Mathematics of computation},
    keywords = {cyclotomic\_polynomial},
    number = {256},
    pages = {2021},
    posted-at = {2012-08-30 23:30:19},
    priority = {2},
    publisher = {American Mathematical Society},
    title = {On the minimal polynomial of Gauss periods for prime powers},
    volume = {75},
    year = {2006}
}

@article{gupta+zagier93,
    author = {Gupta, S. and Zagier, D.},
    citeulike-article-id = {11159182},
    journal = {Mathematics of computation},
    keywords = {cyclotomic\_polynomial},
    number = {201},
    posted-at = {2012-08-30 23:30:19},
    priority = {2},
    publisher = {American Mathematical Society},
    title = {On the coefficients of the minimal polynomials of Gaussian periods},
    volume = {60},
    year = {1993}
}

@article{gurak06,
    author = {Gurak, Stanley},
    citeulike-article-id = {11159159},
    journal = {Acta Arithmetica},
    keywords = {cyclotomic\_polynomial},
    number = {3},
    pages = {233},
    posted-at = {2012-08-30 23:26:37},
    priority = {2},
    publisher = {Institute of Mathematics Polish Academy of Sciences},
    title = {Minimal polynomials for Gauss periods with f=2},
    volume = {121},
    year = {2006}
}

@article{couveignes+lercier11,
    author = {Couveignes, Jean-Marc and Lercier, Reynald},
    citeulike-article-id = {11159134},
    howpublished = {Eprint arXiv:0905.1642},
    journal = {To appear in the Israel Journal of Mathematics},
    keywords = {elliptic\_curve, finite\_field, irreducible\_polynomials},
    month = jul,
    posted-at = {2012-08-30 23:25:40},
    priority = {0},
    title = {Fast construction of irreducible polynomials over finite fields.},
    year = {2011}
}

@book{voskresenskii98,
    author = {Voskresenski\u{i}, Valentin E.},
    citeulike-article-id = {11159127},
    posted-at = {2012-08-30 23:10:34},
    priority = {2},
    publisher = {American Mathematical Society},
    title = {Algebraic groups and their birational invariants},
    volume = {179},
    year = {1998}
}

@inproceedings{rubin+silverberg03,
    abstract = {Abstract. We give a mathematical interpretation in terms of algebraic tori of Lucas-based cryptosystems, {XTR}, and the conjectural generalizations in [2]. We show that the varieties underlying these systems are quotients of algebraic tori by actions of products of symmetric groups. Further, we use these varieties to disprove conjectures from [2]. 1},
    author = {Rubin, Karl and Silverberg, Alice},
    booktitle = {In High Primes and Misdemeanours: Lectures in Honour of the 60th birthday of Hugh Cowie Williams},
    citeulike-article-id = {11159117},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.112.3935},
    howpublished = {Available on the Internet from \url{http://supertech.csail.mit.edu/papers. html}},
    keywords = {algebraic\_geometry, cryptography, finite\_field},
    posted-at = {2012-08-30 23:04:04},
    priority = {2},
    publisher = {AMS},
    series = {Fields Institute Communications},
    title = {Algebraic Tori in Cryptography},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.112.3935},
    volume = {41},
    year = {2004}
}

@inproceedings{rubin-silverberg+crypto03,
    abstract = {We introduce the concept of torus-based cryptography, give a new public key system called {CEILIDH}, and compare it to other discrete log based systems including Lucas-based systems and {XTR}. Like those systems, we obtain small key sizes. While Lucas-based systems and {XTR} are essentially restricted to exponentiation, we are able to perform multiplication as well. We also disprove the open conjectures from [2], and give a new algebro-geometric interpretation of the approach in that paper and of {LUC} and {XTR}.},
    address = {Berlin, Heidelberg},
    author = {Rubin, Karl and Silverberg, Alice},
    booktitle = {Advances in Cryptology - CRYPTO 2003},
    chapter = {21},
    citeulike-article-id = {11159115},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-45146-4_21},
    citeulike-linkout-1 = {http://www.springerlink.com/content/jvg4de9ff5fn3ec3},
    doi = {10.1007/978-3-540-45146-4_21},
    editor = {Boneh, Dan},
    isbn = {978-3-540-40674-7},
    keywords = {algebraic\_geometry, cryptography, finite\_field},
    pages = {349--365},
    posted-at = {2012-08-30 23:02:40},
    priority = {0},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {{Torus-Based} Cryptography},
    url = {http://dx.doi.org/10.1007/978-3-540-45146-4_21},
    volume = {2729},
    year = {2003}
}

@electronic{defeo+jao+plut12,
    abstract = {We present new candidates for quantum-resistant public-key cryptosystems based on the conjectured difficulty of finding isogenies between supersingular elliptic curves.  The main technical idea in our scheme is that we transmit the images of torsion bases under the isogeny in order to allow the parties to construct a shared commutative square despite the noncommutativity of the endomorphism ring.  Our work is motivated by the recent development of a subexponential-time quantum algorithm for constructing isogenies between ordinary elliptic curves. In the supersingular case, by contrast, the fastest known quantum attack remains exponential, since the noncommutativity of the endomorphism ring means that the approach used in the ordinary case does not apply. We give a precise formulation of the necessary computational assumptions along with a discussion of their validity, and prove the security of our protocols under these assumptions. In addition, we present implementation results showing that our protocols are multiple orders of magnitude faster than previous isogeny-based cryptosystems over ordinary curves. <P> This paper is an extended version of\~{}\cite{pqcrypto}. We add a new zero-knowledge identification scheme, and detailed security proofs for the protocols. We also present a new, asymptotically faster, algorithm for key generation, a thorough study of its optimization, and new experimental data.},
    author = {De Feo, Luca and Jao, David and Pl\^{u}t, J\'{e}r\^{o}me},
    citeulike-article-id = {10862817},
    citeulike-linkout-0 = {http://eprint.iacr.org/2011/506},
    day = {15},
    howpublished = {Cryptology ePrint Archive, Report 2011/506},
    keywords = {cryptography, elliptic\_curve, isogenies, supersingular},
    month = sep,
    posted-at = {2012-07-06 23:48:30},
    priority = {0},
    title = {Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies},
    url = {http://eprint.iacr.org/2011/506},
    year = {2011}
}

@misc{oeis,
    author = {{OEIS Foundation Inc.}},
    citeulike-article-id = {10862511},
    howpublished = {\\url{http://oeis.org/A130715}},
    posted-at = {2012-07-06 21:19:48},
    priority = {2},
    title = {The {On-Line} Encyclopedia of Integer Sequences},
    year = {2012}
}

@proceedings{DBLP:conf/pqcrypto/2011,
    booktitle = {PQCrypto},
    citeulike-article-id = {10862510},
    editor = {Yang, Bo-Yin},
    posted-at = {2012-07-06 21:19:48},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes in Computer Science},
    title = {{Post-Quantum} Cryptography - 4th International Workshop, {PQCrypto} 2011, Taipei, Taiwan, November 29 - December 2, 2011. Proceedings},
    volume = {7071},
    year = {2011}
}

@inproceedings{twisted-edwards,
    abstract = {{This paper introduces ” twisted Edwards curves,” a generalization of the recently introduced Edwards curves; shows that twisted Edwards curves include more curves over finite fields, and in particular every elliptic curve in Montgomery form; shows how to cover even more curves via isogenies; presents fast explicit formulas for twisted Edwards curves in projective and inverted coordinates; and shows that twisted Edwards curves save time for many curves that were already expressible as Edwards curves.}},
    author = {Bernstein, Daniel and Birkner, Peter and Joye, Marc and Lange, Tanja and Peters, Christiane},
    booktitle = {Progress in Cryptology – AFRICACRYPT 2008},
    citeulike-article-id = {10862508},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-68164-9_26},
    doi = {10.1007/978-3-540-68164-9_26},
    keywords = {cryptography, edwards, elliptic\_curve},
    posted-at = {2012-07-06 21:19:48},
    priority = {2},
    title = {{Twisted Edwards Curves}},
    url = {http://dx.doi.org/10.1007/978-3-540-68164-9_26},
    year = {2008}
}

@article{1981-knuth-transpose,
    author = {Knuth, Donald E. and Papadimitriou, Christos H.},
    citeulike-article-id = {10862507},
    journal = {Bulletin of the European Association for Theoretical Computer Science},
    posted-at = {2012-07-06 21:19:48},
    priority = {2},
    title = {{Duality in addition chains}},
    volume = {13},
    year = {1981}
}

@inproceedings{1976-pippenger,
    author = {Pippenger, Nicholas},
    citeulike-article-id = {10862506},
    posted-at = {2012-07-06 21:19:48},
    priority = {2},
    title = {{On the evaluation of powers and related problems (preliminary version)}},
    year = {1976}
}

@manual{Pari,
    address = {Bordeaux},
    citeulike-article-id = {10862505},
    comment = {available from \url{http://pari.math.u-bordeaux.fr/}},
    organization = {{The PARI\~{}Group}},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    title = {{PARI/GP, version {\\tt 2.4.3}}},
    year = {2008}
}


@inproceedings{zhang,
    abstract = {{This paper gives a quantum algorithm to search in an set S for a k -tuple satisfying some predefined relation, with the promise that some components of a desired k -tuple are in some subsets of S . In particular when k =2, we show a tight bound of the quantum query complexity for the Claw Finding problem, improving previous upper and lower bounds by Buhrman, Durr, Heiligman, Hoyer, Magniez, Santha and de Wolf [7]. We also consider the distributed scenario, where two parties each holds an n -element set, and they want to decide whether the two sets share a common element. We show a family of protocols s . t . q ( P ) 3/2 . c ( P )= O ( n 2 log n ), where q ( P ) and c ( P ) are the number of quantum queries and the number of communication qubits that the protocol P makes, respectively. This implies that we can pay more for quantum queries to save on quantum communication, and vice versa. To our knowledge, it is the first result about the tradeoff between the two resources.}},
    address = {Berlin, Heidelberg},
    author = {Zhang, Shengyu},
    booktitle = {Proceedings of the Eleventh Annual International Conference on Computing and Combinatorics},
    chapter = {44},
    citeulike-article-id = {10862503},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/11533719_44},
    doi = {10.1007/11533719_44},
    keywords = {cryptography, quantum},
    posted-at = {2012-07-06 21:19:47},
    priority = {1},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {{Promised and Distributed Quantum Search Computing and Combinatorics}},
    url = {http://dx.doi.org/10.1007/11533719_44},
    volume = {3595},
    year = {2005}
}

@unpublished{tani,
    abstract = {{The claw finding problem has been studied in terms of query complexity as one of the problems closely connected to cryptography. For given two functions, f and g, as an oracle which have domains of size N and M (N<=M), respectively, and the same range, the goal of the problem is to find x and y such that f(x)=g(y). This paper describes an optimal algorithm using quantum walk that solves this problem. Our algorithm can be generalized to find a claw of k functions for any constant integer k>1, where the domains of the functions may have different size.}},
    archivePrefix = {arXiv},
    author = {Tani, Seiichiro},
    citeulike-article-id = {10862502},
    citeulike-linkout-0 = {http://arxiv.org/abs/0708.2584},
    comment = {\url{http://arxiv.org/abs/0708.2584}},
    day = {3},
    eprint = {0708.2584},
    eprint = {0708.2584},
    keywords = {cryptography, quantum},
    month = mar,
    posted-at = {2012-07-06 21:19:47},
    priority = {1},
    title = {{Claw Finding Algorithms Using Quantum Walk}},
    url = {http://arxiv.org/abs/0708.2584},
    year = {2008}
}

@article{montgomery,
    abstract = {Since 1974, several algorithms have been developed that attempt to factor a large number \\$N\\$ by doing extensive computations modulo \\$N\\$ and occasionally taking {GCDs} with \\$N\\$. These began with Pollard's \\$p - 1\\$ and Monte Carlo methods. More recently, Williams published a \\$p + 1\\$ method, and Lenstra discovered an elliptic curve method ({ECM}). We present ways to speed all of these. One improvement uses two tables during the second phases of \\$p \\pm 1\\$ and {ECM}, looking for a match. Polynomial preconditioning lets us search a fixed table of size \\$n\\$ with \\$n/2 + o(n)\\$ multiplications. A parametrization of elliptic curves lets Step 1 of {ECM} compute the \\$x\\$-coordinate of \\${nP}\\$ from that of \\$P\\$ in about \\$9.3 \\log\_2 n\\$ multiplications for arbitrary \\$P\\$.},
    author = {Montgomery, Peter L.},
    citeulike-article-id = {10862501},
    citeulike-linkout-0 = {http://dx.doi.org/10.2307/2007888},
    doi = {10.2307/2007888},
    journal = {Math. Comp.},
    keywords = {elliptic\_curve, elliptic\_curve\_method, finite\_field, montgomery\_curve},
    number = {177},
    posted-at = {2012-07-06 21:19:47},
    priority = {0},
    publisher = {American Mathematical Society},
    title = {Speeding the Pollard and Elliptic Curve Methods of Factorization},
    url = {http://dx.doi.org/10.2307/2007888},
    volume = {48},
    year = {1987}
}

@incollection{joux,
    address = {Berlin},
    author = {Joux, Antoine},
    booktitle = {Algorithmic number theory ({S}ydney, 2002)},
    citeulike-article-id = {10862500},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/3-540-45455-1_3},
    doi = {10.1007/3-540-45455-1_3},
    mrnumber = {2041071 (2005a:14030)},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes in Comput. Sci.},
    title = {The {W}eil and {T}ate pairings as building blocks for public key cryptosystems},
    url = {http://dx.doi.org/10.1007/3-540-45455-1_3},
    volume = {2369},
    year = {2002}
}

@article{Tate,
    author = {Tate, John},
    citeulike-article-id = {10862498},
    journal = {Invent. Math.},
    mrnumber = {MR0206004 (34 \#5829)},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    title = {Endomorphisms of abelian varieties over finite fields},
    volume = {2},
    year = {1966}
}

@article{Stol,
    author = {Stolbunov, Anton},
    citeulike-article-id = {10862497},
    journal = {Adv. Math. Commun.},
    number = {2},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    title = {Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves},
    volume = {4},
    year = {2010}
}

@inproceedings{stolbunov-red,
    author = {Stolbunov, Anton},
    booktitle = {Norsk informasjonssikkerhetskonferanse (NISK)},
    citeulike-article-id = {10862496},
    editor = {Mj{\o}lsnes, Stig F.},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    title = {Reductionist Security Arguments for {Public-Key} Cryptographic Schemes Based on Group Action},
    year = {2009}
}

@book{Sil,
    address = {New York},
    author = {Silverman, Joseph H.},
    citeulike-article-id = {10862495},
    comment = {Corrected reprint of the 1986 original},
    mrnumber = {MR1329092 (95m:11054)},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    publisher = {Springer-Verlag},
    series = {Graduate Texts in Mathematics},
    title = {The arithmetic of elliptic curves},
    volume = {106},
    year = {1992}
}

@article{Shumow,
    author = {Shumow, Daniel},
    citeulike-article-id = {10862494},
    journal = {CoRR},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    title = {Isogenies of Elliptic Curves: A Computational Approach},
    volume = {abs/0910.5370},
    year = {2009}
}

@book{Sarnak,
    address = {Cambridge},
    author = {Sarnak, Peter},
    citeulike-article-id = {10862493},
    mrnumber = {1102679 (92k:11045)},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    publisher = {Cambridge University Press},
    series = {Cambridge Tracts in Mathematics},
    title = {Some applications of modular forms},
    volume = {99},
    year = {1990}
}

@misc{R&S,
    author = {Rostovtsev, Alexander and Stolbunov, Anton},
    citeulike-article-id = {10862492},
    citeulike-linkout-0 = {http://eprint.iacr.org/2006/145/},
    comment = {\url{http://eprint.iacr.org/2006/145/}},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    title = {Public-key cryptosystem based on isogenies},
    url = {http://eprint.iacr.org/2006/145/},
    year = {2006}
}

@incollection{pizer2,
    address = {Providence, RI},
    author = {Pizer, Arnold K.},
    booktitle = {Computational perspectives on number theory ({C}hicago, {IL}, 1995)},
    citeulike-article-id = {10862491},
    mrnumber = {1486836 (99b:11046)},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    publisher = {Amer. Math. Soc.},
    series = {AMS/IP Stud. Adv. Math.},
    title = {Ramanujan graphs},
    volume = {7},
    year = {1998}
}

@article{pizer1,
    author = {Pizer, Arnold K.},
    citeulike-article-id = {10862490},
    citeulike-linkout-0 = {http://dx.doi.org/10.1090/S0273-0979-1990-15918-X},
    doi = {10.1090/S0273-0979-1990-15918-X},
    journal = {Bull. Amer. Math. Soc. (N.S.)},
    mrnumber = {1027904 (90m:11063)},
    number = {1},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    title = {Ramanujan graphs and {H}ecke operators},
    url = {http://dx.doi.org/10.1090/S0273-0979-1990-15918-X},
    volume = {23},
    year = {1990}
}

@inproceedings{Mestre,
    address = {Nagoya},
    author = {Mestre, Jean-Fran\c{c}ois},
    booktitle = {Proceedings of the international conference on class numbers and fundamental units of algebraic number fields ({K}atata, 1986)},
    citeulike-article-id = {10862489},
    mrnumber = {891898 (88e:11025)},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    publisher = {Nagoya Univ.},
    title = {La m\'{e}thode des graphes. {E}xemples et applications},
    year = {1986}
}

@article{LubPS,
    author = {Lubotzky, Alexander and Phillips, Ralph and Sarnak, Peter},
    citeulike-article-id = {10862488},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/BF02126799},
    doi = {10.1007/BF02126799},
    journal = {Combinatorica},
    mrnumber = {963118 (89m:05099)},
    number = {3},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    title = {Ramanujan graphs},
    url = {http://dx.doi.org/10.1007/BF02126799},
    volume = {8},
    year = {1988}
}

@book{Lub,
    address = {Basel},
    author = {Lubotzky, Alexander},
    citeulike-article-id = {10862487},
    comment = {With an appendix by Jonathan D. Rogawski},
    mrnumber = {1308046 (96g:22018)},
    posted-at = {2012-07-06 21:19:47},
    priority = {2},
    publisher = {Birkh\"{a}user Verlag},
    series = {Progress in Mathematics},
    title = {Discrete groups, expanding graphs and invariant measures},
    volume = {125},
    year = {1994}
}

@incollection{lo,
    address = {London},
    author = {Lagarias, Jeffrey C. and Odlyzko, Andrew M.},
    booktitle = {Algebraic number fields: {\$L\$}-functions and {G}alois properties ({P}roc. {S}ympos., {U}niv. {D}urham, {D}urham, 1975)},
    citeulike-article-id = {10862486},
    mrnumber = {0447191 (56 \#5506)},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    publisher = {Academic Press},
    title = {Effective versions of the {C}hebotarev density theorem},
    year = {1977}
}

@article{Kup,
    author = {Kuperberg, Greg},
    citeulike-article-id = {10862485},
    eprint = {quant-ph/0302112},
    journal = {SIAM J. Comput.},
    number = {1},
    pages = {170--188},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    title = {A subexponential-time quantum algorithm for the dihedral hidden subgroup problem},
    volume = {35},
    year = {2005}
}

@incollection{JS,
    address = {Berlin},
    author = {Jao, David and Soukharev, Vladimir},
    booktitle = {Algorithmic number theory},
    citeulike-article-id = {10862483},
    mrnumber = {2721423 (2011h:11144)},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes in Comput. Sci.},
    title = {A subexponential algorithm for evaluating large degree isogenies},
    volume = {6197},
    year = {2010}
}

@article{JMV,
    author = {Jao, David and Miller, Stephen D. and Venkatesan, Ramarathnam},
    citeulike-article-id = {10862482},
    citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.jnt.2008.11.006},
    doi = {10.1016/j.jnt.2008.11.006},
    journal = {J. Number Theory},
    mrnumber = {2521489 (2010m:05141)},
    number = {6},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    title = {Expander graphs based on {GRH} with an application to elliptic curve cryptography},
    url = {http://dx.doi.org/10.1016/j.jnt.2008.11.006},
    volume = {129},
    year = {2009}
}

@incollection{Ihara,
    address = {Providence, R.I.},
    author = {Ihara, Yasutaka},
    booktitle = {Algebraic {G}roups and {D}iscontinuous {S}ubgroups ({P}roc. {S}ympos. {P}ure {M}ath., {B}oulder, {C}olo., 1965)},
    citeulike-article-id = {10862481},
    mrnumber = {0205952 (34 \#5777)},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    publisher = {Amer. Math. Soc.},
    title = {Discrete subgroups of {\${\\rm PL}(2,\\,k\_{\\wp })\$}},
    year = {1966}
}

@article{Igusa,
    author = {Igusa, Jun-ichi},
    citeulike-article-id = {10862480},
    journal = {Amer. J. Math.},
    mrnumber = {0104669 (21 \#3422)},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    title = {Fibre systems of {J}acobian varieties. {III}. {F}ibre systems of elliptic curves},
    volume = {81},
    year = {1959}
}

@article{hoory,
    author = {Hoory, Shlomo and Linial, Nathan and Wigderson, Avi},
    citeulike-article-id = {10862479},
    citeulike-linkout-0 = {http://dx.doi.org/10.1090/S0273-0979-06-01126-8},
    doi = {10.1090/S0273-0979-06-01126-8},
    journal = {Bull. Amer. Math. Soc. (N.S.)},
    mrnumber = {2247919 (2007h:68055)},
    number = {4},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    title = {Expander graphs and their applications},
    url = {http://dx.doi.org/10.1090/S0273-0979-06-01126-8},
    volume = {43},
    year = {2006}
}

@incollection{Gross,
    address = {Providence, RI},
    author = {Gross, Benedict H.},
    booktitle = {Number theory ({M}ontreal, {Q}ue., 1985)},
    citeulike-article-id = {10862478},
    mrnumber = {894322 (89c:11082)},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    publisher = {Amer. Math. Soc.},
    series = {CMS Conf. Proc.},
    title = {Heights and the special values of {\$L\$}-series},
    volume = {7},
    year = {1987}
}

@misc{gs,
    archivePrefix = {arXiv},
    author = {Galbraith, Steven D. and Stolbunov, Anton},
    citeulike-article-id = {10862477},
    citeulike-linkout-0 = {http://arxiv.org/abs/1105.6331/},
    comment = {\url{http://arxiv.org/abs/1105.6331/}},
    eprint = {1105.6331/},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    title = {Improved Algorithm for the Isogeny Problem for Ordinary Elliptic Curves},
    url = {http://arxiv.org/abs/1105.6331/},
    year = {2011}
}

@article{Gal,
    author = {Galbraith, Steven D.},
    citeulike-article-id = {10862476},
    journal = {LMS J. Comput. Math.},
    mrnumber = {MR1728955 (2001k:11113)},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    title = {Constructing isogenies between elliptic curves over finite fields},
    volume = {2},
    year = {1999}
}

@incollection{volcano,
    address = {Berlin},
    author = {Fouquet, M. and Morain, F.},
    booktitle = {Algorithmic number theory ({S}ydney, 2002)},
    citeulike-article-id = {10862475},
    mrnumber = {MR2041091 (2005c:11077)},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes in Comput. Sci.},
    title = {Isogeny volcanoes and the {SEA} algorithm},
    volume = {2369},
    year = {2002}
}

@article{Eichler,
    author = {Eichler, Martin},
    citeulike-article-id = {10862474},
    journal = {Arch. Math.},
    mrnumber = {0063406 (16,116d)},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    title = {Quatern\"{a}re quadratische {F}ormen und die {R}iemannsche {V}ermutung f\"{u}r die {K}ongruenzzetafunktion},
    volume = {5},
    year = {1954}
}

@book{DSV,
    address = {Cambridge},
    author = {Davidoff, Giuliana and Sarnak, Peter and Valette, Alain},
    citeulike-article-id = {10862473},
    citeulike-linkout-0 = {http://dx.doi.org/10.1017/CBO9780511615825},
    doi = {10.1017/CBO9780511615825},
    mrnumber = {1989434 (2004f:11001)},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    publisher = {Cambridge University Press},
    series = {London Mathematical Society Student Texts},
    title = {Elementary number theory, group theory, and {R}amanujan graphs},
    url = {http://dx.doi.org/10.1017/CBO9780511615825},
    volume = {55},
    year = {2003}
}

@misc{CJS,
    archivePrefix = {arXiv},
    author = {Childs, Andrew and Jao, David and Soukharev, Vladimir},
    citeulike-article-id = {10862472},
    citeulike-linkout-0 = {http://arxiv.org/abs/1012.4019/},
    comment = {\url{http://arxiv.org/abs/1012.4019/}},
    eprint = {1012.4019/},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    title = {Constructing elliptic curve isogenies in quantum subexponential time},
    url = {http://arxiv.org/abs/1012.4019/},
    year = {2010}
}

@article{CGL,
    author = {Charles, Denis X. and Lauter, Kristin E. and Goren, Eyal Z.},
    citeulike-article-id = {10862471},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s00145-007-9002-x},
    doi = {10.1007/s00145-007-9002-x},
    journal = {Journal of Cryptology},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    title = {Cryptographic Hash Functions from Expander Graphs},
    url = {http://dx.doi.org/10.1007/s00145-007-9002-x},
    volume = {22},
    year = {2009}
}

@incollection{CouvMorain,
    address = {Berlin},
    author = {Couveignes, J. M. and Morain, F.},
    booktitle = {Algorithmic number theory ({I}thaca, {NY}, 1994)},
    citeulike-article-id = {10862470},
    mrnumber = {MR1322711 (95m:11147)},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes in Comput. Sci.},
    title = {Schoof's algorithm and isogeny cycles},
    volume = {877},
    year = {1994}
}

@misc{Couv,
    author = {Couveignes, Jean-Marc},
    citeulike-article-id = {10862469},
    citeulike-linkout-0 = {http://eprint.iacr.org/2006/291/},
    comment = {\url{http://eprint.iacr.org/2006/291/}},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    title = {Hard Homogeneous Spaces},
    url = {http://eprint.iacr.org/2006/291/},
    year = {2006}
}

@proceedings{DBLP:conf/eurocrypt/2001,
    booktitle = {EUROCRYPT},
    citeulike-article-id = {10862468},
    editor = {Pfitzmann, Birgit},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes in Computer Science},
    title = {Advances in Cryptology - {EUROCRYPT} 2001, International Conference on the Theory and Application of Cryptographic Techniques, Innsbruck, Austria, May 6-10, 2001, Proceeding},
    volume = {2045},
    year = {2001}
}

@inproceedings{canetti,
    author = {Canetti, Ran and Krawczyk, Hugo},
    booktitle = {EUROCRYPT},
    citeulike-article-id = {10862467},
    editor = {Pfitzmann, Birgit},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes in Computer Science},
    title = {Analysis of {Key-Exchange} Protocols and Their Use for Building Secure Channels},
    volume = {2045},
    year = {2001}
}

@article{broker-ss,
    author = {Br{\"{o}}ker, Reinier},
    citeulike-article-id = {10862466},
    journal = {J. Comb. Number Theory},
    mrnumber = {2681311 (2011g:11116)},
    number = {3},
    posted-at = {2012-07-06 21:19:46},
    priority = {2},
    title = {Constructing supersingular elliptic curves},
    volume = {1},
    year = {2009}
}

@article{Bostan,
    author = {Bostan, Alin and Morain, Fran\c{c}ois and Salvy, Bruno and Schost, {\'{E}}ric},
    citeulike-article-id = {10862465},
    citeulike-linkout-0 = {http://dx.doi.org/10.1090/S0025-5718-08-02066-8},
    doi = {10.1090/S0025-5718-08-02066-8},
    journal = {Math. Comp.},
    mrnumber = {MR2398793 (2009k:11207)},
    number = {263},
    posted-at = {2012-07-06 21:19:45},
    priority = {2},
    title = {Fast algorithms for computing isogenies between elliptic curves},
    url = {http://dx.doi.org/10.1090/S0025-5718-08-02066-8},
    volume = {77},
    year = {2008}
}

@proceedings{QUANTUMCOMM2009,
    booktitle = {Quantum Communication and Quantum Networking: First International Conference, QuantumComm 2009},
    citeulike-article-id = {10862463},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-642-11731-2},
    doi = {10.1007/978-3-642-11731-2},
    editor = {Sergienki, Alexander and Pascazio, Saverio and Villoresi, Paolo},
    posted-at = {2012-07-06 21:19:45},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering},
    title = {Quantum Communication and Quantum Networking: First International Conference, {QuantumComm} 2009},
    url = {http://dx.doi.org/10.1007/978-3-642-11731-2},
    volume = {36},
    year = {2010}
}

@inproceedings{sml09,
    author = {Stebila, Douglas and Mosca, Michele and L{\"{u}}tkenhaus, Norbert},
    booktitle = {Quantum Communication and Quantum Networking: First International Conference, QuantumComm 2009},
    citeulike-article-id = {10862462},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-642-11731-2_35},
    doi = {10.1007/978-3-642-11731-2_35},
    editor = {Sergienki, Alexander and Pascazio, Saverio and Villoresi, Paolo},
    eprint = {\\href{http://arxiv.org/abs/0902.2839/}{arXiv:0902.2839}, \\url{http://eprint.iacr.org/2009/082/}},
    posted-at = {2012-07-06 21:19:45},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering},
    title = {The Case for Quantum Key Distribution},
    url = {http://dx.doi.org/10.1007/978-3-642-11731-2_35},
    volume = {36},
    year = {2010}
}

@inproceedings{quis,
    address = {Berlin, Heidelberg},
    author = {Petit, Christophe and Lauter, Kristin and Quisquater, Jean-Jacques},
    booktitle = {Proceedings of the 6th international conference on Security and Cryptography for Networks},
    citeulike-article-id = {10862461},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-85855-3_18},
    location = {Amalfi, Italy},
    posted-at = {2012-07-06 21:19:45},
    priority = {2},
    publisher = {Springer-Verlag},
    series = {SCN '08},
    title = {Full Cryptanalysis of {LPS} and {M}orgenstern Hash Functions},
    url = {http://dx.doi.org/10.1007/978-3-540-85855-3_18},
    year = {2008}
}

@article{teske-ph,
    author = {Teske, Edlyn},
    citeulike-article-id = {10862460},
    citeulike-linkout-0 = {http://www.sciencedirect.com/science/article/pii/S0747717199902791},
    journal = {J. Symbolic Comput.},
    number = {6},
    posted-at = {2012-07-06 21:19:45},
    priority = {2},
    title = {The {Pohlig-Hellman} Method Generalized for Group Structure Computation},
    url = {http://www.sciencedirect.com/science/article/pii/S0747717199902791},
    volume = {27},
    year = {1999}
}

@article{galbraith99,
    abstract = {Let E1 and E2 be ordinary elliptic curves over a finite field Fp such that \#{E1(Fp}) = \#{E2(Fp}). Tate's isogeny theorem states that there is an isogeny from E1 to E2 which is defined over Fp. The goal of this paper is to describe a probabilistic algorithm for constructing such an {isogeny.The} algorithm proposed in this paper has exponential complexity in the worst case. Nevertheless, it is efficient in certain situations (that is, when the class number of the endomorphism ring is small). The significance of these results to elliptic curve cryptography is discussed.},
    author = {Galbraith, Steven D.},
    citeulike-article-id = {10862291},
    citeulike-linkout-0 = {http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6560320},
    citeulike-linkout-1 = {http://dx.doi.org/10.1112/S1461157000000097},
    doi = {10.1112/S1461157000000097},
    journal = {LMS Journal of Computation and Mathematics},
    keywords = {isogenies},
    pages = {118--138},
    posted-at = {2012-07-06 19:50:44},
    priority = {3},
    title = {Constructing Isogenies between Elliptic Curves Over Finite Fields},
    url = {http://dx.doi.org/10.1112/S1461157000000097},
    volume = {2},
    year = {1999}
}

@inproceedings{joux+vitse12,
    abstract = {We present a new  ” cover and decomposition” attack on the elliptic curve discrete logarithm problem, that combines Weil descent and decomposition-based index calculus into a single discrete logarithm algorithm. This attack applies, at least theoretically, to all composite degree extension fields, and is particularly well-suited for curves defined over . We give a real-size example of discrete logarithm computations on a curve over a 151-bit degree 6 extension field, which would not have been practically attackable using previously known algorithms.},
    address = {Berlin, Heidelberg},
    author = {Joux, Antoine and Vitse, Vanessa},
    booktitle = {Advances in Cryptology -- EUROCRYPT 2012},
    chapter = {3},
    citeulike-article-id = {10862193},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-642-29011-4_3},
    citeulike-linkout-1 = {http://www.springerlink.com/content/g1xn2357622l3xu3},
    doi = {10.1007/978-3-642-29011-4_3},
    editor = {Pointcheval, David and Johansson, Thomas},
    isbn = {978-3-642-29010-7},
    keywords = {discrete\_log, elliptic\_curve, weil\_descent},
    pages = {9--26},
    posted-at = {2012-07-06 19:34:31},
    priority = {2},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Cover and Decomposition Index Calculus on Elliptic Curves Made Practical},
    url = {http://dx.doi.org/10.1007/978-3-642-29011-4_3},
    volume = {7237},
    year = {2012}
}

@inproceedings{fouquet+morain02,
    abstract = {Recently, Kohel gave algorithms to compute the conductor of the endomorphism ring of an ordinary elliptic curve, given the cardinality of the curve. Using his work, we give a complete description of the structure of curves related via rational ℓ-degree isogenies, a structure we call a volcano. We explain how we can travel through this structure using modular polynomials. The computation of the structure is possible without knowing the cardinality of the curve, and that as a result, we deduce information on the cardinality.},
    address = {Berlin, Heidelberg},
    author = {Fouquet, Mireille and Morain, Fran\c{c}ois},
    booktitle = {Algorithmic Number Theory Symposium},
    chapter = {23},
    citeulike-article-id = {10861974},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/3-540-45455-1_23},
    citeulike-linkout-1 = {http://www.springerlink.com/content/8kv0nfj4x0ldpm0y},
    doi = {10.1007/3-540-45455-1_23},
    editor = {Fieker, Claus and Kohel, David R.},
    isbn = {978-3-540-43863-2},
    keywords = {isogenies, isogeny\_cycles},
    pages = {47--62},
    posted-at = {2012-07-06 16:42:26},
    priority = {0},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Isogeny Volcanoes and the {SEA} Algorithm},
    url = {http://dx.doi.org/10.1007/3-540-45455-1_23},
    volume = {2369},
    year = {2002}
}

@inproceedings{mestre86,
    address = {Nagoya},
    author = {Mestre, Jean-Fran\c{c}ois},
    booktitle = {Proceedings of the international conference on class numbers and fundamental units of algebraic number fields ({K}atata, 1986)},
    citeulike-article-id = {10853480},
    citeulike-linkout-0 = {http://boxen.math.washington.edu/msri06/refs/mestre-method-of-graphs/mestre-fr.pdf},
    keywords = {elliptic\_curve, isogenies, supersingular},
    mrnumber = {891898 (88e:11025)},
    posted-at = {2012-07-03 14:47:54},
    priority = {3},
    publisher = {Nagoya Univ.},
    title = {La m\'{e}thode des graphes. {E}xemples et applications},
    url = {http://boxen.math.washington.edu/msri06/refs/mestre-method-of-graphs/mestre-fr.pdf},
    year = {1986}
}

@article{waterhouse69,
    author = {Waterhouse, William C.},
    citeulike-article-id = {10778159},
    journal = {Annales Scientifiques de l'\'{E}cole Normale Sup\'{e}rieure},
    keywords = {elliptic\_curve, finite\_field},
    number = {4},
    pages = {521--560},
    posted-at = {2012-06-11 10:15:24},
    priority = {3},
    title = {Abelian varieties over finite fields},
    volume = {2},
    year = {1969}
}

@article{deuring41,
    author = {Deuring, Max},
    citeulike-article-id = {10778147},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/BF02940746},
    citeulike-linkout-1 = {http://www.springerlink.com/content/c145767156682h73},
    day = {1},
    doi = {10.1007/BF02940746},
    issn = {0025-5858},
    journal = {Abhandlungen aus dem Mathematischen Seminar der Universit\"{a}t Hamburg},
    keywords = {complex\_multiplication, elliptic\_curve},
    month = dec,
    number = {1},
    pages = {197--272},
    posted-at = {2012-06-11 10:10:38},
    priority = {1},
    publisher = {Springer Berlin / Heidelberg},
    title = {Die {T}ypen der {M}ultiplikatorenringe elliptischer {F}unktionenk\"{o}rper},
    url = {http://dx.doi.org/10.1007/BF02940746},
    volume = {14},
    year = {1941}
}

@phdthesis{belding08-thesis,
    abstract = {We present new algorithms related to both theoretical and practical questions in the area of elliptic curves and class field theory. The dissertation has two main parts, as described below. Let O be an imaginary quadratic order of discriminant D < 0, and let K = √ Q( D). The class polynomial {HD} of O is the polynomial whose roots are precisely the j-invariants of elliptic curves with complex multiplication by O. Computing this polynomial is useful in constructing elliptic curves suitable for cryptography, as well as in the context of explicit class field theory. In the first part of the dissertation, we present an algorithm to compute {HD} p-adically where p is a prime inert in K and not dividing D. ̃ This involves computing the canonical lift E of a pair (E, f ) where E is a supersingular elliptic curve and f is an embedding of O into the endomorphism ring of E. We also present an algorithm to compute {HD} modulo p for p inert which is used in the Chinese remainder theorem algorithm to compute {HD} . For an elliptic curve E over any field K, the Weil pairing en is a bilinear map on the points of order n of E. The Weil pairing is a useful tool in both the theory of elliptic curves and the application of elliptic curves to cryptography. However, for K of characteristic p, the classical Weil pairing on the points of order p is trivial. In the second part of the dissertation, we consider E over the dual numbers K[ ] and define a non-degenerate ” Weil pairing on p-torsion.” We show that this pairing satisfies many of the same properties of the classical pairing. Moreover, we show that it directly relates to recent attacks on the discrete logarithm problem on the p-torsion subgroup of an elliptic curve over the finite field Fq . We also present a new attack on the discrete logarithm problem on anomalous curves using a lift of E over Fp [ ].},
    author = {Belding, Juliana V.},
    citeulike-article-id = {10772351},
    keywords = {elliptic\_curve, pairing, supersingular},
    posted-at = {2012-06-08 19:30:41},
    priority = {2},
    school = {University of Maryland},
    title = {Number Theoretic Algorithms for Elliptic Curves},
    year = {2008}
}

@unpublished{citeulike:10772345,
    abstract = {We present a deterministic and explicit algorithm to compute the endomorphism
rings of supersingular elliptic curves. As an example we compute the
endomorphism rings of all supersingular elliptic curves defined over
characteristic p=29,...,97.},
    archivePrefix = {arXiv},
    author = {Cervi\~{n}o, Juan M.},
    citeulike-article-id = {10772345},
    citeulike-linkout-0 = {http://arxiv.org/abs/math/0404538},
    citeulike-linkout-1 = {http://arxiv.org/pdf/math/0404538},
    day = {29},
    eprint = {math/0404538},
    keywords = {elliptic\_curve, supersingular},
    month = apr,
    posted-at = {2012-06-08 19:26:29},
    priority = {2},
    title = {On the Correspondence between Supersingular Elliptic Curves and maximal quaternionic Orders},
    url = {http://arxiv.org/abs/math/0404538},
    year = {2004}
}

@misc{cervino04,
    abstract = {We present a deterministic and explicit algorithm to compute the endomorphism rings of supersingular elliptic curves. As an example we compute the endomorphism rings of all supersingular elliptic curves defined over characteristic p=29,...,97.},
    archivePrefix = {arXiv},
    author = {Cervi\~{n}o, Juan M.},
    citeulike-article-id = {10772339},
    citeulike-linkout-0 = {http://arxiv.org/abs/math/0404538},
    citeulike-linkout-1 = {http://www.uni-math.gwdg.de/tschinkel/SS04/cervino.pdf},
    comment = {\url{http://arxiv.org/abs/math/0404538}},
    day = {29},
    eprint = {math/0404538},
    eprint = {math/0404538},
    journal = {Mathematisches Institut der Georg-August Universit{\"{a}}t G{\"{o}}ttingen},
    keywords = {elliptic\_curve, supersingular},
    month = apr,
    pages = {53--60},
    posted-at = {2012-06-08 19:23:37},
    priority = {2},
    title = {On the Correspondence between Supersingular Elliptic Curves and maximal quaternionic Orders},
    url = {http://arxiv.org/abs/math/0404538},
    year = {2004}
}

@inproceedings{sakemi+etal12-pkc,
    abstract = {A discrete logarithm problem with auxiliary input ({DLPwAI}) is a problem to find α from G , {αG} , α d G in an additive cyclic group generated by an element G of prime order r , and a positive integer d satisfying d |( r − 1). The infeasibility of this problem assures the security of some cryptographic schemes. In 2006, Cheon proposed a novel algorithm for solving {DLPwAI} (Cheon's algorithm). This paper reports our experimental results of Cheon's algorithm by implementing it with some speeding-up techniques. In fact, we have succeeded to solve {DLPwAI} on a pairing-friendly elliptic curve of 160-bit order in 1314 core days. Implications of our experiments on cryptographic schemes are also discussed.},
    address = {Berlin, Heidelberg},
    author = {Sakemi, Yumi and Hanaoka, Goichiro and Izu, Tetsuya and Takenaka, Masahiko and Yasuda, Masaya},
    booktitle = {Public Key Cryptography 2012},
    chapter = {35},
    citeulike-article-id = {10772326},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-642-30057-8_35},
    citeulike-linkout-1 = {http://www.springerlink.com/content/y571262l54081k1p},
    doi = {10.1007/978-3-642-30057-8_35},
    editor = {Fischlin, Marc and Buchmann, Johannes and Manulis, Mark},
    isbn = {978-3-642-30056-1},
    keywords = {cryptography, discrete\_log, elliptic\_curve},
    pages = {595--608},
    posted-at = {2012-06-08 19:19:21},
    priority = {4},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Solving a Discrete Logarithm Problem with Auxiliary Input on a {160-Bit} Elliptic Curve Public Key Cryptography},
    url = {http://dx.doi.org/10.1007/978-3-642-30057-8_35},
    volume = {7293},
    year = {2012}
}

@inproceedings{stern94-CLE,
    abstract = {In the last few years, there have been several attempts to build identification protocols that do not rely on arithmetical operations with large numbers but only use simple operations (see [ 10 , 8 ]). One was presented at the {CRYPTO} 89 rump session ([ 8 ]) and depends on the so-called Permuted Kernel problem ({PKP}). Another appeared in the {CRYPTO} 93 proceedings and is based on the syndrome decoding problem ({SD}) form the theory of error correcting codes ([ 11 ]). In this paper, we introduce a new scheme of the same family with the distinctive character that both the secret key and the public identification key can be taken to be of short length. By short, we basically mean the usual size of conventional symmetric cryptosystems. As is known, the possibility of using short keys has been a challenge in public key cryptography and has practical applications. Our scheme relies on a combinatorial problem which we call Constrained Linear Equations ({CLE} in short) and which consists of solving a set of linear equations modulo some small prime q , the unknowns being subject to belong to a specific subset of the integers mod q . Thus, we enlarge the set of tools that can be used in cryptography.},
    address = {Berlin, Heidelberg},
    author = {Stern, Jacques},
    booktitle = {Advances in Cryptology — CRYPTO '94},
    chapter = {18},
    citeulike-article-id = {10751498},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/3-540-48658-5_18},
    citeulike-linkout-1 = {http://www.springerlink.com/content/hd14uxt5a96pphjw},
    doi = {10.1007/3-540-48658-5_18},
    editor = {Desmedt, Yvo G.},
    isbn = {978-3-540-58333-2},
    keywords = {cryptography, protocols, zero\_knowledge},
    pages = {164--173},
    posted-at = {2012-06-06 14:30:07},
    priority = {2},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Designing Identification Schemes with Keys of Short Size},
    url = {http://dx.doi.org/10.1007/3-540-48658-5_18},
    volume = {839},
    year = {1994}
}

@inproceedings{stern94-SD,
    abstract = {Zero-knowledge proofs were introduced in 1985, in a paper by Goldwasser, Micali and Rackoff ([ 6 ]). Their practical significance was soon demonstrated in the work of Fiat and Shamir ([ 4 ]), who turned zero-knowledge proofs of quadratic residuosity into efficient means of establishing user identities. Still, as is almost always the case in public-key cryptography, the {Fiat-Shamir} scheme relied on arithmetic operations on large numbers. In 1989, there were two attempts to build identification protocols that only use simple operations (see [ 11 , 10 ]). One appeared in the {EUROCRYPT} proceedings and relies on the intractability of some coding problems, the other was presented at the {CRYPTO} rump session and depends on the so-called Permuted Kernel problem ({PKP}). Unfortunately, the first of the schemes was not really practical. In the present paper, we propose a new identification scheme, based on error-correcting codes, which is zero-knowledge and is of practical value. Furthermore, we describe several variants, including one which has an identity based character. The security of our scheme depends on the hardness of decoding a word of given syndrome w.r.t. some binary linear error-correcting code.},
    address = {Berlin, Heidelberg},
    author = {Stern, Jacques},
    booktitle = {Advances in Cryptology — CRYPTO' 93},
    chapter = {2},
    citeulike-article-id = {10751497},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/3-540-48329-2_2},
    citeulike-linkout-1 = {http://www.springerlink.com/content/4a27n7223h43edkp},
    doi = {10.1007/3-540-48329-2_2},
    editor = {Stinson, Douglas R.},
    isbn = {978-3-540-57766-9},
    keywords = {cryptography, protocols, zero\_knowledge},
    pages = {13--21},
    posted-at = {2012-06-06 14:29:22},
    priority = {2},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {A new identification scheme based on syndrome decoding},
    url = {http://dx.doi.org/10.1007/3-540-48329-2_2},
    volume = {773},
    year = {1994}
}

@inproceedings{pointcheval95-pp,
    abstract = {Identification is a useful cryptographic tool. Since zero-know- ledge theory appeared [ 3 ], several interactive identification schemes have been proposed (in particular Fiat-Shamir [ 2 ] and its variants [ 8 , 5 , 4 ], Schnorr [ 9 ]). These identifications are based on number theoretical prob- lems. More recently, new schemes appeared with the peculiarity that they are more efficient from the computational point of view and that their security is based on NP -complete problems: PKP (Permuted Ker- nels Problem) [ 10 ], SD (Syndrome Decoding) [ 12 ] and CLE (Constrained Linear Equations) [ 13 ]. We present a new NP -complete linear problem which comes from learn- ing machines: the Perceptrons Problem. We have some constraints, m vectors X i of {−1, +1} n , and we want to find a vector V of {−1, +1} n such that X i · V ≥ 0 for all i . Next, we provide some zero-knowledge interactive identification protocols based on this problem, with an evaluation of their security. Eventually, those protocols are well suited for smart card applications.},
    address = {Berlin, Heidelberg},
    author = {Pointcheval, David},
    booktitle = {Advances in Cryptology — EUROCRYPT '95},
    chapter = {26},
    citeulike-article-id = {10751496},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/3-540-49264-X_26},
    citeulike-linkout-1 = {http://www.springerlink.com/content/hwb0ea5bc92mfxbb},
    doi = {10.1007/3-540-49264-X_26},
    editor = {Guillou, Louis C. and Quisquater, Jean-Jacques},
    isbn = {978-3-540-59409-3},
    keywords = {cryptography, protocols, zero\_knowledge},
    pages = {319--328},
    posted-at = {2012-06-06 14:28:10},
    priority = {2},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {A New Identification Scheme Based on the Perceptrons Problem},
    url = {http://dx.doi.org/10.1007/3-540-49264-X_26},
    volume = {921},
    year = {1995}
}

@article{goldreich+micali+widgerson91,
    abstract = {An abstract is not available.},
    address = {New York, NY, USA},
    author = {Goldreich, Oded and Micali, Silvio and Wigderson, Avi},
    citeulike-article-id = {785219},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=116852},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/116825.116852},
    doi = {10.1145/116825.116852},
    issn = {0004-5411},
    journal = {Journal of the Association for Computing Machinery},
    keywords = {cryptography, protocols, zero\_knowledge},
    month = jul,
    number = {3},
    pages = {690--728},
    posted-at = {2012-06-06 14:14:36},
    priority = {0},
    publisher = {ACM},
    title = {Proofs that yield nothing but their validity or all languages in {NP} have zero-knowledge proof systems},
    url = {http://dx.doi.org/10.1145/116825.116852},
    volume = {38},
    year = {1991}
}

@article{feige+fiat+shamir88,
    abstract = {In this paper we extend the notion of interactive proofs of assertions to interactive proofs of knowledge. This leads to the definition of unrestricted input zero-knowledge proofs of knowledge in which the prover demonstrates possession of knowledge without revealing any computational information whatsoever (not even the one bit revealed in zero-knowledge proofs of assertions). We show the relevance of these notions to identification schemes, in which parties prove their identity by demonstrating their knowledge rather than by proving the validity of assertions. We describe a novel scheme which is provably secure if factoring is difficult and whose practical implementations are about two orders of magnitude faster than {RSA}-based identification schemes. The advantages of thinking in terms of proofs of knowledge rather than proofs of assertions are demonstrated in two efficient variants of the scheme: unrestricted input zero-knowledge proofs of knowledge are used in the construction of a scheme which needs no directory; a version of the scheme based on parallel interactive proofs (which are not known to be zero knowledge) is proved secure by observing that the identification protocols are proofs of knowledge.},
    author = {Feige, Uriel and Fiat, Amos and Shamir, Adi},
    citeulike-article-id = {6875730},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/BF02351717},
    citeulike-linkout-1 = {http://www.springerlink.com/content/9gl5112723626462},
    day = {10},
    doi = {10.1007/BF02351717},
    issn = {0933-2790},
    journal = {Journal of Cryptology},
    keywords = {cryptography, protocols, zero\_knowledge},
    month = jun,
    number = {2},
    pages = {77--94},
    posted-at = {2012-06-06 14:13:11},
    priority = {0},
    publisher = {Springer New York},
    title = {Zero-knowledge proofs of identity},
    url = {http://dx.doi.org/10.1007/BF02351717},
    volume = {1},
    year = {1988}
}

@inproceedings{fiat+shamir87,
    abstract = {In this paper we describe simple identification and signature schemes which enable any user to prove his identity and the authenticity of his messages to any other user without shared or public keys. The schemes are provably secure against any known or chosen message attack if factoring is difficult, and typical implementations require only 1\% to 4\% of the number of modular multiplications required by the {RSA} scheme. Due to their simplicity, security and speed, these schemes are ideally suited for microprocessor-based devices such as smart cards, personal computers, and remote control systems.},
    address = {Berlin, Heidelberg},
    author = {Fiat, Amos and Shamir, Adi},
    booktitle = {Advances in Cryptology — CRYPTO' 86},
    chapter = {12},
    citeulike-article-id = {943028},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/3-540-47721-7_12},
    citeulike-linkout-1 = {http://www.springerlink.com/content/cp9tqrxmgdr55rv5},
    doi = {10.1007/3-540-47721-7_12},
    editor = {Odlyzko, Andrew M.},
    isbn = {978-3-540-18047-0},
    journal = {Lecture Notes in Computer Science : Advances in Cryptology - CRYPTO '86: Proceedings},
    keywords = {cryptography, protocols, zero\_knowledge},
    pages = {186--194},
    posted-at = {2012-06-06 14:10:33},
    priority = {0},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {How To Prove Yourself: Practical Solutions to Identification and Signature Problems},
    url = {http://dx.doi.org/10.1007/3-540-47721-7_12},
    volume = {263},
    year = {1987}
}

@inproceedings{shamir89-pkp,
    abstract = {In 1985 Goldwasser Micali and Rackoff proposed a new type of interactive proof system which reveals no knowledge whatsoever about the assertion except its validity. The practical significance of these proofs was demonstrated in 1986 by Fiat and Shamir, who showed how to use efficient zero knowledge proofs of quadratic residuosity to establish user identities and to digitally sign messages. In this paper we propose a new zero knowledge identification scheme, which is even faster than the {Fiat-Shamir} scheme, using a small number of communicated bits, simple 8-bit arithmetic operations, and compact public and private keys. The security of the new scheme depends on an {NP}-complete algebraic problem rather than on factoring, and thus it widens the basis of public key cryptography, which has become dangerously dependent on the difficulty of a single problem.},
    address = {New York, NY, USA},
    author = {Shamir, Adi},
    booktitle = {Proceedings on Advances in cryptology},
    citeulike-article-id = {10751396},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=118263},
    citeulike-linkout-1 = {http://portal.acm.org/citation.cfm?id=118263},
    isbn = {0-387-97317-6},
    keywords = {cryptography, protocols, zero\_knowledge},
    location = {Santa Barbara, California, United States},
    pages = {606--609},
    posted-at = {2012-06-06 13:56:01},
    priority = {0},
    publisher = {Springer-Verlag New York, Inc.},
    series = {CRYPTO '89},
    title = {An efficient identification scheme based on permuted kernels (extended abstract)},
    url = {http://portal.acm.org/citation.cfm?id=118263},
    year = {1989}
}

@electronic{leiserson98-bit,
    author = {Leiserson, Charles E. and Prokop, Harald and Randall, Keith H.},
    citeulike-article-id = {10682872},
    citeulike-linkout-0 = { http://supertech.csail.mit.edu/papers/debruijn.pdf},
    howpublished = {Available on the Internet from \url{http://supertech.csail.mit.edu/papers. html}},
    keywords = {computer\_science},
    posted-at = {2012-05-18 17:40:22},
    priority = {0},
    title = {Using de {B}ruijn sequences to index a 1 in a computer word},
    url = { http://supertech.csail.mit.edu/papers/debruijn.pdf},
    year = {1998}
}

@inproceedings{kocher+jaffe+jun99,
    abstract = {{Cryptosystem designers frequently assume that secrets will be manipulated in closed, reliable computing environments. Unfortunately, actual computers and microchips leak information about the operations they process. This paper examines specific methods for analyzing power consumption measurements to find secret keys from tamper resistant devices. We also discuss approaches for building cryptosystems that can operate securely in existing hardware that leaks information.}},
    address = {Berlin, Heidelberg},
    author = {Kocher, Paul and Jaffe, Joshua and Jun, Benjamin},
    booktitle = {Advances in Cryptology — CRYPTO' 99},
    chapter = {25},
    citeulike-article-id = {9696410},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/3-540-48405-1_25},
    citeulike-linkout-1 = {http://www.springerlink.com/content/kx35ub53vtrkh2nx},
    day = {16},
    doi = {10.1007/3-540-48405-1_25},
    editor = {Wiener, Michael},
    isbn = {978-3-540-66347-8},
    issn = {0302-9743},
    keywords = {cryptography},
    month = dec,
    pages = {388--397},
    posted-at = {2012-05-03 12:06:38},
    priority = {1},
    publisher = {Springer Berlin Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {{Differential Power Analysis}},
    url = {http://dx.doi.org/10.1007/3-540-48405-1_25},
    volume = {1666},
    year = {1999}
}

@article{df+schost12,
    abstract = {An {Artin-Schreier} tower over the finite field Fp is a tower of field extensions generated by polynomials of the form {Xp−X}−α. Following Cantor and Couveignes, we give algorithms with quasi-linear time complexity for arithmetic operations in such towers. As an application, we present an implementation of Couveignes' algorithm for computing isogenies between elliptic curves using the p-torsion.},
    author = {De Feo, Luca and Schost, {\'E}ric},
    citeulike-article-id = {10175624},
    citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.jsc.2011.12.008},
    doi = {10.1016/j.jsc.2011.12.008},
    issn = {07477171},
    journal = {J. Symbolic Comput.},
    keywords = {artin-schreier, computer\_algebra, finite\_field, trace\_formulas, transposed\_algorithms},
    number = {7},
    pages = {771--792},
    posted-at = {2012-04-04 12:50:03},
    priority = {0},
    title = {Fast arithmetics in {A}rtin-{S}chreier towers over finite fields},
    url = {http://dx.doi.org/10.1016/j.jsc.2011.12.008},
    volume = {47},
    year = {2012}
}

@article{brassard+hoyer+kalach+kaplan+laplante+salvail11,
    abstract = {In 1974, Ralph Merkle proposed the first unclassified scheme for secure
communications over insecure channels. When legitimate communicating parties
are willing to spend an amount of computational effort proportional to some
parameter N, an eavesdropper cannot break into their communication without
spending a time proportional to N^2, which is quadratically more than the
legitimate effort. We showed in an earlier paper that Merkle's schemes are
completely insecure against a quantum adversary, but that their security can be
partially restored if the legitimate parties are also allowed to use quantum
computation: the eavesdropper needed to spend a time proportional to N^{3/2} to
break our earlier quantum scheme. Furthermore, all previous classical schemes
could be broken completely by the onslaught of a quantum eavesdropper and we
conjectured that this is unavoidable.


We give two novel key establishment schemes in the spirit of Merkle's. The
first one can be broken by a quantum adversary that makes an effort
proportional to N^{5/3} to implement a quantum random walk in a Johnson graph
reminiscent of Andris Ambainis' quantum algorithm for the element distinctness
problem. This attack is optimal up to logarithmic factors. Our second scheme is
purely classical, yet it cannot be broken by a quantum eavesdropper who is only
willing to expend effort proportional to that of the legitimate parties.},
    archivePrefix = {arXiv},
    author = {Brassard, Gilles and Hoyer, Peter and Kalach, Kassem and Kaplan, Marc and Laplante, Sophie and Salvail, Louis},
    citeulike-article-id = {9648219},
    citeulike-linkout-0 = {http://arxiv.org/abs/1108.2316},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1108.2316},
    day = {11},
    eprint = {1108.2316},
    keywords = {cryptography, quantum},
    month = aug,
    posted-at = {2012-02-23 17:17:20},
    priority = {4},
    title = {Merkle Puzzles in a Quantum World},
    url = {http://arxiv.org/abs/1108.2316},
    year = {2011}
}

@article{dinh+moore+rusell10,
    abstract = {Quantum computers can break the RSA and El Gamal public-key cryptosystems,
since they can factor integers and extract discrete logarithms. If we believe
that quantum computers will someday become a reality, we would like to have
\emph{post-quantum} cryptosystems which can be implemented today with classical
computers, but which will remain secure even in the presence of quantum
attacks.


In this article we show that the McEliece cryptosystem over
\emph{well-permuted, well-scrambled} linear codes resists precisely the attacks
to which the RSA and El Gamal cryptosystems are vulnerable---namely, those
based on generating and measuring coset states. This eliminates the approach of
strong Fourier sampling on which almost all known exponential speedups by
quantum algorithms are based. Specifically, we show that the natural case of
the Hidden Subgroup Problem to which the McEliece cryptosystem reduces cannot
be solved by strong Fourier sampling, or by any measurement of a coset state.
We start with recent negative results on quantum algorithms for Graph
Isomorphism, which are based on particular subgroups of size two, and extend
them to subgroups of arbitrary structure, including the automorphism groups of
linear codes. This allows us to obtain the first rigorous results on the
security of the McEliece cryptosystem in the face of quantum adversaries,
strengthening its candidacy for post-quantum cryptography.},
    archivePrefix = {arXiv},
    author = {Dinh, Hang and Moore, Cristopher and Russell, Alexander},
    citeulike-article-id = {7679501},
    citeulike-linkout-0 = {http://arxiv.org/abs/1008.2390},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1008.2390},
    day = {15},
    eprint = {1008.2390},
    keywords = {coding\_theory, cryptography, quantum},
    month = oct,
    posted-at = {2012-02-23 17:14:40},
    priority = {3},
    title = {The {McEliece} Cryptosystem Resists Quantum Fourier Sampling Attacks},
    url = {http://arxiv.org/abs/1008.2390},
    year = {2010}
}

@article{dinh+morre+russel11,
    abstract = {The Code Equivalence problem is that of determining whether two given linear
codes are equivalent to each other up to a permutation of the coordinates. This
problem has a direct reduction to a nonabelian hidden subgroup problem ({HSP}),
suggesting a possible quantum algorithm analogous to Shor's algorithms for
factoring or discrete log. However, we recently showed that in many cases of
interest---including Goppa codes---solving this case of the {HSP} requires rich,
entangled measurements. Thus, solving these cases of Code Equivalence via
Fourier sampling appears to be out of reach of current families of quantum
algorithms.


Code equivalence is directly related to the security of {McEliece}-type
cryptosystems in the case where the private code is known to the adversary.
However, for many codes the support splitting algorithm of Sendrier provides a
classical attack in this case. We revisit the claims of our previous article in
the light of these classical attacks, and discuss the particular case of the
Sidelnikov cryptosystem, which is based on {Reed-Muller} codes.},
    archivePrefix = {arXiv},
    author = {Dinh, Hang and Moore, Cristopher and Russell, Alexander},
    citeulike-article-id = {10353349},
    citeulike-linkout-0 = {http://arxiv.org/abs/1111.4382},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1111.4382},
    day = {18},
    eprint = {1111.4382},
    keywords = {coding\_theory, cryptography, quantum},
    month = nov,
    posted-at = {2012-02-23 17:14:06},
    priority = {3},
    title = {Quantum Fourier sampling, Code Equivalence, and the quantum security of the {McEliece} and Sidelnikov cryptosystems},
    url = {http://arxiv.org/abs/1111.4382},
    year = {2011}
}

@inproceedings{jao+defeo2011,
    abstract = {We present new candidates for quantum-resistant public-key cryptosystems based on the conjectured difficulty of finding isogenies between supersingular elliptic curves. The main technical idea in our scheme is that we transmit the images of torsion bases under the isogeny in order to allow the two parties to arrive at a common shared key despite the noncommutativity of the endomorphism ring. Our work is motivated by the recent development of a subexponential-time quantum algorithm for constructing isogenies between ordinary elliptic curves. In the supersingular case, by contrast, the fastest known quantum attack remains exponential, since the noncommutativity of the endomorphism ring means that the approach used in the ordinary case does not apply. We give a precise formulation of the necessary computational assumption along with a discussion of its validity. In addition, we present implementation results showing that our protocols are multiple orders of magnitude faster than previous isogeny-based cryptosystems over ordinary curves.},
    address = {Berlin, Heidelberg},
    author = {Jao, David and De Feo, Luca},
    booktitle = {Post-Quantum Cryptography},
    chapter = {2},
    citeulike-article-id = {10133770},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-642-25405-5_2},
    citeulike-linkout-1 = {http://www.springerlink.com/content/g906v31619h30364},
    doi = {10.1007/978-3-642-25405-5_2},
    editor = {Yang, Bo-Yin},
    isbn = {978-3-642-25404-8},
    keywords = {elliptic\_curve, isogenies, quantum},
    location = {Taipei, Taiwan},
    pages = {19--34},
    posted-at = {2011-12-16 13:16:08},
    priority = {0},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Towards {Quantum-Resistant} Cryptosystems from Supersingular Elliptic Curve Isogenies},
    url = {http://dx.doi.org/10.1007/978-3-642-25405-5_2},
    volume = {7071},
    year = {2011}
}

@misc{efd,
    author = {Bernstein, Daniel J. and Lange, Tanja},
    citeulike-article-id = {9976969},
    citeulike-linkout-0 = {http://www.hyperelliptic.org/EFD/index.html},
    comment = {\url{http://www.hyperelliptic.org/EFD/index.html}},
    keywords = {cryptography, elliptic\_curve, multiplication},
    posted-at = {2011-11-01 22:42:06},
    priority = {0},
    title = {{Explicit-Formulas Database}},
    url = {http://www.hyperelliptic.org/EFD/index.html},
    year = {2007}
}

@inproceedings{joye07,
    abstract = {{This papers introduces several binary scalar multiplication algorithms with applications to cryptography. Remarkably, the proposed algorithms regularly repeat the same pattern when evaluating a scalar multiplication in an (additively written) abelian group. Furthermore, they are generic and feature the following properties: • no dummy operation is involved; • no precomputation nor prior recoding is needed; • a small number of temporary registers / code memory is required; • the scalar is processed right-to-left. As a result, they are particularly well fitted for implementing cryptosystems in constrained devices, in an efficient yet secure way.}},
    address = {Berlin, Heidelberg},
    author = {Joye, Marc},
    booktitle = {Cryptographic Hardware and Embedded Systems - CHES 2007},
    chapter = {10},
    citeulike-article-id = {9976141},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-74735-2_10},
    citeulike-linkout-1 = {http://www.springerlink.com/content/q872n615uq7255q1},
    doi = {10.1007/978-3-540-74735-2_10},
    editor = {Paillier, Pascal and Verbauwhede, Ingrid},
    isbn = {978-3-540-74734-5},
    issn = {0302-9743},
    keywords = {montgomery\_curve, multiplication},
    pages = {135--147},
    posted-at = {2011-11-01 16:31:41},
    priority = {0},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {{Highly Regular Right-to-Left Algorithms for Scalar Multiplication}},
    url = {http://dx.doi.org/10.1007/978-3-540-74735-2_10},
    volume = {4727},
    year = {2007}
}

@inproceedings{antipa+brown+gallant+lambert+struik+vanstone06,
    abstract = {{Verification of ECDSA signatures is considerably slower than generation of ECDSA signatures. This paper describes a method that can be used to accelerate verification of ECDSA signatures by more than 40\% with virtually no added implementation complexity. The method can also be used to accelerate verification for other ElGamal-like signature algorithms, including DSA.}},
    address = {Berlin, Heidelberg},
    author = {Antipa, Adrian and Brown, Daniel and Gallant, Robert and Lambert, Rob and Struik, Ren\'{e} and Vanstone, Scott},
    booktitle = {Selected Areas in Cryptography 2005},
    chapter = {21},
    citeulike-article-id = {9971276},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/11693383_21},
    citeulike-linkout-1 = {http://www.springerlink.com/content/t0303h226u13ulk6},
    doi = {10.1007/11693383_21},
    editor = {Preneel, Bart and Tavares, Stafford},
    isbn = {978-3-540-33108-7},
    keywords = {elliptic\_curve, multiplication, naf},
    pages = {307--318},
    posted-at = {2011-10-31 13:11:17},
    priority = {0},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Accelerated Verification of {ECDSA} Signatures},
    url = {http://dx.doi.org/10.1007/11693383_21},
    volume = {3897},
    year = {2006}
}

@techreport{solinas01,
    author = {Solinas, Jerome A.},
    citeulike-article-id = {9971253},
    institution = {National Security Agency, USA},
    keywords = {elliptic\_curve, multiplication, naf},
    posted-at = {2011-10-31 13:06:41},
    priority = {0},
    title = {{Low-Weight} Binary Representations for Pairs of Integers},
    year = {2001}
}

@inproceedings{zhang05,
    abstract = {This paper gives a quantum algorithm to search in an set S for a k -tuple satisfying some predefined relation, with the promise that some components of a desired k -tuple are in some subsets of S . In particular when k =2, we show a tight bound of the quantum query complexity for the Claw Finding problem, improving previous upper and lower bounds by Buhrman, Durr, Heiligman, Hoyer, Magniez, Santha and de Wolf [7]. We also consider the distributed scenario, where two parties each holds an n -element set, and they want to decide whether the two sets share a common element. We show a family of protocols s . t . q ( P ) 3/2 . c ( P )= O ( n 2 log n ), where q ( P ) and c ( P ) are the number of quantum queries and the number of communication qubits that the protocol P makes, respectively. This implies that we can pay more for quantum queries to save on quantum communication, and vice versa. To our knowledge, it is the first result about the tradeoff between the two resources.},
    address = {Berlin, Heidelberg},
    author = {Zhang, Shengyu},
    booktitle = {Proceedings of the Eleventh Annual Inter- national Conference on Computing and Combinatorics},
    chapter = {44},
    citeulike-article-id = {9737644},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/11533719_44},
    citeulike-linkout-1 = {http://www.springerlink.com/content/7qdp96j9096y18ak},
    doi = {10.1007/11533719_44},
    editor = {Wang, Lusheng},
    isbn = {978-3-540-28061-3},
    keywords = {cryptography, quantum},
    pages = {430--439},
    posted-at = {2011-09-03 20:31:08},
    priority = {1},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Promised and Distributed Quantum Search Computing and Combinatorics},
    url = {http://dx.doi.org/10.1007/11533719_44},
    volume = {3595},
    year = {2005}
}

@unpublished{tani08,
    abstract = {The claw finding problem has been studied in terms of query complexity as one
of the problems closely connected to cryptography. For given two functions, f
and g, as an oracle which have domains of size N and M ({N<=M}), respectively,
and the same range, the goal of the problem is to find x and y such that
f(x)=g(y). This paper describes an optimal algorithm using quantum walk that
solves this problem. Our algorithm can be generalized to find a claw of k
functions for any constant integer k>1, where the domains of the functions may
have different size.},
    archivePrefix = {arXiv},
    author = {Tani, Seiichiro},
    citeulike-article-id = {9737642},
    citeulike-linkout-0 = {http://arxiv.org/abs/0708.2584},
    citeulike-linkout-1 = {http://arxiv.org/pdf/0708.2584},
    day = {3},
    eprint = {0708.2584},
    keywords = {cryptography, quantum},
    month = mar,
    posted-at = {2011-09-03 20:27:51},
    priority = {1},
    title = {Claw Finding Algorithms Using Quantum Walk},
    url = {http://arxiv.org/abs/0708.2584},
    year = {2008}
}

@article{bernstein+sorenson07,
    abstract = {Fix pairwise coprime positive integers p1 , p2 , . . . , ps . We propose
representing integers u modulo m, where m is any positive integer up to
√
roughly p1 p2 · · · ps , as vectors (u mod p1 , u mod p2 , . . . , u mod ps ). We use
this representation to obtain a new result on the parallel complexity of mod-
ular exponentiation: there is an algorithm for the Common {CRCW} {PRAM}
that, given positive integers x, e, and m in binary, of total bit length n,
computes xe mod m in time O(n/lg lg n) using {nO}(1) processors. For com-
parison, a parallelization of the standard binary algorithm takes superlinear
time; Adleman and Kompella gave an O((lg n)3 ) expected time algorithm us-
√
ing {exp(O}( n lg n)) processors; von zur Gathen gave an {NC} algorithm for the
highly special case that m is polynomially smooth.},
    author = {Bernstein, Daniel J. and Sorenson, Jonathan P.},
    citeulike-article-id = {9679560},
    citeulike-linkout-0 = {http://www.ams.org/journals/mcom/2007-76-257/S0025-5718-06-01849-7/},
    journal = {Math. Comp.},
    keywords = {chinese\_remainder\_theorem, computer\_algebra},
    month = jan,
    number = {257},
    pages = {443--454},
    posted-at = {2011-08-18 19:22:27},
    priority = {3},
    title = {Modular exponentiation via the {C}hinese remainder theorem},
    url = {http://www.ams.org/journals/mcom/2007-76-257/S0025-5718-06-01849-7/},
    volume = {76},
    year = {2007}
}

@inproceedings{heath+loehr99,
    abstract = {An abstract is not available.},
    address = {Philadelphia, PA, USA},
    author = {Heath, Lenwood S. and Loehr, Nicholas A.},
    booktitle = {Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms},
    citeulike-article-id = {9608507},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=314596},
    isbn = {0-89871-434-6},
    keywords = {conway\_polynomial, finite\_field},
    location = {Baltimore, Maryland, United States},
    pages = {429--437},
    posted-at = {2011-08-05 01:37:27},
    priority = {3},
    publisher = {Society for Industrial and Applied Mathematics},
    series = {SODA '99},
    title = {New algorithms for generating {C}onway polynomials over finite fields},
    url = {http://portal.acm.org/citation.cfm?id=314596},
    year = {1999}
}

@article{scheerhorn92,
    abstract = {J. H. Conway suggested to use norm-compatible polynomials when constructing extension fields in order to get finite field embeddings, which are computationally easy to handle. Analogous to norm-compatibility for the multiplicative group of ¯Fq we exhibit the notion of trace-compatibility for its additive group. By this we get computationally simple embeddings in the case of normal basis representations of finite fields. We give an algorithm for computing tracecompatible polynomials and count the number of distinct compatible polynomials of a fixed degree of both kinds.},
    author = {Scheerhorn, Alfred},
    citeulike-article-id = {9608506},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/BF01268660},
    citeulike-linkout-1 = {http://www.springerlink.com/content/j483773txj8w4842},
    day = {1},
    doi = {10.1007/BF01268660},
    issn = {0938-1279},
    journal = {Applicable Algebra in Engineering, Communication and Computing},
    keywords = {conway\_polynomial, normal\_base},
    month = sep,
    number = {3},
    pages = {199--209},
    posted-at = {2011-08-05 01:35:19},
    priority = {4},
    publisher = {Springer Berlin / Heidelberg},
    title = {Trace- and norm-compatible extensions of finite fields},
    url = {http://dx.doi.org/10.1007/BF01268660},
    volume = {3},
    year = {1992}
}

@article{skachek+roth08,
    abstract = {A probabilistic algorithm is presented for finding a basis of the root space of a linearized polynomial     \$\$L(x) = \sum\_{i=0}^t L\_i x^{q^i}\$\$    over \$\$\mathbb {F}\_{q^n}\$\$ . The expected time complexity of the presented algorithm is O(n t) operations in \$\$\mathbb {F}\_{q^n}\$\$ .},
    author = {Skachek, Vitaly and Roth, Ron},
    citeulike-article-id = {1952941},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s10623-007-9125-y},
    citeulike-linkout-1 = {http://www.springerlink.com/content/4811757628402r58},
    day = {1},
    doi = {10.1007/s10623-007-9125-y},
    issn = {0925-1022},
    journal = {Designs, Codes and Cryptography},
    keywords = {coding\_theory, gabidulin},
    month = jan,
    number = {1},
    pages = {17--23},
    posted-at = {2011-07-12 02:57:57},
    priority = {2},
    publisher = {Springer Netherlands},
    title = {Probabilistic algorithm for finding roots of linearized polynomials},
    url = {http://dx.doi.org/10.1007/s10623-007-9125-y},
    volume = {46},
    year = {2008}
}

@unpublished{sutherland11-supersingular,
    abstract = {Given an elliptic curve E over a field of characteristic p, we consider how
to efficiently determine whether E is ordinary or supersingular. We analyze the
complexity of several existing algorithms and then present a new approach that
exploits structural differences between ordinary and supersingular isogeny
graphs. This yields a simple algorithm that, given E and a suitable non-residue
in F\_p^2, determines the supersingularity of E in O(n^3 log^2 n) time and O(n)
space, where {n=O}(log p). Both these complexity bounds are significant
improvements over existing methods, as we demonstrate with some practical
computations.},
    archivePrefix = {arXiv},
    author = {Sutherland, Andrew V.},
    citeulike-article-id = {9521780},
    citeulike-linkout-0 = {http://arxiv.org/abs/1107.1140},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1107.1140},
    day = {6},
    eprint = {1107.1140},
    keywords = {elliptic\_curve, isogenies, supersingular},
    month = jul,
    posted-at = {2011-07-07 18:00:20},
    priority = {0},
    title = {Identifying supersingular elliptic curves},
    url = {http://arxiv.org/abs/1107.1140},
    year = {2011}
}

@unpublished{mathscheme11,
    abstract = {We present some of the experiments we have performed to best test our design
for a library for {MathScheme}, the mechanized mathematics software system we are
building. We wish for our library design to use and reflect, as much as
possible, the mathematical structure present in the objects which populate the
library.},
    archivePrefix = {arXiv},
    author = {Carette, Jacques and Farmer, William M. and Jeremic, Filip and Maccio, Vincent and O'Connor, Russell and Tran, Quang M.},
    citeulike-article-id = {9496690},
    citeulike-linkout-0 = {http://arxiv.org/abs/1106.1862},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1106.1862},
    day = {9},
    eprint = {1106.1862},
    keywords = {algebraic\_data\_types, categories, computer\_algebra},
    month = jun,
    posted-at = {2011-07-03 20:40:32},
    priority = {0},
    title = {The {MathScheme} Library: Some Preliminary Experiments},
    url = {http://arxiv.org/abs/1106.1862},
    year = {2011}
}

@unpublished{birkner+sica11,
    abstract = {The GLV method of Gallant, Lambert and Vanstone (CRYPTO 2001) computes any
multiple \$kP\$ of a point \$P\$ of prime order \$n\$ lying on an elliptic curve with
a low-degree endomorphism \$\Phi\$ (called GLV curve) over \$\mathbb{F}\_p\$ as


\[kP = k\_1P + k\_2\Phi(P), \quad\text{with} \max\{|k\_1|,|k\_2|\}\leq C\sqrt n\]


for some explicit constant \$C>0\$. Recently, Galbraith, Lin and Scott
(EUROCRYPT 2009) extended this method to all curves over \$\mathbb{F}\_{p^2}\$
which are twists of curves defined over \$\mathbb{F}\_p\$. These are examples of
two-dimensional decompositions (with two new scalars), and the GLS approach
shows that for curves with many automorphisms (cubic and quartic twists) one
can achieve a four-dimensional decomposition as well.


We show in this work how to merge the two approaches in order to get, for
twists of any GLV curve over \$\mathbb{F}\_{p^2}\$, a four-dimensional
decomposition together with fast endomorphisms \$\Phi, \Psi\$ over
\$\mathbb{F}\_{p^2}\$ acting on the group generated by a point \$P\$ of prime order
\$n\$, resulting in a proved decomposition for any scalar \$k\in[1,n]\$ \$\$ kP=k\_1P+
k\_2\Phi(P)+ k\_3\Psi(P) + k\_4\Psi\Phi(P)\quad \text{with} \max\_i (|k\_i|)< C
n^{1/4} \$\$ for some explicitly computable \$C>0\$. If parallel computation is
available, then the computation of \$kP\$ can possibly be implemented up to four
times as fast as a traditional scalar multiplication.


We also derive new families of GLV curves, corresponding to those curves with
degree 3 endomorphisms.


Finally, we present a faster approach to produce a better proven
decomposition inspired by the original GLV method, which computes the reduced
kernel lattice basis using a truncated form of the Euclidean algorithm instead
of the LLL algorithm.},
    archivePrefix = {arXiv},
    author = {Birkner, Peter and Sica, Francesco},
    citeulike-article-id = {9496430},
    citeulike-linkout-0 = {http://arxiv.org/abs/1106.5149},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1106.5149},
    day = {25},
    eprint = {1106.5149},
    keywords = {elliptic\_curve, multiplication},
    month = jun,
    posted-at = {2011-07-03 19:13:45},
    priority = {3},
    title = {Four-Dimensional {G}allant-{L}ambert-{V}anstone Scalar Multiplication},
    url = {http://arxiv.org/abs/1106.5149},
    year = {2011}
}

@unpublished{schicho+sevilla,
    abstract = {We define the concept of {Tschirnhaus-Weierstrass} curve, named after the
Weierstrass form of an elliptic curve and Tschirnhaus transformations. Every
pointed curve has a {Tschirnhaus-Weierstrass} form, and this representation is
unique up to a scaling of variables. This is useful for computing isomorphisms
between curves.},
    archivePrefix = {arXiv},
    author = {Schicho, Josef and Sevilla, David},
    citeulike-article-id = {3169063},
    citeulike-linkout-0 = {http://arxiv.org/abs/0808.3038},
    citeulike-linkout-1 = {http://arxiv.org/pdf/0808.3038},
    day = {9},
    eprint = {0808.3038},
    keywords = {elliptic\_curve, hyperelliptic\_curves, weierstrass},
    month = jun,
    posted-at = {2011-07-03 19:11:30},
    priority = {3},
    title = {{T}schirnhaus-{W}eierstrass curves},
    url = {http://arxiv.org/abs/0808.3038},
    year = {2011}
}

@electronic{galbraith+stolbunov11,
    abstract = {A low storage algorithm for constructing isogenies between ordinary elliptic
curves was proposed by Galbraith, Hess and Smart ({GHS}). We give an improvement
of this algorithm by modifying the pseudorandom walk so that lower-degree
isogenies are used more frequently. This is motivated by the fact that high
degree isogenies are slower to compute than low degree ones. We analyse the
running time of the parallel collision search algorithm when the partitioning
is uneven. We also give experimental results. We conclude that our algorithm is
around 14 times faster than the {GHS} algorithm when constructing horizontal
isogenies between random isogenous elliptic curves over a 160-bit prime field.


The results apply to generic adding walks and the more general group action
inverse problem; a speed-up is obtained whenever the cost of computing edges in
the graph varies significantly.},
    archivePrefix = {arXiv},
    author = {Galbraith, Steven and Stolbunov, Anton},
    citeulike-article-id = {9369852},
    citeulike-linkout-0 = {http://arxiv.org/abs/1105.6331},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1105.6331},
    day = {31},
    eprint = {1105.6331},
    keywords = {elliptic\_curve, isogenies},
    month = may,
    posted-at = {2011-06-03 13:56:14},
    priority = {4},
    title = {Improved Algorithm for the Isogeny Problem for Ordinary Elliptic Curves},
    url = {http://arxiv.org/abs/1105.6331},
    year = {2011}
}

@article{koetter+kschischang08,
    abstract = {The problem of error-control in random linear network coding is considered. A ldquononcoherentrdquo or ldquochannel obliviousrdquo model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic. Motivated by the property that linear network coding is vector-space preserving, information transmission is modeled as the injection into the network of a basis for a vector space  V  and the collection by the receiver of a basis for a vector space  U . A metric on the projective geometry associated with the packet space is introduced, and it is shown that a minimum-distance decoder for this metric achieves correct decoding if the dimension of the space  V  cap U  is sufficiently large. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or, equivalently, a subset of the vertices of the corresponding Grassmann graph. Sphere-packing and sphere-covering bounds as well as a generalization of the singleton bound are provided for such codes. Finally, a {Reed-Solomon}-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style ldquolist-1rdquo minimum-distance decoding algorithm is provided.},
    author = {Koetter, Ralf and Kschischang, Frank R.},
    booktitle = {Information Theory, IEEE Transactions on},
    citeulike-article-id = {3023290},
    citeulike-linkout-0 = {http://dx.doi.org/10.1109/TIT.2008.926449},
    citeulike-linkout-1 = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4567581},
    day = {16},
    doi = {10.1109/TIT.2008.926449},
    issn = {0018-9448},
    journal = {IEEE Transactions on Information Theory},
    keywords = {gabidulin, network\_coding},
    month = aug,
    number = {8},
    pages = {3579--3591},
    posted-at = {2011-05-18 22:36:50},
    priority = {2},
    title = {Coding for Errors and Erasures in Random Network Coding},
    url = {http://dx.doi.org/10.1109/TIT.2008.926449},
    volume = {54},
    year = {2008}
}

@article{silva+kschi+koetter08,
    abstract = {The problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of Rotter and Kschischang. A large class of constant-dimension subspace codes is investigated. It is shown that codes in this class can be easily constructed from rank-metric codes, while preserving their distance properties. Moreover, it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank-metric codes where partial information about the error is available. This partial information may be in the form of erasures (knowledge of an error location but not its value) and deviations (knowledge of an error value but not its location). Taking erasures and deviations into account (when they occur) strictly increases the error correction capability of a code: if mu erasures and delta deviations occur, then errors of rank t can always be corrected provided that 2t les d - 1 + mu + delta, where d is the minimum rank distance of the code. For Gabidulin codes, an important family of maximum rank distance codes, an efficient decoding algorithm is proposed that can properly exploit erasures and deviations. In a network coding application, where n packets of length M over F(q) are transmitted, the complexity of the decoding algorithm is given by {O(dM}) operations in an extension field F(q n ).},
    author = {Silva, Danilo and Kschischang, Frank R. and Koetter, Ralf},
    citeulike-article-id = {6591209},
    citeulike-linkout-0 = {http://dx.doi.org/10.1109/TIT.2008.928291},
    citeulike-linkout-1 = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4608992},
    day = {26},
    doi = {10.1109/TIT.2008.928291},
    issn = {0018-9448},
    journal = {IEEE Transactions on Information Theory},
    keywords = {gabidulin, network\_coding},
    month = sep,
    number = {9},
    pages = {3951--3967},
    posted-at = {2011-05-18 22:34:35},
    priority = {2},
    title = {A {Rank-Metric} Approach to Error Control in Random Network Coding},
    url = {http://dx.doi.org/10.1109/TIT.2008.928291},
    volume = {54},
    year = {2008}
}

@inproceedings{harvey+roche10,
    abstract = {The truncated Fourier transform ({TFT}) was introduced by van der Hoeven in 2004 as a means of smoothing the "jumps" in running time of the ordinary {FFT} algorithm that occur at power-of-two input sizes. However, the {TFT} still introduces these jumps in memory usage. We describe in-place variants of the forward and inverse {TFT} algorithms, achieving time complexity O(n log n) with only O(1) auxiliary space. As an application, we extend the second author's results on space-restricted {FFT}-based polynomial multiplication to polynomials of arbitrary degree.},
    address = {New York, NY, USA},
    author = {Harvey, David and Roche, Daniel S.},
    booktitle = {Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation},
    citeulike-article-id = {9275399},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1837996},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1837934.1837996},
    doi = {10.1145/1837934.1837996},
    isbn = {978-1-4503-0150-3},
    keywords = {computer\_algebra, polynomial\_multiplication},
    location = {Munich, Germany},
    pages = {325--329},
    posted-at = {2011-05-11 11:03:03},
    priority = {0},
    publisher = {ACM},
    series = {ISSAC '10},
    title = {An in-place truncated {F}ourier transform and applications to polynomial multiplication},
    url = {http://dx.doi.org/10.1145/1837934.1837996},
    year = {2010}
}

@unpublished{bostan+chowdhury+hoeven+schost,
    abstract = {We study the cost of multiplication modulo triangular families of
polynomials. Following previous work by Li, Moreno Maza and Schost, we propose
an algorithm that relies on homotopy and fast evaluation-interpolation
techniques. We obtain a quasi-linear time complexity for substantial families
of examples, for which no such result was known before. Applications are given
to notably addition of algebraic numbers in small characteristic.},
    archivePrefix = {arXiv},
    author = {Bostan, Alin and Chowdhury, Muhammad and van der Hoeven, Joris and Schost, Eric},
    citeulike-article-id = {9271510},
    citeulike-linkout-0 = {http://arxiv.org/abs/0901.3657},
    citeulike-linkout-1 = {http://arxiv.org/pdf/0901.3657},
    day = {23},
    eprint = {0901.3657},
    keywords = {multivariate\_polynomial, polynomial\_multiplication, triangular\_set},
    month = jan,
    posted-at = {2011-05-10 18:14:18},
    priority = {0},
    title = {Homotopy methods for multiplication modulo triangular sets},
    url = {http://arxiv.org/abs/0901.3657},
    year = {2009}
}

@article{gao+gathen+panario+shoup00,
    author = {Gao, Shuhong and Joachim von zur Gathen and Panario, Daniel and Shoup, Victor},
    citeulike-article-id = {71317},
    citeulike-linkout-0 = {http://citeseer.ist.psu.edu/gao00algorithms.html},
    citeulike-linkout-1 = {http://citeseer.lcs.mit.edu/gao00algorithms.html},
    citeulike-linkout-2 = {http://citeseer.ifi.unizh.ch/gao00algorithms.html},
    citeulike-linkout-3 = {http://citeseer.comp.nus.edu.sg/gao00algorithms.html},
    journal = {J. Symbolic Comput.},
    keywords = {finite\_field, normal\_base},
    number = {6},
    pages = {879--889},
    posted-at = {2011-05-09 17:45:42},
    priority = {3},
    title = {Algorithms for Exponentiation in Finite Fields},
    url = {http://citeseer.ist.psu.edu/gao00algorithms.html},
    volume = {29},
    year = {2000}
}

@article{couveignes_lercier09,
    abstract = {We construct two new families of basis for finite field extensions. Bases in the first family, the so-called elliptic bases, are not quite normal bases, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Bases in the second family, the so-called normal elliptic bases are normal bases and allow fast (quasi-linear) arithmetic. We prove that all extensions admit models of this kind.},
    author = {Couveignes, Jean-Marc and Lercier, Reynald},
    citeulike-article-id = {9268493},
    citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.ffa.2008.07.004},
    doi = {10.1016/j.ffa.2008.07.004},
    issn = {10715797},
    journal = {Finite Fields and Their Applications},
    keywords = {elliptic\_curve, finite\_field, normal\_base},
    month = feb,
    number = {1},
    pages = {1--22},
    posted-at = {2011-05-09 16:59:28},
    priority = {0},
    title = {Elliptic periods for finite fields},
    url = {http://dx.doi.org/10.1016/j.ffa.2008.07.004},
    volume = {15},
    year = {2009}
}

@unpublished{arnault+pickett+vinatier,
    abstract = {Recent work of Pickett has given a construction of self-dual normal bases for
extensions of finite fields, whenever they exist. In this article we present
these results in an explicit and constructive manner and apply them, through
computer search, to identify the lowest complexity of self-dual normal bases
for extensions of low degree. Comparisons to similar searches amongst normal
bases show that the lowest complexity is often achieved from a self-dual normal
basis.},
    archivePrefix = {arXiv},
    author = {Arnault, Fran\c{c}ois and Pickett, Erik J. and Vinatier, St\'{e}phane},
    citeulike-article-id = {9268484},
    citeulike-linkout-0 = {http://arxiv.org/abs/1007.4899},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1007.4899},
    day = {6},
    eprint = {1007.4899},
    keywords = {finite\_field, normal\_base},
    month = may,
    posted-at = {2011-05-09 16:55:53},
    priority = {4},
    title = {Construction of self-dual normal bases and their complexity},
    url = {http://arxiv.org/abs/1007.4899},
    year = {2011}
}

@inproceedings{wachter+afanassiev+sidorenko11,
    author = {Wachter, Antonia and Afanassiev, Valentin and Sidorenko, Vladimir},
    booktitle = {The Seventh International Workshop on Coding and Cryptography 2011 (WCC 2011)},
    citeulike-article-id = {9175738},
    citeulike-linkout-0 = {http://tait.e-technik.uni-ulm.de/~wachter/papers/WachterAfanSido-FastDecGabidulin-WCC.pdf},
    keywords = {coding\_theory, gabidulin, normal\_base},
    location = {Paris, France},
    month = apr,
    posted-at = {2011-04-20 17:24:57},
    priority = {0},
    title = {Fast Decoding of {G}abidulin Codes},
    url = {http://tait.e-technik.uni-ulm.de/~wachter/papers/WachterAfanSido-FastDecGabidulin-WCC.pdf},
    year = {2011}
}

@inproceedings{wachter+sidorenko+bossert10,
    author = {Wachter, Antonia and Sidorenko, Vladimir and Bossert, Martin},
    booktitle = {Twelfth International Workshop on Algebraic and Combinatorial Coding Theory (ACCT 2010)},
    citeulike-article-id = {9175731},
    citeulike-linkout-0 = {http://tait.e-technik.uni-ulm.de/~wachter/papers/WachterSidorenkoBossert_Acct.pdf},
    keywords = {coding\_theory, gabidulin},
    location = {Novosibirsk, Russia.},
    month = sep,
    posted-at = {2011-04-20 17:23:00},
    priority = {0},
    title = {A Fast Linearized {E}uclidean Algorithm for Decoding {G}abidulin Codes},
    url = {http://tait.e-technik.uni-ulm.de/~wachter/papers/WachterSidorenkoBossert_Acct.pdf},
    year = {2010}
}

@article{gabidulin85,
    author = {Gabidulin, Ernst M.},
    citeulike-article-id = {9175722},
    journal = {Problems of Information Transmission},
    keywords = {coding\_theory, gabidulin},
    pages = {1--12},
    posted-at = {2011-04-20 17:17:53},
    priority = {2},
    title = {Theory of codes with maximum rank distance},
    volume = {21},
    year = {1985}
}

@inproceedings{silva+kschi09,
    abstract = {Gabidulin codes are the rank-metric analogs of {Reed-Solomon} codes and have a major role in practical error control for network coding. This paper presents new encoding and decoding algorithms for Gabidulin codes based on low-complexity normal bases. In addition, a new decoding algorithm is proposed based on a transform-domain approach. Together, these represent the fastest known algorithms for encoding and decoding Gabidulin codes.},
    address = {Piscataway, NJ, USA},
    author = {Silva, Danilo and Kschischang, Frank R.},
    booktitle = {Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 4},
    citeulike-article-id = {9175717},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1701108},
    isbn = {978-1-4244-4312-3},
    keywords = {coding\_theory, gabidulin},
    location = {Coex, Seoul, Korea},
    pages = {2858--2862},
    posted-at = {2011-04-20 17:15:51},
    priority = {0},
    publisher = {IEEE Press},
    series = {ISIT'09},
    title = {Fast encoding and decoding of {G}abidulin codes},
    url = {http://portal.acm.org/citation.cfm?id=1701108},
    year = {2009}
}

@unpublished{cardella11,
    abstract = {This article is a short introduction to the theory of the groups of points of
elliptic curves over finite fields. This note is concerned with the elementary
theory and practice of elliptic curves cryptography, the new generation of
public key systems. The material and coverage are focused on the groups of
points of elliptic curves and algebraic curves, but not exclusively. The level
of detail ranges from the very simple to the very difficult and open problems.
The detailed proofs of many simple results are included or discussed. However,
the long and difficult proofs of difficult results are not included, but
relevant references are given. The reader should consult the literature for
further elaboration on difficult concepts and ideas. Now and again, a concept
is stated without explanation, but references to the literature are provided.
An elementary background in finite fields analysis and number theory at the
level of [{MC87}], [{LN97}], [{ST92}], [{WN03}] etc, should be sufficient to read the
material and will be assumed.},
    archivePrefix = {arXiv},
    author = {Carella, N. A.},
    citeulike-article-id = {9053260},
    citeulike-linkout-0 = {http://arxiv.org/abs/1103.4560},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1103.4560},
    day = {22},
    eprint = {1103.4560},
    keywords = {education, elliptic\_curve},
    month = mar,
    posted-at = {2011-03-24 12:41:45},
    priority = {2},
    title = {Topic In Elliptic Curves Over Finite Fields: The Groups of Points},
    url = {http://arxiv.org/abs/1103.4560},
    year = {2011}
}

@article{mpfr,
    abstract = {This article presents a multiple-precision binary floating-point library, written in the {ISO} C language, and based on the {GNU} {MP} library. Its particularity is to extend to arbitrary-precision, ideas from the {IEEE} 754 standard, by providing correct rounding and exceptions. We demonstrate how these strong semantics are achieved---with no significant slowdown with respect to other arbitrary-precision tools---and discuss a few applications where such a library can be useful.},
    address = {New York, NY, USA},
    author = {Fousse, Laurent and Hanrot, Guillaume and Lef{\`{e}}vre, Vincent and P{\'{e}}lissier, Patrick and Zimmermann, Paul},
    citeulike-article-id = {2869457},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1236463.1236468},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1236463.1236468},
    doi = {10.1145/1236463.1236468},
    issn = {0098-3500},
    journal = {ACM Transactions on Mathematical Software},
    month = jun,
    number = {2},
    pages = {13+},
    posted-at = {2011-03-22 18:54:38},
    priority = {1},
    publisher = {ACM},
    title = {{MPFR}: A multiple-precision binary floating-point library with correct rounding},
    url = {http://dx.doi.org/10.1145/1236463.1236468},
    volume = {33},
    year = {2007}
}

@article{thome02,
    abstract = {This paper describes a new algorithm for computing linear generators (vector generating polynomials) for matrix sequences, running in subquadratic time. This algorithm applies in particular to the sequential stage of Coppersmith's block Wiedemann algorithm. Experiments showed that our method can be substituted in place of the quadratic one proposed by Coppersmith, yielding important speedups even for realistic matrix sizes. The base fields we were interested in were finite fields of large characteristic. As an example, we have been able to compute a linear generator for a sequence of 4  ×  4 matrices of length 242 304 defined over F 2607 − 1 in less than 2 days on one 667 {MHz} alpha ev67 {CPU}.},
    author = {Thom{\'{e}}, Emmanuel},
    citeulike-article-id = {9043129},
    citeulike-linkout-0 = {http://dx.doi.org/10.1006/jsco.2002.0533},
    doi = {10.1006/jsco.2002.0533},
    issn = {07477171},
    journal = {J. Symbolic Comput.},
    keywords = {factorization},
    month = may,
    number = {5},
    pages = {757--775},
    posted-at = {2011-03-22 18:52:43},
    priority = {2},
    title = {Subquadratic Computation of Vector Generating Polynomials and Improvement of the Block Wiedemann Algorithm},
    url = {http://dx.doi.org/10.1006/jsco.2002.0533},
    volume = {33},
    year = {2002}
}

@inproceedings{akss08,
    abstract = {We give an {O(N} • log N • {2O}({log*N})) algorithm for multiplying two N-bit integers that improves the {O(N} • log N • log log N) algorithm by {Sch\"{o}nhage-Strassen}. Both these algorithms use modular arithmetic. Recently, F\"{u}rer gave an {O(N} • log N • {2O}({log*N})) algorithm which however uses arithmetic over complex numbers as opposed to modular arithmetic. In this paper, we use multivariate polynomial multiplication along with ideas from F\"{u}rer's algorithm to achieve this improvement in the modular setting. Our algorithm can also be viewed as a p-adic version of F\"{u}rer's algorithm. Thus, we show that the two seemingly different approaches to integer multiplication, modular and complex arithmetic, are similar.},
    address = {New York, NY, USA},
    author = {De, Anindya and Kurur, Piyush P. and Saha, Chandan and Saptharishi, Ramprasad},
    booktitle = {Proceedings of the 40th annual ACM symposium on Theory of computing},
    citeulike-article-id = {8349225},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1374447},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1374376.1374447},
    doi = {10.1145/1374376.1374447},
    isbn = {978-1-60558-047-0},
    keywords = {multiplication},
    location = {Victoria, British Columbia, Canada},
    pages = {499--506},
    posted-at = {2011-03-15 00:11:44},
    priority = {3},
    publisher = {ACM},
    series = {STOC '08},
    title = {Fast integer multiplication using modular arithmetic},
    url = {http://dx.doi.org/10.1145/1374376.1374447},
    year = {2008}
}

@article{furer09,
    author = {F{\"{u}}rer, Martin},
    citeulike-article-id = {5713660},
    citeulike-linkout-0 = {http://dx.doi.org/10.1137/070711761},
    citeulike-linkout-1 = {http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SMJCAT000039000003000979000001&idtype=cvips&gifs=yes},
    citeulike-linkout-2 = {http://link.aip.org/link/?SMJ/39/979},
    doi = {10.1137/070711761},
    journal = {SIAM Journal on Computing},
    keywords = {polynomial\_multiplication},
    number = {3},
    pages = {979--1005},
    posted-at = {2011-03-15 00:00:14},
    priority = {3},
    publisher = {SIAM},
    title = {Faster Integer Multiplication},
    url = {http://dx.doi.org/10.1137/070711761},
    volume = {39},
    year = {2009}
}

@incollection{couveignes+henocq02,
    abstract = {We study the action of modular correspondences in the p-adic neighborhood of CM points. We deduce and prove two stable and efficient p-adic analytic methods for computing singular values of modular functions. On the way we prove a non trivial lower bound for the density of smooth numbers in imaginary quadratic rings and show that the canonical lift of an elliptic curve over  \$\$\mathbb{F}\_q\$\$  can be computed in probabilistic time ≪ exp((log q)1/2+ε) under GRH. We also extend the notion of canonical lift to supersingular elliptic curves and show how to compute it in that case.},
    address = {Berlin, Heidelberg},
    author = {Couveignes, Jean-Marc and Henocq, Thierry},
    booktitle = {Algorithmic Number Theory},
    chapter = {19},
    citeulike-article-id = {8989468},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/3-540-45455-1_19},
    citeulike-linkout-1 = {http://www.springerlink.com/content/15krnvgulecqjd86},
    day = {21},
    doi = {10.1007/3-540-45455-1_19},
    isbn = {978-3-540-43863-2},
    keywords = {elliptic\_curve, modular\_curves, p-adic},
    month = jun,
    pages = {57--69},
    posted-at = {2011-03-14 14:40:32},
    priority = {3},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Action of Modular Correspondences around {CM} Points},
    url = {http://dx.doi.org/10.1007/3-540-45455-1_19},
    volume = {2369},
    year = {2002}
}

@article{broker08,
    author = {Br{\"{o}}ker, Reinier},
    citeulike-article-id = {8989461},
    citeulike-linkout-0 = {http://dx.doi.org/10.1090/S0025-5718-08-02091-7},
    doi = {10.1090/S0025-5718-08-02091-7},
    journal = {Math. Comp.},
    keywords = {elliptic\_curve, p-adic},
    number = {264},
    pages = {2417--2435},
    posted-at = {2011-03-14 14:36:06},
    priority = {4},
    title = {A \(p\)-adic algorithm to compute the {H}ilbert class polynomial},
    url = {http://dx.doi.org/10.1090/S0025-5718-08-02091-7},
    volume = {77},
    year = {2008}
}

@inproceedings{belding+broker+enge+lauter08,
    abstract = {We present and analyze two algorithms for computing {theHilbert} class polynomial {HD}. The first is a p-adic lifting algorithm forinert primes p in the order of discriminant D < 0. The second is an {improvedChinese} remainder algorithm which uses the class group action {onCM}-curves over finite fields. Our run time analysis gives tighter boundsfor the complexity of all known algorithms for computing {HD}, and weshow that all methods have comparable run times.},
    address = {Berlin, Heidelberg},
    author = {Belding, Juliana and Br{\"{o}}ker, Reinier and Enge, Andreas and Lauter, Kristin},
    booktitle = {Proceedings of the 8th international conference on Algorithmic number theory},
    citeulike-article-id = {8989400},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1789741},
    citeulike-linkout-1 = {http://portal.acm.org/citation.cfm?id=1789741},
    isbn = {3-540-79455-7, 978-3-540-79455-4},
    keywords = {elliptic\_curve, p-adic},
    location = {Banff, Canada},
    pages = {282--295},
    posted-at = {2011-03-14 14:17:42},
    priority = {4},
    publisher = {Springer-Verlag},
    series = {ANTS-VIII'08},
    title = {Computing {H}ilbert class polynomials},
    url = {http://portal.acm.org/citation.cfm?id=1789741},
    year = {2008}
}

@manual{sage4.6.2,
    author = {Stein, William A. and {others}},
    citeulike-article-id = {8989164},
    citeulike-linkout-0 = {http://www.sagemath.org/},
    comment = {{\tt http://www.sagemath.org}},
    key = {Sage},
    keywords = {computer\_algebra},
    organization = {The Sage Development Team},
    posted-at = {2011-03-14 13:09:41},
    priority = {2},
    title = {{S}age {M}athematics {S}oftware ({V}ersion 4.6.2)},
    url = {http://www.sagemath.org/},
    year = {2011}
}

@article{bostan+schost04,
    abstract = {We compare the complexities of multipoint polynomial evaluation and interpolation. We show that, over a field of characteristic zero, both questions have equivalent complexities, up to a constant number of polynomial multiplications.},
    author = {Bostan, Alin and Schost, \'{E}ric},
    citeulike-article-id = {8979545},
    citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.tcs.2004.09.002},
    day = {13},
    doi = {10.1016/j.tcs.2004.09.002},
    issn = {03043975},
    journal = {Theoretical Computer Science},
    keywords = {polynomial\_evaluation, polynomial\_interpolation, transposition},
    month = dec,
    number = {1-3},
    pages = {223--235},
    posted-at = {2011-03-11 13:55:18},
    priority = {0},
    title = {On the complexities of multipoint evaluation and interpolation},
    url = {http://dx.doi.org/10.1016/j.tcs.2004.09.002},
    volume = {329},
    year = {2004}
}

@article{enge09,
    author = {Enge, Andreas},
    citeulike-article-id = {8976948},
    journal = {Math. Comp.},
    keywords = {elliptic\_curve},
    number = {267},
    pages = {1809--1824},
    posted-at = {2011-03-10 14:53:26},
    priority = {2},
    title = {Computing modular polynomials in quasi-linear time},
    volume = {78},
    year = {2009}
}

@phdthesis{streng-phd,
    address = {Leiden, The Netherlands},
    author = {Streng, Marco},
    citeulike-article-id = {8976136},
    citeulike-linkout-0 = {http://www.warwick.ac.uk/~masjap/thesis.pdf},
    keywords = {algebraic\_geometry, complex\_multiplication},
    month = jun,
    posted-at = {2011-03-10 11:54:46},
    priority = {4},
    school = {Thomas Stieltjes Institute for Mathematics, Leiden University},
    title = {Complex multiplication of abelian surfaces},
    url = {http://www.warwick.ac.uk/~masjap/thesis.pdf},
    year = {2010}
}

@article{rubin+silverberg10,
    author = {Rubin, Karl and Silverberg, Alice},
    citeulike-article-id = {8972921},
    citeulike-linkout-0 = {http://dx.doi.org/10.1090/S0025-5718-09-02266-2},
    doi = {10.1090/S0025-5718-09-02266-2},
    journal = {Math. Comp.},
    keywords = {complex\_multiplication, elliptic\_curve},
    pages = {545--561},
    posted-at = {2011-03-09 20:29:08},
    priority = {3},
    title = {Choosing the correct elliptic curve in the {CM} method},
    url = {http://dx.doi.org/10.1090/S0025-5718-09-02266-2},
    volume = {79},
    year = {2010}
}

@article{bisson+sutherland11,
    abstract = {We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field . Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of , while our bound for the second algorithm depends primarily on {log|DE}|, where {DE} is the discriminant of the order isomorphic to {End(E}). As a byproduct, our method yields a short certificate that may be used to verify that the endomorphism ring is as claimed.},
    author = {Bisson, Gaetan and Sutherland, Andrew V.},
    citeulike-article-id = {6411804},
    citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.jnt.2009.11.003},
    day = {14},
    doi = {10.1016/j.jnt.2009.11.003},
    issn = {0022314X},
    journal = {Journal of Number Theory},
    keywords = {elliptic\_curve, isogenies, isogeny\_cycles},
    month = may,
    number = {5},
    pages = {815--831},
    posted-at = {2011-03-09 19:58:26},
    priority = {0},
    title = {Computing the endomorphism ring of an ordinary elliptic curve over a finite field},
    url = {http://dx.doi.org/10.1016/j.jnt.2009.11.003},
    volume = {131},
    year = {2011}
}

@unpublished{childs+jao+soukharev10,
    abstract = {Given two elliptic curves over a finite field having the same cardinality and
endomorphism ring, it is known that the curves admit an isogeny between them,
but finding such an isogeny is believed to be computationally difficult. The
fastest known classical algorithm takes exponential time, and prior to our work
no faster quantum algorithm was known. Recently, public-key cryptosystems based
on the presumed hardness of this problem have been proposed as candidates for
post-quantum cryptography. In this paper, we give a subexponential-time quantum
algorithm for constructing isogenies, assuming the Generalized Riemann
Hypothesis (but with no other assumptions). This result suggests that
isogeny-based cryptosystems may be uncompetitive with more mainstream
quantum-resistant cryptosystems such as lattice-based cryptosystems. As part of
our algorithm, we also obtain a second result of independent interest: we
provide a new subexponential-time classical algorithm for evaluating a
horizontal isogeny given its kernel ideal, assuming (only) {GRH}, eliminating the
heuristic assumptions required by prior algorithms.},
    archivePrefix = {arXiv},
    author = {Childs, Andrew M. and Jao, David and Soukharev, Vladimir},
    citeulike-article-id = {8971379},
    citeulike-linkout-0 = {http://arxiv.org/abs/1012.4019},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1012.4019},
    day = {17},
    eprint = {1012.4019},
    keywords = {elliptic\_curve, isogenies, quantum},
    month = dec,
    posted-at = {2011-03-09 14:55:54},
    priority = {5},
    title = {Constructing elliptic curve isogenies in quantum subexponential time},
    url = {http://arxiv.org/abs/1012.4019},
    year = {2010}
}

@article{jao+miller+venkatesan09,
    abstract = {We present a construction of expander graphs obtained from Cayley graphs of narrow ray class groups, whose eigenvalue bounds follow from the Generalized Riemann Hypothesis. Our result implies that the Cayley graph of with respect to small prime generators is an expander. As another application, we show that the graph of small prime degree isogenies between ordinary elliptic curves achieves nonnegligible eigenvalue separation, and explain the relationship between the expansion properties of these graphs and the security of the elliptic curve discrete logarithm problem. For a video summary of this paper, please visit {http://www.youtube.com/watch?v=7jwxmKWWsyM}.},
    author = {Jao, David and Miller, Stephen D. and Venkatesan, Ramarathnam},
    citeulike-article-id = {8969286},
    citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.jnt.2008.11.006},
    doi = {10.1016/j.jnt.2008.11.006},
    issn = {0022314X},
    journal = {Journal of Number Theory},
    keywords = {elliptic\_curve, isogenies},
    month = jun,
    number = {6},
    pages = {1491--1504},
    posted-at = {2011-03-09 13:26:40},
    priority = {5},
    title = {Expander graphs based on {GRH} with an application to elliptic curve cryptography},
    url = {http://dx.doi.org/10.1016/j.jnt.2008.11.006},
    volume = {129},
    year = {2009}
}

@inproceedings{jao+soukharev10,
    abstract = {An isogeny between elliptic curves is an algebraic morphism which is a group homomorphism. Many applications in cryptography require evaluating large degree isogenies between elliptic curves efficiently. For ordinary curves of the same endomorphism ring, the previous best known algorithm has a worst case running time which is exponential in the length of the input. In this paper we show this problem can be solved in subexponential time under reasonable heuristics. Our approach is based on factoring the ideal corresponding to the kernel of the isogeny, modulo principal ideals, into a product of smaller prime ideals for which the isogenies can be computed directly. Combined with previous work of Bostan et al., our algorithm yields equations for large degree isogenies in quasi-optimal time given only the starting curve and the kernel.},
    address = {Berlin, Heidelberg},
    author = {Jao, David and Soukharev, Vladimir},
    booktitle = {ANTS IX: Proceedings of the Algorithmic Number Theory 9th International Symposium},
    chapter = {19},
    citeulike-article-id = {8969249},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-642-14518-6_19},
    citeulike-linkout-1 = {http://www.springerlink.com/content/726480228q711123},
    doi = {10.1007/978-3-642-14518-6_19},
    isbn = {978-3-642-14517-9},
    keywords = {cryptography, elliptic\_curve, isogenies, isogeny\_cycles},
    pages = {219--233},
    posted-at = {2011-03-09 13:21:33},
    priority = {5},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {A Subexponential Algorithm for Evaluating Large Degree Isogenies},
    url = {http://dx.doi.org/10.1007/978-3-642-14518-6_19},
    volume = {6197},
    year = {2010}
}

@article{vzgathen+shoup92:journal,
    abstract = {. A new probabilistic algorithm for factoring univariate polynomials over finite fields is presented. To factor a polynomial of degree  n over F q , the number of arithmetic operations in F q is O((n  2  +n log q) \Delta  (log n)  2  loglog n). The main technical innovation is a new way to compute Frobenius and trace maps in the ring of polynomials modulo the polynomial to be factored.  Subject classifications. {68Q40}; {11Y16}, {12Y05}. 1. Introduction We consider the problem of factoring a univariate polynomial over a finite field. This problem plays a central role in computational algebra. Indeed, many of the efficient algorithms for factoring univariate and multivariate polynomials over finite fields, the field of rational numbers, and finite extensions of the rationals solve as a subproblem the problem of factoring univariate polynomials over finite fields (Kaltofen 1990). This problem also has important applications in number theory (Buchmann 1990), coding theory (Berlekamp 1968), and ...},
    author = {von zur Gathen, Joachim and Shoup, Victor},
    citeulike-article-id = {8907398},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.9544},
    journal = {Computational Complexity},
    keywords = {computer\_algebra, finite\_field, polynomial\_factorization, polynomials\_over\_finite\_fields, probabilistic\_algorithm, transposed\_algorithms, transposition},
    pages = {187--224},
    posted-at = {2011-03-01 18:41:52},
    priority = {0},
    title = {Computing {F}robenius Maps And Factoring Polynomials},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.9544},
    volume = {2},
    year = {1992}
}

@unpublished{citeulike:8881285,
    abstract = {In this paper, a linear univariate representation for the roots of a
zero-dimensional polynomial equation system is presented, where the roots of
the equation system are represented as linear combinations of roots of several
univariate polynomial equations. The main advantage of this representation is
that the precision of the roots can be easily controlled. In fact, based on the
linear univariate representation, we can give the exact precisions needed for
roots of the univariate equations in order to obtain the roots of the equation
system to a given precision. As a consequence, a root isolation algorithm for a
zero-dimensional polynomial equation system can be easily derived from its
linear univariate representation.},
    archivePrefix = {arXiv},
    author = {Cheng, Jin-San and Gao, Xiao-Shan and Guo, Leilei},
    citeulike-article-id = {8881285},
    citeulike-linkout-0 = {http://arxiv.org/abs/1102.4681},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1102.4681},
    day = {23},
    eprint = {1102.4681},
    keywords = {trace\_formulas},
    month = feb,
    posted-at = {2011-02-24 16:12:44},
    priority = {3},
    title = {Root Isolation of Zero-dimensional Polynomial Systems with Linear Univariate Representation},
    url = {http://arxiv.org/abs/1102.4681},
    year = {2011}
}

@unpublished{beecken+mittman+saxena11,
    abstract = {Algebraic independence is an advanced notion in commutative algebra that
generalizes independence of linear polynomials to higher degree. Polynomials
{f\_1, ..., f\_m} \subset \F[x\_1, ..., x\_n] are called algebraically independent
if there is no non-zero polynomial F such that F(f\_1, ..., f\_m) = 0. The
transcendence degree, trdeg{f\_1, ..., f\_m}, is the maximal number r of
algebraically independent polynomials in the set. In this paper we design
blackbox and efficient linear maps \phi that reduce the number of variables
from n to r but maintain trdeg{\phi(f\_i)}\_i = r, assuming f\_i's sparse and
small r. We apply these fundamental maps to solve several cases of blackbox
identity testing:


(1) Given a polynomial-degree circuit C and sparse polynomials f\_1, ..., f\_m
with trdeg r, we can test blackbox D := C(f\_1, ..., f\_m) for zeroness in
poly(size(D))^r time.


(2) Define a spsp\_\delta(k,s,n) circuit C to be of the form \sum\_{i=1}^k
\prod\_{j=1}^s f\_{i,j}, where f\_{i,j} are sparse n-variate polynomials of degree
at most \delta. For k = 2 we give a poly(sn\delta)^{\delta^2} time blackbox
identity test.


(3) For a general depth-4 circuit we define a notion of rank. Assuming there
is a rank bound R for minimal simple spsp\_\delta(k,s,n) identities, we give a
poly(snR\delta)^{Rk\delta^2} time blackbox identity test for spsp\_\delta(k,s,n)
circuits. This partially generalizes the state of the art of depth-3 to depth-4
circuits.


The notion of trdeg works best with large or zero characteristic, but we also
give versions of our results for arbitrary fields.},
    archivePrefix = {arXiv},
    author = {Beecken, Malte and Mittmann, Johannes and Saxena, Nitin},
    citeulike-article-id = {8823306},
    citeulike-linkout-0 = {http://arxiv.org/abs/1102.2789},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1102.2789},
    day = {14},
    eprint = {1102.2789},
    keywords = {black\_box, circuit\_complexity},
    month = feb,
    posted-at = {2011-02-15 14:52:16},
    priority = {3},
    title = {Algebraic Independence and Blackbox Identity Testing},
    url = {http://arxiv.org/abs/1102.2789},
    year = {2011}
}

@unpublished{streng-igusa,
    abstract = {We give an algorithm that computes the genus-two class polynomials of a
primitive quartic {CM}-field K, and we give a running time bound and a proof of
correctness of this algorithm. This is the first proof of correctness and the
first running time bound of any algorithm that computes these polynomials. Our
algorithm is based on the complex analytic method of Spallek and van Wamelen
and runs in time {Otilde(Delta}^(7/2)), where Delta is the discriminant of K.},
    archivePrefix = {arXiv},
    author = {Streng, Marco},
    citeulike-article-id = {8798895},
    citeulike-linkout-0 = {http://arxiv.org/abs/0903.4766},
    citeulike-linkout-1 = {http://arxiv.org/pdf/0903.4766},
    day = {9},
    eprint = {0903.4766},
    keywords = {elliptic\_curve, genus\_2, igusa},
    month = feb,
    posted-at = {2011-02-10 12:09:59},
    priority = {2},
    title = {Computing {I}gusa class polynomials},
    url = {http://arxiv.org/abs/0903.4766},
    year = {2011}
}

@unpublished{ritzenthaler11,
    abstract = {In this survey, we discuss the problem of the maximum number of points of
curves of genus 1,2 and 3 over finite fields},
    archivePrefix = {arXiv},
    author = {Ritzenthaler, Christophe},
    citeulike-article-id = {8737261},
    citeulike-linkout-0 = {http://arxiv.org/abs/1101.5871},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1101.5871},
    day = {31},
    eprint = {1101.5871},
    keywords = {algebraic\_curves, algebraic\_geometry, coding\_theory, genus\_2, hyperelliptic\_curves},
    month = jan,
    posted-at = {2011-02-01 12:44:32},
    priority = {4},
    title = {Optimal curves of genus 1,2 and 3},
    url = {http://arxiv.org/abs/1101.5871},
    year = {2011}
}

@unpublished{castryck+folsom+hubrechts+suthreland11,
    abstract = {In 2000, Galbraith and {McKee} heuristically derived a formula that estimates
the probability that a randomly chosen elliptic curve over a fixed finite prime
field has a prime number of rational points. We show how their heuristics can
be generalized to Jacobians of curves of higher genus. We then elaborate this
in genus 2 and study various related issues, such as the probability of
cyclicity and the probability of primality of the number of points on the curve
itself. Finally, we discuss the asymptotic behavior as the genus tends to
infinity.},
    archivePrefix = {arXiv},
    author = {Castryck, Wouter and Folsom, Amanda and Hubrechts, Hendrik and Sutherland, Andrew V.},
    citeulike-article-id = {8693359},
    citeulike-linkout-0 = {http://arxiv.org/abs/1101.4792},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1101.4792},
    day = {25},
    eprint = {1101.4792},
    keywords = {cryptography, genus\_2, hyperelliptic\_curves},
    month = jan,
    posted-at = {2011-01-26 13:04:31},
    priority = {3},
    title = {The probability that the number of points on the {J}acobian of a genus 2 curve is prime},
    url = {http://arxiv.org/abs/1101.4792},
    year = {2011}
}

@unpublished{bisson11,
    abstract = {We design a probabilistic algorithm for computing endomorphism rings of
ordinary elliptic curves defined over finite fields that we prove has a
subexponential runtime in the size of the base field, assuming solely the
generalized Riemann hypothesis.


Additionally, we improve the asymptotic complexity of previously known,
heuristic, subexponential methods by describing a faster isogeny-computing
routine.},
    archivePrefix = {arXiv},
    author = {Bisson, Gaetan},
    citeulike-article-id = {8690741},
    citeulike-linkout-0 = {http://arxiv.org/abs/1101.4323},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1101.4323},
    day = {22},
    eprint = {1101.4323},
    keywords = {elliptic\_curve, isogenies, isogeny\_cycles},
    month = jan,
    posted-at = {2011-01-25 14:18:11},
    priority = {5},
    title = {Computing endomorphism rings of elliptic curves under the {GRH}},
    url = {http://arxiv.org/abs/1101.4323},
    year = {2011}
}

@unpublished{strzebonski+tsigaridas11,
    abstract = {We present algorithmic, complexity and implementation results for the problem
of isolating the real roots of a univariate polynomial in \$B\_{\alpha} \in
L[y]\$, where \$L=\QQ(\alpha)\$ is a simple algebraic extension of the rational
numbers. We consider two approaches for tackling the problem. In the first
approach using resultant computations we perform a reduction to a polynomial
with integer coefficients. We compute separation bounds for the roots, and
using them we deduce that we can isolate the real roots of \$B\_{\alpha}\$ in
\$\sOB(N^{10})\$, where \$N\$ is an upper bound on all the quantities (degree and
bitsize) of the input polynomials. In the second approach we isolate the real
roots working directly on the polynomial of the input. We compute improved
separation bounds for real roots and we prove that they are optimal, under mild
assumptions. For isolating the roots we consider a modified Sturm's algorithm,
and a modified version of \func{descartes}' algorithm introduced by Sagraloff.
For the former we prove a complexity bound of \$\sOB(N^8)\$ and for the latter a
bound of \$\sOB(N^{7})\$. We implemented the algorithms in \func{C} as part of
the core library of \mathematica and we illustrate their efficiency over
various data sets. Finally, we present complexity results for the general case
of the first approach, where the coefficients belong to multiple extensions.},
    archivePrefix = {arXiv},
    author = {Strzebonski, Adam and Tsigaridas, Elias},
    citeulike-article-id = {8690740},
    citeulike-linkout-0 = {http://arxiv.org/abs/1101.4369},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1101.4369},
    day = {23},
    eprint = {1101.4369},
    keywords = {computer\_algebra, polynomial\_factorization},
    month = jan,
    posted-at = {2011-01-25 14:16:56},
    priority = {4},
    title = {Univariate real root isolation in an extension field},
    url = {http://arxiv.org/abs/1101.4369},
    year = {2011}
}

@unpublished{vogel97,
    abstract = {There exists a graded Z-algebra acting in a natural way on many modules of 3-valent diagrams. Every simple Lie superalgebra with a nontrivial invariant bilinear form induces a character on . Classical and exceptional Lie algebras and the Lie superalgebra D(2; 1; ff) produce eight distinct characters on and eight distinct families of weight functions on chord diagrams. As a consequence we prove that weight functions coming from semisimple Lie superalgebras do not detect every element in the module A of chord diagrams.},
    author = {Vogel, Pierre},
    citeulike-article-id = {8632669},
    citeulike-linkout-0 = {http://www.math.jussieu.fr/~vogel/diagrams.pdf},
    citeulike-linkout-1 = {http://www.math.jussieu.fr/~vogel/},
    citeulike-linkout-2 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.5848},
    journal = {UNIVERSIT E PARIS VII},
    keywords = {circuit\_complexity, transposition},
    posted-at = {2011-01-17 17:02:18},
    priority = {2},
    title = {Algebraic Structures on Modules of Diagrams},
    url = {http://www.math.jussieu.fr/~vogel/diagrams.pdf},
    year = {1997}
}

@article{moenck76,
    abstract = {The current proposals for applying the so called  fast  {O(N} {logaN}) algorithms to multivariate polynomials is that the univariate methods be applied recursively, much in the way more conventional algorithms are used. Since the size of the problems is rather large for which a  fastrd algorithm is more efficient than a classical one, the recursive approach compounds this size completely out of any practical range .},
    author = {Moenck, Robert T.},
    citeulike-article-id = {8598493},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/BF00268498},
    citeulike-linkout-1 = {http://www.springerlink.com/content/p0062t11v5606838},
    day = {1},
    doi = {10.1007/BF00268498},
    issn = {0001-5903},
    journal = {Acta Informatica},
    keywords = {multivariate\_polynomial, polynomial\_multiplication},
    month = jun,
    number = {2},
    pages = {153--169},
    posted-at = {2011-01-13 17:55:47},
    priority = {1},
    publisher = {Springer Berlin / Heidelberg},
    title = {Another polynomial homomorphism},
    url = {http://dx.doi.org/10.1007/BF00268498},
    volume = {6},
    year = {1976}
}

@incollection{carette+sexton+sorge+watt10,
    abstract = {Decomposing the domain of a function into parts has many uses in mathematics. A domain may naturally be a union of pieces, a function may be defined by cases, or different boundary conditions may hold on different regions. For any particular problem the domain can be given explicitly, but when dealing with a family of problems given in terms of symbolic parameters, matters become more difficult. This article shows how hybrid sets, that is multisets allowing negative multiplicity, may be used to express symbolic domain decompositions in an efficient, elegant and uniform way, simplifying both computation and reasoning. We apply this theory to the arithmetic of piecewise functions and symbolic matrices and show how certain operations may be reduced from exponential to linear complexity.},
    address = {Berlin, Heidelberg},
    author = {Carette, Jacques and Sexton, Alan and Sorge, Volker and Watt, Stephen},
    booktitle = {Intelligent Computer Mathematics},
    chapter = {16},
    citeulike-article-id = {8511511},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-642-14128-7_16},
    citeulike-linkout-1 = {http://www.springerlink.com/content/p8231723028t7v5j},
    doi = {10.1007/978-3-642-14128-7_16},
    editor = {Autexier, Serge and Calmet, Jacques and Delahaye, David and Ion, Patrick and Rideau, Laurence and Rioboo, Renaud and Sexton, Alan},
    isbn = {978-3-642-14127-0},
    keywords = {computer\_algebra, functional\_language},
    pages = {172--188},
    posted-at = {2011-01-06 13:05:22},
    priority = {4},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Symbolic Domain Decomposition},
    url = {http://dx.doi.org/10.1007/978-3-642-14128-7_16},
    volume = {6167},
    year = {2010}
}

@inproceedings{carette07,
    abstract = {We define a canonical form for piecewise defined functions. We show that the domains and ranges for which these functions are defined is larger than in previous work. Also, our canonical form algorithm is linear in the number of breakpoints instead of exponential. These results rely on the linear structure of the underlying domain of definition.},
    address = {New York, NY, USA},
    author = {Carette, Jacques},
    booktitle = {Proceedings of the 2007 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {8511507},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1277560},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1277548.1277560},
    doi = {10.1145/1277548.1277560},
    isbn = {978-1-59593-743-8},
    keywords = {computer\_algebra, functional\_language},
    location = {Waterloo, Ontario, Canada},
    pages = {77--84},
    posted-at = {2011-01-06 13:03:57},
    priority = {4},
    publisher = {ACM},
    series = {ISSAC '07},
    title = {A canonical form for piecewise defined functions},
    url = {http://dx.doi.org/10.1145/1277548.1277560},
    year = {2007}
}

@proceedings{MPC2006,
    booktitle = {Mathematics of Program Construction, {MPC 2006}},
    citeulike-article-id = {8511503},
    citeulike-linkout-0 = {http://link.springer.de/link/service/series/0558/tocs/t4014.htm},
    editor = {Uustalu, Tarmo},
    keywords = {functional\_language},
    posted-at = {2011-01-06 13:01:25},
    priority = {2},
    title = {Mathematics of Program Construction, {MPC 2006, Kuressaare, Estonia}},
    url = {http://link.springer.de/link/service/series/0558/tocs/t4014.htm},
    volume = {4014},
    year = {2006}
}

@inproceedings{Kahl-Carette-Ji-2006a,
    author = {Kahl, Wolfram and Carette, Jacques and Ji, Xiaoheng},
    booktitle = {Mathematics of Program Construction, {MPC 2006}},
    citeulike-article-id = {8511502},
    citeulike-linkout-0 = {http://link.springer.de/link/service/series/0558/tocs/t4014.htm},
    editor = {Uustalu, Tarmo},
    keywords = {functional\_language},
    pages = {253--273},
    posted-at = {2011-01-06 13:01:25},
    priority = {5},
    title = {Bimonadic Semantics for Basic Pattern Matching Calculi},
    url = {http://link.springer.de/link/service/series/0558/tocs/t4014.htm},
    volume = {4014},
    year = {2006}
}

@techreport{Kahl-Carette-Ji-2006b,
    abstract = {The pattern matching calculi introduced by the first author are a refinement of the \$\\lambda\$-calculus that integrates mechanisms appropriate for fine-grained modelling of non-strict pattern matching. While related work in the literature only uses a single monad, typically \\textsf{Maybe}, for matchings, we present an axiomatic approach to semantics of these pattern matching calculi using two monads, one for expressions and one for matchings. Although these two monads only need to be relatively lightly coupled, this semantics implies soundness of all core \$\\PMC\$ rules, and is a useful tool for exploration of the design space for pattern matching calculi. Using lifting and \\textsf{Maybe} monads, we obtain standard Haskell semantics, and by adding another level of \\textsf{Maybe} to both, we obtain a denotational semantics of the ``matching failure as exceptions'' approach of Erwig and Peyton Jones. Using list-like monads opens up interesting extensions in the direction of functional-logic programming. A short version of this report appears as \\cite{Kahl-Carette-Ji-2006a}.},
    author = {Kahl, Wolfram and Carette, Jacques and Ji, Xiaoheng},
    citeulike-article-id = {8511501},
    comment = {long version of \cite{Kahl-Carette-Ji-2006a}, available from \textsf{\url{http://sqrl.mcmaster.ca/sqrl\_reports.html}}},
    institution = {Software Quality Research Laboratory, McMaster Univ.},
    keywords = {functional\_language},
    month = jun,
    number = {33},
    pages = {32},
    posted-at = {2011-01-06 13:01:25},
    priority = {5},
    title = {Bimonadic Semantics for Basic Pattern Matching Calculi},
    type = {SQRL Report},
    year = {2006}
}

@phdthesis{df+thesis,
    abstract = {{D}ans cette th{\`{e}}se nous appliquons des techniques provenant du calcul formel et de la th{\'{e}}orie des langages afin d'am{\'{e}}liorer les op{\'{e}}rations {\'{e}}l{\'{e}}mentaires dans certaines tours de corps finis. {N}ous appliquons notre construction au probl{\`{e}}me du calcul d'isog{\'{e}}nies entre courbes elliptiques et obtenons une variante plus rapide ({\`{a}} la fois en th{\'{e}}orie et en pratique) de l'algorithme de {C}ouveignes. {L}e document est divis{\'{e}} en quatre parties. {D}ans la partie {I} nous faisons des rappels d'alg{\`{e}}bre et de th{\'{e}}orie de la complexit{\'{e}}. {L}a partie {II} traite du principe de transposition : nous g{\'{e}}n{\'{e}}ralisons des id{\'{e}}es de {B}ostan, {S}chost et {L}ecerf et nous montrons qu'il est possible de transposer automatiquement des programmes sans pertes en complexit{\'{e}}-temps et avec une petite perte en complexit{\'{e}}-espace. {L}a partie {III} combine les r{\'{e}}sultats sur le principe de transposition avec des techniques classiques en th{\'{e}}orie de l'{\'{e}}limination ; nous appliquons ces id{\'{e}}es pour obtenir des algorithmes asymptotiquement optimaux pour l'arithm{\'{e}}tique des tours d'{A}rtin-{S}chreier de corps finis. {N}ous d{\'{e}}crivons aussi une implantation de ces algorithmes. {E}nfin, dans la partie {IV} nous utilisons les r{\'{e}}sultats pr{\'{e}}c{\'{e}}dents afin d'acc{\'{e}}l{\'{e}}rer l'algorithme de {C}ouveignes et de comparer le r{\'{e}}sultat avec les autres algorithmes pour le calcul d'isog{\'{e}}nies qui font l'{\'{e}}tat de l'art. {N}ous pr{\'{e}}sentons aussi une nouvelle g{\'{e}}n{\'{e}}ralisation de l'algorithme de {C}ouveignes qui calcule des isog{\'{e}}nies de degr{\'{e}} inconnu.},
    author = {De Feo, Luca},
    citeulike-article-id = {8428864},
    citeulike-linkout-0 = {http://tel.archives-ouvertes.fr/tel-00547034/en/},
    day = {13},
    keywords = {computer\_algebra, elliptic\_curve, finite\_field, isogenies, transposition},
    month = dec,
    posted-at = {2010-12-15 17:37:01},
    priority = {0},
    school = {Ecole Polytechnique X},
    title = {{A}lgorithmes {R}apides pour les {T}ours de {C}orps {F}inis et les {I}sog{\'{e}}nies},
    url = {http://tel.archives-ouvertes.fr/tel-00547034/en/},
    year = {2010}
}

@techreport{beda59,
    address = {Moscow, USSR},
    author = {Beda, L. M. and Korolev, L. N. and Sukkikh, N. V. and Frolova, T. S.},
    citeulike-article-id = {8405429},
    institution = {Institute for Precise Mechanics and Computation Techniques, Academy of Science},
    keywords = {automatic\_differentiation},
    posted-at = {2010-12-10 16:49:56},
    priority = {1},
    title = {Programs for automatic differentiation for the machine {BESM}},
    year = {1959}
}

@inproceedings{khodadad+monagan06,
    abstract = {Let F be a field and let f and g be polynomials in F[t] satisfying deg f > deg g. Recall that on input of f and g the extended Euclidean algorithm computes a sequence of polynomials (si, ti, ri) satisfying sif + tig = ri. Thus for i with gcd(ti, f) = 1, we obtain rational functions ri/ti ∈ F(t) satisfying ri/ti ≡ g (mod {f).In} this paper we modify the fast extended Euclidean algorithm to compute the smallest ri/ti, that is, an ri/ti minimizing deg ri + deg ti. This means that in an output sensitive modular algorithm when we are recovering rational functions in F(t) from their images modulo f(t) where f(t) is increasing in degree, we can recover them as soon as the degree of f is large enough and we can do this {fast.We} have implemented our modified fast Euclidean algorithm for F = Zp, p a word sized prime, in Java. Our fast algorithm beats the ordinary Euclidean algorithm around degree 200. This has application to polynomial gcd computation and linear algebra over Zp(t).},
    address = {New York, NY, USA},
    author = {Khodadad, Sara and Monagan, Michael},
    booktitle = {Proceedings of the 2006 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {8385740},
    citeulike-linkout-0 = {http://www.cecm.sfuhttp://www.cecm.sfu.ca/~monaganm/papers/fastRFR.pdf.ca/~monaganm/papers/fastRFR.pdf},
    citeulike-linkout-1 = {http://portal.acm.org/citation.cfm?id=1145801},
    citeulike-linkout-2 = {http://dx.doi.org/10.1145/1145768.1145801},
    doi = {10.1145/1145768.1145801},
    isbn = {1-59593-276-3},
    keywords = {computer\_algebra},
    location = {Genoa, Italy},
    pages = {184--190},
    posted-at = {2010-12-08 16:29:13},
    priority = {0},
    publisher = {ACM},
    series = {ISSAC '06},
    title = {Fast rational function reconstruction},
    url = {http://www.cecm.sfuhttp://www.cecm.sfu.ca/~monaganm/papers/fastRFR.pdf.ca/~monaganm/papers/fastRFR.pdf},
    year = {2006}
}

@incollection{strak73,
    abstract = {Let E be an elliptic curve     \$\$y^2 = 4x^3 - g\_2 x - g\_3 ,\Delta = g\_2^3 - 27g\_3^2 \ne 0,\$\$     in Weierstrass normal form. The curve may be parametrized by the Weierstrass ℘-function, x = ℘(z), y = ℘′(z). The function ℘(z) is a doubly periodic function whose periods form a lattice     \$\$\Lambda = \{ \omega \_1 ,\omega \_2 \} = \{ a\omega \_1 + b\omega \_2 |a,b \in \mathbb{Z}\} \$\$     where ω1/ω2 ∉ ℝ and for convenience we assume that ω1 and ω2 are so ordered that Im(ω1/ω2) > 0. We then have the relations     \$\$g\_2 = 60\sum\limits\_\omega {'\omega ^{ - 4} } ,g\_3 = 140\sum\limits\_\omega {'\omega ^{ - 6} } ,\$\$  (1)    where the summations are over all ω ∈ Λ other than ω = 0, and     \$\$\wp (z) = \frac{1}{{z^2 }} + \frac{{g\_2 }}{{20}}z^2 + \frac{{g\_3 }}{{28}}z^4 + \frac{{g\_2^2 }}{{1200}}z^6 + ....\$\$  (2)   .},
    address = {Berlin, Heidelberg},
    author = {Stark, Harold M.},
    booktitle = {Modular Functions of One Variable I},
    chapter = {5},
    citeulike-article-id = {8385519},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-38509-7_5},
    citeulike-linkout-1 = {http://www.springerlink.com/content/lm33275055606310},
    doi = {10.1007/978-3-540-38509-7_5},
    editor = {Kuijk, Willem},
    isbn = {978-3-540-06219-6},
    issn = {0075-8434},
    keywords = {elliptic\_curve, isogenies},
    pages = {153--174},
    posted-at = {2010-12-08 15:59:29},
    priority = {2},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Mathematics},
    title = {{Class-Numbers} of Complex Quadratic Fields},
    url = {http://dx.doi.org/10.1007/978-3-540-38509-7_5},
    volume = {320},
    year = {1973}
}

@article{ostrowski+al71,
    author = {Ostrowski, G. M. and Wolin, J. M. and Borisow, W. W.},
    citeulike-article-id = {8334663},
    journal = {Wissenschaftliche Zeitschrift der Technischen Hochschule f\"{u}r Chemie, Leuna-Merseburg},
    keywords = {automatic\_differentiation},
    number = {4},
    pages = {382--384},
    posted-at = {2010-11-30 21:35:00},
    priority = {2},
    title = {{\"{U}}ber die {B}erechnung von {A}bleitungen},
    volume = {13},
    year = {1971}
}

@book{griewank2008evaluating,
    author = {Griewank, Andreas and Walther, Andrea},
    citeulike-article-id = {8334638},
    edition = {Second},
    keywords = {automatic\_differentiation},
    posted-at = {2010-11-30 21:28:13},
    priority = {3},
    publisher = {Society for Industrial and Applied Mathematics (SIAM)},
    title = {Evaluating derivatives: principles and techniques of algorithmic differentiation},
    year = {2008}
}

@article{kaltofen87,
    address = {Palo Alto, California},
    author = {Kaltofen, Erich},
    citeulike-article-id = {8334422},
    citeulike-linkout-0 = {http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_annrev.pdf},
    journal = {Annual Review in Computer Science},
    keywords = {computer\_algebra},
    pages = {91--118},
    posted-at = {2010-11-30 20:48:13},
    priority = {2},
    publisher = {Annual Reviews Inc.},
    title = {Computer algebra algorithms},
    url = {http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_annrev.pdf},
    volume = {2},
    year = {1987}
}

@phdthesis{kohel,
    author = {Kohel, David},
    citeulike-article-id = {8290577},
    keywords = {elliptic\_curve, isogenies},
    posted-at = {2010-11-21 22:45:34},
    priority = {3},
    school = {University of California at Berkley},
    title = {Endomorphism rings of elliptic curves over finite fields},
    year = {1996}
}

@techreport{nist07-2,
    author = {Barker, Elaine and Barker, William and Burr, William and Polk, William and Smid, Miles},
    citeulike-article-id = {8266883},
    citeulike-linkout-0 = {http://csrc.nist.gov/publications/nistpubs/800-57/sp800-57-Part1-revised2_Mar08-2007.pdf},
    howpublished = {NIST Special Publication 800-57},
    keywords = {cryptography, elliptic\_curve},
    month = mar,
    posted-at = {2010-11-17 14:11:39},
    priority = {2},
    title = {Recommendation for Key Management},
    url = {http://csrc.nist.gov/publications/nistpubs/800-57/sp800-57-Part1-revised2_Mar08-2007.pdf},
    year = {2007}
}

@techreport{nist07,
    author = {Barker, Elaine and Johnson, Don and Smid, Miles},
    citeulike-article-id = {8255046},
    citeulike-linkout-0 = {http://csrc.nist.gov/publications/nistpubs/800-56A/SP800-56A_Revision1_Mar08-2007.pdf},
    howpublished = {NIST Special Publication 800-56A},
    keywords = {cryptography, elliptic\_curve},
    month = mar,
    posted-at = {2010-11-16 16:38:06},
    priority = {2},
    title = {Recommendations for {Pair-Wise} Key Establishment Schemes Using Discrete Logarithm Cryptography},
    url = {http://csrc.nist.gov/publications/nistpubs/800-56A/SP800-56A_Revision1_Mar08-2007.pdf},
    year = {2007}
}

@inproceedings{gallant+lambert+vanstone01,
    abstract = {The fundamental operation in elliptic curve cryptographic schemes is that of point multiplication of an elliptic curve point by an integer. This paper describes a new method for accelerating this operation on classes of elliptic curves that have efficiently-computable endomorphisms. One advantage of the new method is that it is applicable to a larger class of curves than previous such methods.},
    address = {London, UK},
    author = {Gallant, Robert P. and Lambert, Robert J. and Vanstone, Scott A.},
    booktitle = {CRYPTO '01: Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology},
    citeulike-article-id = {8119380},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.2004},
    isbn = {3-540-42456-3},
    keywords = {elliptic\_curve, isogenies},
    pages = {190--200},
    posted-at = {2010-10-24 20:48:47},
    priority = {1},
    publisher = {Springer-Verlag},
    title = {Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.2004},
    year = {2001}
}

@article{shum-et-al01,
    abstract = {Since the proof in 1982, by Tsfasman Vladut and Zink of theexistence of algebraic-geometric ({AG}) codes with asymptotic performanceexceeding the {Gilbert-Varshamov} ({G-V}) bound, one of the challenges incoding theory has been to provide explicit constructions for thesecodes. In a major step forward during 1995-1996, Garcia and {Stichtenoth(GS}) provided an explicit description of algebraic curves, such that {AGcodes} constructed on them would have a performance better than the {G-Vbound}. We present the first low-complexity algorithm for obtaining thegenerator matrix for {AG} codes on the curves of {GS}. The symbol alphabetof the {AG} code is the finite field of q 2 ,q 2 \&ges;49, elements. The complexity of the algorithm, asmeasured in terms of multiplications and divisions over the finite {fieldGF}(q 2 ), is upper-bounded by [Nlog q (N)] 3 where N is the length of the code. An example of code construction usingthe above algorithm is presented. By concatenating the {AG} code withshort binary block codes, it is possible to obtain binary codes withasymptotic performance close to the {G-V} bound. Some examples of suchconcatenation are included},
    author = {Shum, Kenneth W. and Aleshnikov, Ilia and Kumar, P. Vijay and Stichtenoth, Hennning and Deolalikar, Vinay},
    citeulike-article-id = {493513},
    citeulike-linkout-0 = {http://dx.doi.org/10.1109/18.945244},
    citeulike-linkout-1 = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=945244},
    doi = {10.1109/18.945244},
    issn = {00189448},
    journal = {IEEE Transactions on Information Theory},
    keywords = {algebraic\_curves, algebraic\_geometry, artin-schreier, coding\_theory},
    month = sep,
    number = {6},
    pages = {2225--2241},
    posted-at = {2010-09-16 19:39:26},
    priority = {2},
    title = {A low-complexity algorithm for the construction of algebraic-geometric codes better than the {G}ilbert-{V}arshamov bound},
    url = {http://dx.doi.org/10.1109/18.945244},
    volume = {47},
    year = {2001}
}

@inproceedings{enge+sutherland10,
    author = {Enge, Andreas and Sutherland, Andrew V.},
    booktitle = {ANTS IX: Proceedings of the Algorithmic Number Theory 9th International Symposium},
    citeulike-article-id = {7843054},
    keywords = {elliptic\_curve, isogenies},
    location = {Nancy, France},
    pages = {142--156},
    posted-at = {2010-09-15 14:38:28},
    priority = {2},
    publisher = {Springer--Verlag},
    series = {Lecture Notes in Computer Science},
    title = {Class invariants by the {CRT} method},
    volume = {6197},
    year = {2010}
}

@article{morain07,
    author = {Morain, Fran\c{c}ois},
    citeulike-article-id = {7817531},
    citeulike-linkout-0 = {http://www.lix.polytechnique.fr/Labo/Francois.Morain/Articles/fastecpp-final.pdf},
    journal = {Math. Comp.},
    keywords = {elliptic\_curve, elliptic\_curve\_method},
    pages = {493--505},
    posted-at = {2010-09-13 16:37:38},
    priority = {2},
    title = {Implementing the asymptotically fast version of the elliptic curve primality proving algorithm},
    url = {http://www.lix.polytechnique.fr/Labo/Francois.Morain/Articles/fastecpp-final.pdf},
    volume = {76},
    year = {2007}
}

@article{harvey07,
    author = {Harvey, David},
    citeulike-article-id = {7802544},
    journal = {International Mathematics Research Notices},
    keywords = {algebraic\_curves, p-adic},
    number = {rnm095},
    posted-at = {2010-09-09 13:29:16},
    priority = {2},
    title = {{K}edlaya's Algorithm in Larger Characteristic},
    volume = {2007},
    year = {2007}
}

@article{sutherland10:hilbert,
    author = {Sutherland, Andrew V.},
    citeulike-article-id = {7801161},
    journal = {Math. Comp.},
    keywords = {elliptic\_curve, isogenies, isogeny\_cycles},
    posted-at = {2010-09-08 17:02:06},
    priority = {3},
    publisher = {AMS},
    title = {Computing {H}ilbert class polynomials with the {C}hinese remainder theorem},
    year = {2010}
}

@article{schonhage77,
    abstract = {Polynomial multiplication of degree N can be accomplished in time O (N \^{A}· log N) provided the scalar field contains suitable roots of unity. Otherwise at least O (N \^{A}· log N \^{A}· log log N) is obtained by a modified version of the {Sch\~{A}{\P}nhage-Strassen} multiplication which employs computations modulo 1 + xn (where N = 2n), if the field contains 2-1, or modulo 1 + xn + {x2N}, if 3-1 exists. The latter method covering all fields of characteristic 2 is presented here in detail.},
    author = {Sch{\"{o}}nhage, Arnold},
    citeulike-article-id = {7791194},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/BF00289470},
    citeulike-linkout-1 = {http://www.springerlink.com/content/jt613564113l2013},
    day = {1},
    doi = {10.1007/BF00289470},
    issn = {0001-5903},
    journal = {Acta Informatica},
    keywords = {multiplication, polynomial\_multiplication},
    month = dec,
    number = {4},
    pages = {395--398},
    posted-at = {2010-09-06 16:44:27},
    priority = {1},
    publisher = {Springer Berlin / Heidelberg},
    title = {Schnelle {M}ultiplikation von {P}olynomen {\"u}ber {K}{\"o}rpern der {C}harakteristik 2},
    url = {http://dx.doi.org/10.1007/BF00289470},
    volume = {7},
    year = {1977}
}

@article{cantor+kaltofen91,
    author = {Cantor, David G. and Kaltofen, Erich},
    citeulike-article-id = {7791190},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/BF01178683},
    citeulike-linkout-1 = {http://www.springerlink.com/content/w0tj813x126u0h67},
    day = {1},
    doi = {10.1007/BF01178683},
    issn = {0001-5903},
    journal = {Acta Informatica},
    keywords = {multiplication, polynomial\_multiplication},
    location = {Berlin / Heidelberg},
    month = jul,
    number = {7},
    pages = {693--701},
    posted-at = {2010-09-06 16:39:38},
    priority = {1},
    publisher = {Springer},
    title = {On fast multiplication of polynomials over arbitrary algebras},
    url = {http://dx.doi.org/10.1007/BF01178683},
    volume = {28},
    year = {1991}
}

@electronic{rostovtsev+stolbunov06,
    abstract = {A new general mathematical problem, suitable for public-key cryptosystems, is proposed: morphism computation in a category of Abelian groups. In connection with elliptic curves over finite fields, the problem becomes the following: compute an isogeny (an algebraic homomorphism) between the elliptic curves given. The problem seems to be hard for solving with a quantum computer. {ElGamal} public-key encryption and {Diffie-Hellman} key agreement are proposed for an isogeny cryptosystem. The paper describes theoretical background and a public-key encryption technique, followed by security analysis and consideration of cryptosystem parameters selection. A demonstrative example of encryption is included as well.},
    author = {Rostovtsev, Alexander and Stolbunov, Anton},
    citeulike-article-id = {7764930},
    citeulike-linkout-0 = {http://eprint.iacr.org/2006/145},
    day = {13},
    howpublished = {Cryptology ePrint Archive, Report 2006/145},
    keywords = {cryptography, isogenies},
    month = apr,
    posted-at = {2010-09-03 16:42:56},
    priority = {0},
    title = {Public-key Cryptosystem Based On Isogenies},
    url = {http://eprint.iacr.org/2006/145},
    year = {2006}
}

@article{smith09,
    abstract = {We describe the use of explicit isogenies to translate instances of the Discrete Logarithm Problem\^{A} ({DLP}) from Jacobians of hyperelliptic genus\^{A} 3 curves to Jacobians of non-hyperelliptic genus\^{A} 3 curves, where they are vulnerable to faster index calculus attacks. We provide explicit formulae for isogenies with kernel isomorphic to\^{A} (\^{a}¤/2\^{a}¤)3 (over an algebraic closure of the base field) for any hyperelliptic genus\^{A} 3 curve over a field of characteristic not\^{A} 2 or\^{A} 3. These isogenies are rational for a positive fraction of all hyperelliptic genus\^{A} 3 curves defined over a finite field of characteristic\^{A} p>3. Subject to reasonable assumptions, our constructions give an explicit and efficient reduction of instances of the {DLP} from hyperelliptic to non-hyperelliptic Jacobians for around\^{A} 18.57\% of all hyperelliptic genus\^{A} 3 curves over a given finite field. We conclude with a discussion on extending these ideas to isogenies with more general kernels.},
    author = {Smith, Benjamin},
    citeulike-article-id = {4314077},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s00145-009-9038-1},
    citeulike-linkout-1 = {http://www.springerlink.com/content/u66m211065821nm6},
    day = {1},
    doi = {10.1007/s00145-009-9038-1},
    issn = {0933-2790},
    journal = {Journal of Cryptology},
    keywords = {hyperelliptic\_curves, isogenies},
    month = oct,
    number = {4},
    pages = {505-529--529},
    posted-at = {2010-09-01 17:05:19},
    priority = {2},
    publisher = {Springer New York},
    title = {Isogenies and the Discrete Logarithm Problem in {J}acobians of Genus 3 Hyperelliptic Curves},
    url = {http://dx.doi.org/10.1007/s00145-009-9038-1},
    volume = {22},
    year = {2009}
}

@article{gaudry+hess+smart02,
    abstract = {In this paper we look in detail at the curves which arise in the method of Galbraith and Smart for producing curves in the Weil restriction of an elliptic curve over a finite field of characteristic 2 of composite degree. We explain how this method can be used to construct hyperelliptic cryptosystems which could be as secure as cryptosystems based on the original elliptic curve. On the other hand, we show that the same technique may provide a way of attacking the original elliptic curve cryptosystem using recent advances in the study of the discrete logarithm problem on hyperelliptic curves.  We examine the resulting higher genus curves in some detail and propose an additional check on elliptic curve systems defined over fields of characteristic 2 so as to make them immune from the methods in this paper.},
    author = {Gaudry, Pierrick and Hess, Florian and Smart, Niegel},
    citeulike-article-id = {7751788},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s00145-001-0011-x},
    citeulike-linkout-1 = {http://www.springerlink.com/content/hx8b621p8p4417qv},
    day = {1},
    doi = {10.1007/s00145-001-0011-x},
    issn = {0933-2790},
    journal = {Journal of Cryptology},
    keywords = {cryptography, ghs, hyperelliptic\_curves, weil\_descent},
    month = mar,
    number = {1},
    pages = {19-46--46},
    posted-at = {2010-09-01 15:45:05},
    priority = {3},
    publisher = {Springer New York},
    title = {Constructive and destructive facets of {W}eil descent on elliptic curves},
    url = {http://dx.doi.org/10.1007/s00145-001-0011-x},
    volume = {15},
    year = {2002}
}

@electronic{jones+kiselyov:advanced+overlap08,
    author = {Kiselyov, Oleg and Peyton-Jones, Simon},
    citeulike-article-id = {7750329},
    howpublished = {Last accessed 31-Aug-2010 \url{http://www.haskell.org/haskellwiki/GHC/AdvancedOverlap}},
    keywords = {functional\_language, type\_classes},
    month = apr,
    posted-at = {2010-08-31 19:07:21},
    priority = {0},
    title = {{GHC}/{A}dvanced Overlap},
    year = {2008}
}

@inproceedings{liu+hudak10,
    abstract = {We study a number of embedded {DSLs} for autonomous ordinary differential equations (autonomous {ODEs}) in Haskell. A naive implementation based on the lazy tower of derivatives is straightforward but has serious time and space leaks due to the loss of sharing when handling cyclic and infinite data structures. In seeking a solution to fix this problem, we explore a number of {DSLs} ranging from shallow to deep embeddings, and middle-grounds in between. We advocate a solution based on arrows, an abstract notion of computation that offers both a succinct representation and an effective implementation. Arrows are ubiquitous in their combinator style that happens to capture both sharing and recursion elegantly. We further relate our arrow-based {DSL} to a more constrained form of arrows called causal commutative arrows, the normalization of which leads to a staged compilation technique improving {ODE} performance by orders of magnitude.},
    address = {Berlin, Heidelberg},
    author = {Liu, Hai and Hudak, Paul},
    booktitle = {Practical Aspects of Declarative Languages},
    chapter = {14},
    citeulike-article-id = {7750241},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-642-11503-5_14},
    citeulike-linkout-1 = {http://www.springerlink.com/content/d54k723xt2q81530},
    doi = {10.1007/978-3-642-11503-5_14},
    editor = {Carro, Manuel and Pe\~{n}a, Ricardo},
    isbn = {978-3-642-11502-8},
    keywords = {arrows, functional\_language},
    pages = {152-166--166},
    posted-at = {2010-08-31 17:50:28},
    priority = {0},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {An Ode to Arrows},
    url = {http://dx.doi.org/10.1007/978-3-642-11503-5_14},
    volume = {5937},
    year = {2010}
}

@incollection{pitts01,
    abstract = {This document provides an introduction to the interaction between category theory and mathematical logic which is slanted towards computer scientists.},
    address = {Oxford},
    author = {Pitts, Andrew M.},
    booktitle = {Handbook of Logic in Computer Science, Volume 5: Algebraic and Logical Structures},
    citeulike-article-id = {253615},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.9970},
    editor = {Abramsky, S. and Gabbay, D. M. and Maibaum, T. S. E.},
    keywords = {categories, functional\_language, functor},
    posted-at = {2010-08-31 13:12:41},
    priority = {3},
    publisher = {Clarendon Press},
    title = {Categorical Logic},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.9970},
    year = {2001}
}

@techreport{gilbert+levey+masse91,
    author = {{G}ilbert, {J}ean and {L}e {V}ey, {G}eorges and {M}asse, {J}ohn},
    citeulike-article-id = {7730580},
    citeulike-linkout-0 = {http://hal.inria.fr/inria-00075004/en/},
    comment = {{P}rojet {PROMATH}},
    institution = {INRIA},
    keywords = {automatic\_differentiation},
    number = {{RR}-1557},
    posted-at = {2010-08-29 12:09:11},
    priority = {2},
    title = {{L}a {D}ifferentiation automatique de fonctions representees par des programmes},
    type = {Research Report},
    url = {http://hal.inria.fr/inria-00075004/en/},
    year = {1991}
}

@article{kaltofen88:gcd,
    author = {Kaltofen, Erich},
    citeulike-article-id = {3084957},
    journal = {Journal of the Association for Computing Machinery},
    keywords = {black\_box, polynomials\_over\_finite\_fields},
    month = jan,
    number = {1},
    pages = {231--264},
    posted-at = {2010-08-28 23:55:48},
    priority = {2},
    title = {Greatest Common Divisors of Polynomials Given by {Straight-Line} Programs},
    volume = {35},
    year = {1988}
}

@article{bosma+cannon+steel97,
    address = {Duluth, MN, USA},
    author = {Bosma, Wieb and Cannon, John and Steel, Allan},
    citeulike-article-id = {7729221},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=268921.268929},
    citeulike-linkout-1 = {http://dx.doi.org/10.1006/jsco.1997.0138},
    doi = {10.1006/jsco.1997.0138},
    issn = {0747-7171},
    journal = {J. Symbolic Comput.},
    keywords = {finite\_field},
    number = {3-4},
    pages = {351--369},
    posted-at = {2010-08-28 00:14:32},
    priority = {0},
    publisher = {Academic Press, Inc.},
    title = {Lattices of compatibly embedded finite fields},
    url = {http://dx.doi.org/10.1006/jsco.1997.0138},
    volume = {24},
    year = {1997}
}

@incollection{cesar,
    abstract = {Exstant et ad Ciceronem, item ad familiares domesticis de rebus, in quibus, si qua occultius perferenda erant, per notas scripsit, id est sic structo litterarum ordine, ut nullum uerbum effici posset: quae si qui inuestigare et persequi uelit,quartam elementorum litteram, id est D pro A et perinde reliquas commutet.},
    address = {Paris},
    author = {Suetonius, Caius T.},
    booktitle = {Vies des douze C\'{e}sars},
    citeulike-article-id = {7724813},
    editor = {Ailloud, Henri and L'Yvonnet, Fran\c{c}ois},
    isbn = {978-2-251-79994-0},
    keywords = {cryptography},
    pages = {72},
    posted-at = {2010-08-27 19:28:09},
    priority = {0},
    publisher = {Les belles lettres},
    title = {De vita {C}aesarum, {L}iber {I}, {D}iuus {I}ulius, {LVI}},
    volume = {I},
    year = {2002}
}

@manual{intel-carryless,
    author = {Gueron, Shay and Kounavis, Michael E.},
    citeulike-article-id = {7722704},
    citeulike-linkout-0 = {http://software.intel.com/en-us/articles/intel-carry-less-multiplication-instruction-and-its-usage-for-computing-the-gcm-mode/},
    howpublished = {\url{http://software.intel.com/en-us/articles/intel-carry-less-multiplication-instruction-and-its-usage-for-computing-the-gcm-mode/}},
    keywords = {finite\_field},
    organization = {white paper, Intel Corporation},
    posted-at = {2010-08-27 18:03:02},
    priority = {2},
    title = {Carry-less multiplication and its usage for computing the {GCM} mode},
    url = {http://software.intel.com/en-us/articles/intel-carry-less-multiplication-instruction-and-its-usage-for-computing-the-gcm-mode/},
    year = {2008}
}

@electronic{djb:tellegen,
    author = {Bernstein, Daniel J.},
    citeulike-article-id = {7716333},
    citeulike-linkout-0 = {http://cr.yp.to/transposition.html},
    howpublished = {Last accessed 26-Aug-2010. \url{http://cr.yp.to/transposition.html}},
    keywords = {transposition},
    posted-at = {2010-08-26 22:34:13},
    priority = {0},
    title = {The transposition principle},
    url = {http://cr.yp.to/transposition.html}
}

@article{atkin+morain93:2,
    abstract = {. The aim of this paper is to describe the theory and implementation of the Elliptic Curve {PrimalityProving} {algorithm.Problema}, numeros primos a compositis dignoscendi, hosque in factores suos primos resolvendi, ad gravissima ac utilissimatotius arithmeticae pertinere, et geometrarum tum veterum tum recentiorum industriam ac sagacitatem occupavisse, tamnotum est, ut de hac re copiose loqui superfluum {foret.C}. F. Gauss [37, Art. 329]1 {IntroductionPrimality} testing is one of the most...},
    author = {Atkin, A. O. L. and Morain, Fran\c{c}ois},
    citeulike-article-id = {1195322},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.1146},
    journal = {Math. Comp.},
    keywords = {elliptic\_curve, elliptic\_curve\_method},
    number = {203},
    pages = {29--68},
    posted-at = {2010-08-25 21:32:31},
    priority = {2},
    title = {Elliptic curves and primality proving},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.1146},
    volume = {61},
    year = {1993}
}

@article{lenstra87,
    author = {Lenstra, Hendrik W.},
    citeulike-article-id = {7661377},
    journal = {Annals of Mathematics},
    keywords = {elliptic\_curve\_method},
    pages = {649--673},
    posted-at = {2010-08-25 21:29:07},
    priority = {2},
    title = {Factoring integers with elliptic curves},
    volume = {126},
    year = {1987}
}

@article{wiles+taylor95,
    author = {Taylor, Richard and Wiles, Andrew},
    citeulike-article-id = {7709022},
    citeulike-linkout-0 = {http://dx.doi.org/10.2307/2118560},
    doi = {10.2307/2118560},
    issn = {0003486X},
    journal = {The Annals of Mathematics},
    keywords = {elliptic\_curve},
    month = may,
    number = {3},
    pages = {553+},
    posted-at = {2010-08-25 21:22:43},
    priority = {2},
    title = {Ring-Theoretic Properties of Certain {H}ecke Algebras},
    url = {http://dx.doi.org/10.2307/2118560},
    volume = {141},
    year = {1995}
}

@article{wiles95,
    author = {Wiles, Andrew},
    citeulike-article-id = {7709012},
    citeulike-linkout-0 = {http://dx.doi.org/10.2307/2118559},
    doi = {10.2307/2118559},
    issn = {0003486X},
    journal = {The Annals of Mathematics},
    keywords = {elliptic\_curve},
    month = may,
    number = {3},
    pages = {443+},
    posted-at = {2010-08-25 21:21:40},
    priority = {1},
    title = {Modular Elliptic Curves and {F}ermat's Last Theorem},
    url = {http://dx.doi.org/10.2307/2118559},
    volume = {141},
    year = {1995}
}

@inproceedings{schost05,
    abstract = {We study the multiplication of multivariate power series. We show that over large enough fields, the bilinear complexity of the product modulo a monomial ideal  M  is bounded by the product of the regularity of  M  by the degree of  M . In some special cases, such as partial degree truncation, this estimate carries over to total complexity. This leads to complexity improvements for some basic algorithms with algebraic numbers, and some polynomial system solving algorithms.},
    address = {New York, NY, USA},
    author = {Schost, \'{E}ric},
    booktitle = {ISSAC '05: Proceedings of the 2005 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {7706793},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1073884.1073925},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1073884.1073925},
    doi = {10.1145/1073884.1073925},
    isbn = {1-59593-095-7},
    keywords = {multiplication, multivariate\_polynomial, power\_series},
    location = {Beijing, China},
    pages = {293--300},
    posted-at = {2010-08-25 15:00:30},
    priority = {3},
    publisher = {ACM},
    title = {Multivariate power series multiplication},
    url = {http://dx.doi.org/10.1145/1073884.1073925},
    year = {2005}
}

@article{huang+pan98,
    address = {Orlando, FL, USA},
    author = {Huang, Xiaohan and Pan, Victor Y.},
    citeulike-article-id = {7691105},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=299029.299038},
    citeulike-linkout-1 = {http://dx.doi.org/10.1006/jcom.1998.0476},
    doi = {10.1006/jcom.1998.0476},
    issn = {0885-064X},
    journal = {Journal of Complexity},
    keywords = {matrix\_multiplication},
    number = {2},
    pages = {257--299},
    posted-at = {2010-08-23 15:14:25},
    priority = {1},
    publisher = {Academic Press, Inc.},
    title = {Fast rectangular matrix multiplication and applications},
    url = {http://dx.doi.org/10.1006/jcom.1998.0476},
    volume = {14},
    year = {1998}
}

@article{karatsuba,
    author = {Karatsuba, Anatolii and Ofman, Yuri},
    citeulike-article-id = {4034988},
    journal = {Soviet Physics-Doklady},
    keywords = {multiplication},
    pages = {595--596},
    posted-at = {2010-08-22 22:53:29},
    priority = {2},
    title = {Multiplication of Multidigit Numbers on Automata},
    volume = {7},
    year = {1963}
}

@article{schonage+strassen,
    abstract = {An algorithm is given for computing the product of two N-digit binary numbers by {O(N} {lgN} {lglgN}) steps. Two ways of implementing the algorithm are considered: multitape Turing machines and logical nets (with step = binary logical element).},
    author = {Sch{\"{o}}nhage, Arnold and Strassen, Volker},
    citeulike-article-id = {4486431},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/BF02242355},
    citeulike-linkout-1 = {http://www.springerlink.com/content/y251407745475773},
    day = {23},
    doi = {10.1007/BF02242355},
    issn = {0010-485X},
    journal = {Computing},
    keywords = {multiplication},
    month = sep,
    number = {3},
    pages = {281--292},
    posted-at = {2010-08-22 22:52:44},
    priority = {2},
    publisher = {Springer Wien},
    title = {Schnelle {M}ultiplikation gro{\ss}er {Z}ahlen},
    url = {http://dx.doi.org/10.1007/BF02242355},
    volume = {7},
    year = {1971}
}

@booklet{poly-formel,
    author = {Bostan, Alin and Chyzak, Fr\'{e}d\'{e}ric and Giusti, Marc and Lebreton, Romain and Salvy, Bruno and Schost, \'{E}ric},
    citeulike-article-id = {7687847},
    howpublished = {Lecture notes for the MPRI course 2-22},
    keywords = {computer\_algebra},
    posted-at = {2010-08-22 22:36:38},
    priority = {2},
    title = {Algorithmes en Calcul Formel et en Automatique},
    year = {2010}
}

@article{harvey09,
    abstract = {We present several new algorithms for dense polynomial multiplication in  {inlMMLBox}  based on the Kronecker substitution method. Instead of reducing to a single integer multiplication, we reduce to several smaller multiplications. We describe an implementation of multiplication in  {inlMMLBox}  for a word-sized modulus  n  based on these methods, and compare its performance to that of {NTL} and Magma.},
    author = {Harvey, David},
    citeulike-article-id = {5365958},
    citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.jsc.2009.05.004},
    doi = {10.1016/j.jsc.2009.05.004},
    issn = {07477171},
    journal = {J. Symbolic Comput.},
    keywords = {multivariate\_polynomial, polynomial\_multiplication},
    month = oct,
    number = {10},
    pages = {1502--1510},
    posted-at = {2010-08-19 17:39:26},
    priority = {4},
    title = {Faster polynomial multiplication via multipoint {K}ronecker substitution},
    url = {http://dx.doi.org/10.1016/j.jsc.2009.05.004},
    volume = {44},
    year = {2009}
}

@article{morain95,
    author = {Morain, Fran\c{c}ois},
    citeulike-article-id = {7678503},
    journal = {Journal de Th\'{e}orie des Nombres de Bordeaux},
    keywords = {elliptic\_curve},
    pages = {255--282},
    posted-at = {2010-08-19 00:45:28},
    priority = {2},
    title = {Calcul du nombre de points sur une courbe elliptique dans un corps fini: aspects algorithmiques},
    volume = {7},
    year = {1995}
}

@electronic{sutherland10:modpol,
    abstract = {We present a new algorithm to compute the classical modular polynomial Phi\_nin the rings {Z[X},Y] and ({Z/mZ})[{X,Y}], for a prime n and any positive integer {m.Our} approach uses the graph of n-isogenies to efficiently compute Phi\_n mod pfor many primes p of a suitable form, and then applies the Chinese {RemainderTheorem} ({CRT}). Under the Generalized Riemann Hypothesis ({GRH}), we achieve anexpected running time of O(n^3 (log n)^3 log log n), and compute Phi\_n mod musing O(n^2 (log n)^2 + n^2 log m) space. We have used the new algorithm tocompute Phi\_n with n over 5000, and Phi\_n mod m with n over 20000. We alsoconsider several modular functions g for which Phi\_n^g is smaller than Phi\_n,allowing us to handle n over 60000.},
    archivePrefix = {arXiv},
    author = {Br{\"{o}}ker, Reinier and Lauter, Kristin and Sutherland, Andrew V.},
    citeulike-article-id = {7678482},
    citeulike-linkout-0 = {http://arxiv.org/abs/1001.0402},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1001.0402},
    day = {3},
    eprint = {1001.0402},
    keywords = {elliptic\_curve, isogenies, isogeny\_cycles},
    month = jan,
    posted-at = {2010-08-19 00:13:01},
    priority = {2},
    title = {Modular polynomials via isogeny volcanoes},
    url = {http://arxiv.org/abs/1001.0402},
    year = {2010}
}

@misc{milne1996elliptic,
    author = {Milne, James S.},
    citeulike-article-id = {7664594},
    keywords = {elliptic\_curve},
    posted-at = {2010-08-17 16:52:50},
    priority = {3},
    publisher = {Citeseer},
    title = {Elliptic curves},
    year = {1996}
}

@booklet{connell:elliptic,
    author = {Connell, Ian},
    citeulike-article-id = {7664577},
    howpublished = {Lecture Notes from class at McGill University},
    keywords = {elliptic\_curve},
    posted-at = {2010-08-17 16:47:15},
    priority = {3},
    title = {Elliptic curve handbook},
    year = {1991}
}

@inbook{alonso+becker+roy+wormann,
    abstract = {The motivation for this paper is the following: In several examples,when the radical of an ideal \$\scr I\$ of \$k[X\_1,\cdots,X\_n]\$ iscomputed (in order to compute \$V(\scr I))\$, and one coordinate isexpressed as the zeros of a univariate polynomial with the othercoordinates being polynomial functions of this first coordinate (theshape lemma), then the sizes of the coefficients of this univariatepolynomial are not very big. For the other polynomials involved thisis not the case. And, in any case, the computation is difficult.<P> The authors define the Chow polynomial in order to describe the pointsof \$V(\scr I)\$ through the zeros of a \$u\$-Chow polynomial, where \$u\inA=k[X\_1,\cdots,X\_n]/\scr I\$ is well chosen: for each multiplicity\$µ\$ which may occur for points in \$V(\scr I)\$ the \$u\$-Chowpolynomial gives a rational expression for the coordinates of eachpoint of \$V(\scr I)\$ with multiplicity \$µ\$. This leads to a globalrational computation of the points of \$V(\scr I)\$.<P> The \$u\$-Chow polynomial also gives information on which multiplicitiesmay occur: a rational function on \$V(\scr I)\$ represents themultiplicity \$µ\$ of a zero.<P> Again using the \$u\$-Chow polynomial the authors describe a method forcomputing the decomposition of \$A\$ into its local components, i.e.\for computing the indecomposable idempotents in \$A\$. They apply thesemethods first when \$A\$ is given through a Gr\"{o}bner basis, and thendescribe the computations from the knowledge of the multiplicationtables in \$A\$. Then, when \$A\$ is a complete intersection, adeformation is easily performed in order to compute a Gr\"{o}bner basisin the modified situation and to obtain information for \$A\$.<P>{For the entire collection see <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=1414441\&loc=fromrevtext">MR1414441 (97d:14002)</A>.}},
    address = {Basel},
    author = {Alonso, Mar\'{\i}a-Emilia and Becker, Eberhard and Roy, Marie-Fran\c{c}oise and W{\"{o}}rmann, Thorsten},
    booktitle = {Algorithms in algebraic geometry and applications (Santander, 1994)},
    citeulike-article-id = {244735},
    citeulike-linkout-0 = {http://www.ams.org/mathscinet-getitem?mr=1414442},
    keywords = {algebraic\_geometry, ideal, trace\_formulas, zero\_dimensional},
    mrnumber = {MR1414442},
    pages = {1--15},
    posted-at = {2010-08-17 14:04:18},
    priority = {2},
    publisher = {Birkh\"{a}user},
    series = {Progress in Mathematics},
    title = {Zeros, multiplicities, and idempotents for zero-dimensional systems},
    url = {http://www.ams.org/mathscinet-getitem?mr=1414442},
    volume = {143},
    year = {1996}
}

@article{df+schost10,
    abstract = {We present here transalpyne, a scripting language, to be executed on top of a computer algebra system, that is specifically conceived for automatic transposition of linear functions. Its type system is able to automatically infer all the possible linear functions realized by a computer program. The key feature of transalpyne is its ability to transform a computer program computing a linear function in another computer program computing the transposed linear function. The time and space complexity of the resulting program are similar to the original ones.},
    address = {New York, NY, USA},
    author = {De Feo, Luca and Schost, \'{E}ric},
    citeulike-article-id = {7663840},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1838599.1838624},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1838599.1838624},
    doi = {10.1145/1838599.1838624},
    issn = {0163-5824},
    journal = {SIGSAM Bulletin},
    keywords = {functional\_language, transposition},
    number = {1/2},
    pages = {59--71},
    posted-at = {2010-08-17 12:36:20},
    priority = {0},
    publisher = {ACM},
    title = {transalpyne: a language for automatic transposition},
    url = {http://dx.doi.org/10.1145/1838599.1838624},
    volume = {44},
    year = {2010}
}

@inproceedings{faugere02,
    abstract = {This paper introduces a new efficient algorithm for computing Gr{\"{o}}bner bases. We replace the Buchberger criteria by an optimal criteria. We give a proof that the resulting algorithm (called F5) generates no useless critical pairs ...},
    author = {Faug\`{e}re, Jean-Charles},
    booktitle = {ISSAC '02: Proceedings of the 2002 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {3978844},
    citeulike-linkout-0 = {http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=pubmed&#38;cmd=Retrieve&#38;dopt=AbstractPlus&#38;list_uids=780506.780516},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/780506.780516},
    doi = {10.1145/780506.780516},
    keywords = {groebner\_bases},
    posted-at = {2010-08-15 16:59:49},
    priority = {2},
    title = {A new efficient algorithm for computing {G}r{\"{o}}bner bases without reduction to zero ({F5})},
    url = {http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=pubmed&#38;cmd=Retrieve&#38;dopt=AbstractPlus&#38;list_uids=780506.780516},
    year = {2002}
}

@article{faugere99,
    abstract = {This paper introduces a new efficient algorithm for computing Grobner bases. To avoid as much intermediate computation as possible, the algorithm computes successive truncated Grobner bases and it replaces the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduction of several polynomials. This powerful reduction mechanism is achieved by means of a symbolic precomputation and by extensive use of sparse linear algebra methods. Current techniques in linear algebra used in Computer Algebra are reviewed together with other methods coming from the numerical field. Some previously untractable problems (Cyclic 9) are presented as well as an empirical comparison of a first implementation of this algorithm with other well known programs. This comparison pays careful attention to methodology issues. All the benchmarks and {CPU} times used in this paper are frequently updated and available on a Web page. Even though the new algorithm does not improve the worst case complexity it is several times faster than previous implementations both for integers and modulo p computations.},
    author = {Faug\`{e}re, Jean-Charles},
    citeulike-article-id = {712537},
    citeulike-linkout-0 = {http://dx.doi.org/10.1016/S0022-4049(99)00005-5},
    citeulike-linkout-1 = {http://www.sciencedirect.com/science/article/B6V0K-3XBTSG0-5/2/a7d0e1134137bb94db1d1172abcc2774},
    doi = {10.1016/S0022-4049(99)00005-5},
    journal = {Journal of Pure and Applied Algebra},
    keywords = {groebner\_bases},
    month = jun,
    number = {1-3},
    pages = {61--88},
    posted-at = {2010-08-15 16:59:18},
    priority = {2},
    title = {A new efficient algorithm for computing {G}r\"{o}bner bases ({F4})},
    url = {http://dx.doi.org/10.1016/S0022-4049(99)00005-5},
    volume = {139},
    year = {1999}
}

@phdthesis{buchberger,
    address = {Austria},
    author = {Buchberger, Bruno},
    citeulike-article-id = {3776489},
    keywords = {groebner\_bases},
    posted-at = {2010-08-15 16:43:50},
    priority = {1},
    school = {Institute of Mathematics, University of Innsbruck},
    title = {An Algorithm for Finding a Basis for the Residue Class Ring of a {Zero-Dimensional} Polynomial Ideal (in German)},
    year = {1965}
}

@article{df10,
    abstract = {The problem of computing an explicit isogeny between two given elliptic curves over , originally motivated by point counting, has recently awaken new interest in the cryptology community thanks to the works of Teske and Rostovtsev \& Stolbunov. While the large characteristic case is well understood, only suboptimal algorithms are known in small characteristic; they are due to Couveignes, Lercier, Lercier \& Joux and Lercier \& Sirvent. In this paper we discuss the differences between them and run some comparative experiments. We also present the first complete implementation of Couveignes' second algorithm and present improvements that make it the algorithm having the best asymptotic complexity in the degree of the isogeny.},
    author = {De Feo, Luca},
    citeulike-article-id = {7652683},
    citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.jnt.2010.07.003},
    day = {12},
    doi = {10.1016/j.jnt.2010.07.003},
    issn = {0022-314X},
    journal = {Journal of Number Theory},
    keywords = {elliptic\_curve, finite\_field, isogenies},
    month = may,
    number = {5},
    pages = {873--893},
    posted-at = {2010-08-15 14:14:50},
    priority = {0},
    title = {Fast algorithms for computing isogenies between ordinary elliptic curves in small characteristic},
    url = {http://dx.doi.org/10.1016/j.jnt.2010.07.003},
    volume = {131},
    year = {2011}
}

@book{hachenberger,
    author = {Hachenberger, Dirk},
    citeulike-article-id = {7648858},
    keywords = {finite\_field},
    posted-at = {2010-08-14 03:00:37},
    priority = {0},
    publisher = {Kluwer Academic Pub},
    title = {{Finite fields: normal bases and completely free elements}},
    year = {1997}
}

@book{mourrain+elkadi,
    author = {Elkadi, Mohamed and Mourrain, Bernard},
    citeulike-article-id = {7550324},
    keywords = {algebra\_books, algebraic\_geometry, groebner\_bases, ideal, zero\_dimensional},
    posted-at = {2010-07-29 08:19:43},
    priority = {5},
    publisher = {Springer Verlag},
    title = {Introduction {\`{a}} la r{\'{e}}solution des syst{\`{e}}mes polynomiaux},
    year = {2007}
}

@unpublished{df10-2,
    author = {De Feo, Luca},
    citeulike-article-id = {7548858},
    citeulike-linkout-0 = {http://hal.inria.fr/hal-00505791/},
    howpublished = {\url{http://hal.inria.fr/hal-00505791/}},
    keywords = {elliptic\_curve, isogenies},
    posted-at = {2010-07-28 13:37:51},
    priority = {2},
    title = {On computing isogenies of unknown degree},
    url = {http://hal.inria.fr/hal-00505791/},
    year = {2010}
}

@phdthesis{robert,
    author = {Robert, Damien},
    citeulike-article-id = {7548302},
    citeulike-linkout-0 = {http://www.normalesup.org/~robert/pro/publications/academic/phd.pdf},
    keywords = {cryptography, hyperelliptic\_curves},
    posted-at = {2010-07-28 12:12:16},
    priority = {3},
    school = {LORIA, Nancy},
    title = {Theta functions and applications in cryptography},
    url = {http://www.normalesup.org/~robert/pro/publications/academic/phd.pdf},
    year = {2010}
}

@incollection{nusken+ziegler04,
    abstract = {We generalize univariate multipoint evaluation of polynomials of degree n at sublinear amortized cost per point. More precisely, it is shown how to evaluate a bivariate polynomial p of maximum degree less than n, specified by its n 2 coefficients, simultaneously at n 2 given points using a total of \$\mathcal{O}(n^{2.667})\$ arithmetic operations. In terms of the input size N being quadratic in n, this amounts to an amortized cost of \$\mathcal{O}(N^{0.334})\$ per point.},
    author = {N\"{u}sken, Michael and Ziegler, Martin},
    booktitle = {Algorithms – ESA 2004},
    citeulike-article-id = {7548295},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-30140-0_49},
    citeulike-linkout-1 = {http://www.springerlink.com/content/7b1hfpf6hr9ab5jn},
    doi = {10.1007/978-3-540-30140-0_49},
    keywords = {multivariate\_polynomial, polynomial\_evaluation},
    pages = {544--555},
    posted-at = {2010-07-28 12:03:22},
    priority = {2},
    title = {Fast Multipoint Evaluation of Bivariate Polynomials},
    url = {http://dx.doi.org/10.1007/978-3-540-30140-0_49},
    year = {2004}
}

@article{vzgathen09-2,
    abstract = {A polynomial f (multivariate over a field) is decomposable if f = g(h) with gunivariate of degree at least 2. We determine the dimension (over analgebraically closed field) of the set of decomposables, and an approximationto their number over a finite field. The relative error in our approximationsis exponentially decaying in the input size.},
    archivePrefix = {arXiv},
    author = {von zur Gathen, Joachim},
    citeulike-article-id = {7547602},
    citeulike-linkout-0 = {http://arxiv.org/abs/0811.4726},
    citeulike-linkout-1 = {http://arxiv.org/pdf/0811.4726},
    day = {1},
    eprint = {0811.4726},
    keywords = {composition, multivariate\_polynomial},
    month = jul,
    posted-at = {2010-07-28 09:13:32},
    priority = {2},
    title = {Counting decomposable multivariate polynomials},
    url = {http://arxiv.org/abs/0811.4726},
    year = {2009}
}

@inproceedings{vzgathen09,
    abstract = {A univariate polynomial  f  over a field is  decomposable  if it is the composition  f  =  g  ⊕  h  of two polynomials  g  and  h  whose degree is at least 2. We determine an approximation to the number of decomposable polynomials over a finite field. The tame case, where the field characteristic  p  does not divide the degree  n  of  f , is reasonably well understood, and we obtain exponentially decreasing error bounds.},
    address = {New York, NY, USA},
    author = {von zur Gathen, Joachim},
    booktitle = {ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {7547594},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1576751},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1576702.1576751},
    doi = {10.1145/1576702.1576751},
    isbn = {978-1-60558-609-0},
    keywords = {composition, polynomials\_over\_finite\_fields},
    location = {Seoul, Republic of Korea},
    pages = {359--366},
    posted-at = {2010-07-28 09:10:10},
    priority = {2},
    publisher = {ACM},
    title = {The number of decomposable univariate polynomials. extended abstract},
    url = {http://dx.doi.org/10.1145/1576702.1576751},
    year = {2009}
}

@article{vzgathen+giesbrecht+ziegler10,
    abstract = {The functional decomposition of polynomials has been a topic of greatinterest and importance in pure and computer algebra and their {applications.The} structure of compositions of (suitably normalized) polynomials f=g(h) overfinite fields is well understood in many cases, but quite poorly when thedegrees of both components are divisible by the characteristic p. This workinvestigates the decomposition of polynomials whose degree is a power of p.},
    archivePrefix = {arXiv},
    author = {von zur Gathen, Joachim and Giesbrecht, Mark and Ziegler, Konstantin},
    citeulike-article-id = {7547593},
    citeulike-linkout-0 = {http://arxiv.org/abs/1005.1087},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1005.1087},
    day = {6},
    eprint = {1005.1087},
    keywords = {composition, polynomials\_over\_finite\_fields},
    month = may,
    posted-at = {2010-07-28 09:08:08},
    priority = {4},
    title = {Composition collisions and projective polynomials},
    url = {http://arxiv.org/abs/1005.1087},
    year = {2010}
}

@incollection{elaimani08,
    abstract = {Undeniable signatures were proposed to limit the verification property of ordinary digital signatures. In fact, the verification of such signatures cannot be attained without the help of the signer, via the confirmation/denial protocols. Later, the concept was refined to give the possibility of converting the issued undeniable signatures into ordinary ones by publishing a universal receipt that turns them publicly verifiable. In this paper, we present the first generic construction for universally convertible undeniable signatures from certain weakly secure cryptosystems and any secure digital signature scheme. Next, we give two specific approaches for building universally convertible undeniable signatures from a large class of pairing-based signatures. These methods find a nice and practical instantiation with known encryption and signature schemes. For instance, we achieve the most efficient undeniable signatures with regard to the signature length and cost, the underlying assumption and the security model. We believe these constructions could be an interesting starting point to develop more efficient schemes or give better security analyses of the existing ones.},
    address = {Berlin, Heidelberg},
    author = {{E}l Aimani, Laila},
    booktitle = {Progress in Cryptology - INDOCRYPT 2008},
    chapter = {12},
    citeulike-article-id = {7542473},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-89754-5_12},
    citeulike-linkout-1 = {http://www.springerlink.com/content/k4853p021uvm8307},
    doi = {10.1007/978-3-540-89754-5_12},
    editor = {Chowdhury, Dipanwita R. and Rijmen, Vincent and Das, Abhijit},
    isbn = {978-3-540-89753-8},
    issn = {0302-9743},
    keywords = {cryptography, pairing},
    pages = {145--157},
    posted-at = {2010-07-27 10:39:31},
    priority = {2},
    publisher = {Springer Berlin Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Toward a Generic Construction of Universally Convertible Undeniable Signatures from {Pairing-Based} Signatures},
    url = {http://dx.doi.org/10.1007/978-3-540-89754-5_12},
    volume = {5365},
    year = {2008}
}

@misc{elaimani09,
    author = {{E}l Aimani, Laila},
    citeulike-article-id = {7542470},
    citeulike-linkout-0 = {http://eprint.iacr.org/2009/362},
    comment = {\url{http://eprint.iacr.org/}},
    howpublished = {Cryptology ePrint Archive, Report 2009/362},
    keywords = {cryptography, pairing},
    posted-at = {2010-07-27 10:36:33},
    priority = {2},
    title = {Toward a Generic Construction of Convertible Undeniable Signatures from {Pairing-Based} Signatures},
    url = {http://eprint.iacr.org/2009/362},
    year = {2009}
}

@article{cantor87,
    abstract = {In this paper we present algorithms, suitable for computer use, for computation in the Jacobian of a hyperelliptic curve. We present a reduction algorithm which is asymptotically faster than that of Gauss when the genus \$g\$ is very large.},
    author = {Cantor, David G.},
    citeulike-article-id = {7539872},
    citeulike-linkout-0 = {http://dx.doi.org/10.2307/2007876},
    citeulike-linkout-1 = {http://www.jstor.org/stable/2007876},
    doi = {10.2307/2007876},
    issn = {00255718},
    journal = {Math. Comp.},
    keywords = {hyperelliptic\_curves},
    number = {177},
    pages = {95--101},
    posted-at = {2010-07-26 17:42:23},
    priority = {2},
    publisher = {American Mathematical Society},
    title = {Computing in the {J}acobian of a Hyperelliptic Curve},
    url = {http://dx.doi.org/10.2307/2007876},
    volume = {48},
    year = {1987}
}

@incollection{granger+etal07,
    abstract = {In this paper we show that the Ate pairing, originally defined for elliptic curves, generalises to hyperelliptic curves and in fact to arbitrary algebraic curves. It has the following surprising properties: The loop length in Miller's algorithm can be up to g times shorter than for the Tate pairing, with g the genus of the curve, and the pairing is automatically reduced, i.e. no final exponentiation is needed.},
    address = {Berlin, Heidelberg},
    author = {Granger, R. and Hess, F. and Oyono, R. and Th\'{e}riault, N. and Vercauteren, F.},
    booktitle = {Advances in Cryptology - EUROCRYPT 2007},
    chapter = {25},
    citeulike-article-id = {7539828},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-72540-4_25},
    citeulike-linkout-1 = {http://www.springerlink.com/content/d8711v376761530t},
    doi = {10.1007/978-3-540-72540-4_25},
    editor = {Naor, Moni},
    isbn = {978-3-540-72539-8},
    issn = {0302-9743},
    keywords = {cryptography, hyperelliptic\_curves, pairing},
    pages = {430--447},
    posted-at = {2010-07-26 17:08:01},
    priority = {2},
    publisher = {Springer Berlin Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Ate Pairing on Hyperelliptic Curves},
    url = {http://dx.doi.org/10.1007/978-3-540-72540-4_25},
    volume = {4515},
    year = {2007}
}

@incollection{eisentrager+lauter+montgomery04,
    abstract = {We present algorithms for computing the squared Weil and Tate pairings on elliptic curves and the squared Tate pairing on hyperelliptic curves. The squared pairings introduced in this paper have the advantage that our algorithms for evaluating them are deterministic and do not depend on a random choice of points. Our algorithm to evaluate the squared Weil pairing is about 20\% more efficient than the standard Weil pairing. Our algorithm for the squared Tate pairing on elliptic curves matches the efficiency of the algorithm given by Barreto, Lynn, and Scott in the case of arbitrary base points where their denominator cancellation technique does not apply. Our algorithm for the squared Tate pairing for hyperelliptic curves is the first detailed implementation of the pairing for general hyperelliptic curves of genus 2, and saves an estimated 30\% over the standard algorithm.},
    author = {Eisentr\"{a}ger, Kirsten and Lauter, Kristin and Montgomery, Peter},
    booktitle = {Algorithmic Number Theory},
    citeulike-article-id = {7539825},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-24847-7_12},
    citeulike-linkout-1 = {http://www.springerlink.com/content/5wpapcr7rbte3nb1},
    doi = {10.1007/978-3-540-24847-7_12},
    keywords = {cryptography, hyperelliptic\_curves, pairing},
    pages = {169--183},
    posted-at = {2010-07-26 17:06:47},
    priority = {2},
    title = {Improved {W}eil and {T}ate Pairings for Elliptic and Hyperelliptic Curves},
    url = {http://dx.doi.org/10.1007/978-3-540-24847-7_12},
    year = {2004}
}

@incollection{galbraith01,
    abstract = {Frey and R\"{u}ck gave a method to transform the discrete logarithm problem in the divisor class group of a curve over \$\$ \mathbb{F}\_q \$\$ into a discrete logarithm problem in some finite field extension \$\$ \mathbb{F}\_{q^k } \$\$ . The discrete logarithm problem can therefore be solved using index calculus algorithms as long as k is small. In the elliptic curve case it was shown by Menezes, Okamoto and Vanstone that for supersingular curves one has k ⪯ 6. In this paper curves of higher genus are studied. Bounds on the possible values for k in the case of supersingular curves are given which imply that supersingular curves are weaker than the general case for cryptography. Ways to ensure that a curve is not supersingular are also discussed. A constructive application of supersingular curves to cryptography is given, by generalising an identity-based cryptosystem due to Boneh and Franklin. The generalised scheme provides a significant reduction in bandwidth compared with the original scheme.},
    address = {Berlin, Heidelberg},
    author = {Galbraith, Steven},
    booktitle = {Advances in Cryptology — ASIACRYPT 2001},
    chapter = {29},
    citeulike-article-id = {7539537},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/3-540-45682-1_29},
    citeulike-linkout-1 = {http://www.springerlink.com/content/6071gxe3cwl9b1kq},
    day = {20},
    doi = {10.1007/3-540-45682-1_29},
    editor = {Boyd, Colin},
    isbn = {978-3-540-42987-6},
    keywords = {cryptography, elliptic\_curve, pairing},
    month = nov,
    pages = {495--513},
    posted-at = {2010-07-26 14:06:55},
    priority = {2},
    publisher = {Springer Berlin Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Supersingular Curves in Cryptography},
    url = {http://dx.doi.org/10.1007/3-540-45682-1_29},
    volume = {2248},
    year = {2001}
}

@article{joux04,
    abstract = {Abstract\&nbsp;\&nbsp; In this paper we propose a three participants variation of the {Diffie--Hellman} protocol. This variation is based on the Weil and Tate pairings on elliptic curves, which were first used in cryptography as cryptanalytic tools for reducing the discrete logarithm problem on some elliptic curves to the discrete logarithm problem in a finite field.},
    author = {Joux, Antoine},
    citeulike-article-id = {650284},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s00145-004-0312-y},
    citeulike-linkout-1 = {http://www.ingentaconnect.com/content/klu/145/2004/00000017/00000004/art00003},
    citeulike-linkout-2 = {http://www.springerlink.com/content/cddc57yyva0hburb},
    day = {28},
    doi = {10.1007/s00145-004-0312-y},
    issn = {0933-2790},
    journal = {Journal of Cryptology},
    keywords = {cryptography, elliptic\_curve, pairing},
    month = sep,
    number = {4},
    pages = {263--276},
    posted-at = {2010-07-26 13:58:39},
    priority = {1},
    publisher = {Springer},
    title = {A One Round Protocol for Tripartite {Diffie–Hellman}},
    url = {http://dx.doi.org/10.1007/s00145-004-0312-y},
    volume = {17},
    year = {2004}
}

@article{boneh+lynn+shacham04,
    abstract = {Abstract   We introduce a short signature scheme based on the Computational {Diffie–Hellman} assumption on certain elliptic and hyperelliptic curves. For standard security parameters, the signature length is about half that of a {DSA} signature with a similar level of security. Our short signature scheme is designed for systems where signatures are typed in by a human or are sent over a low-bandwidth channel. We survey a number of properties of our signature scheme such as signature aggregation and batch verification.},
    author = {Boneh, Dan and Lynn, Ben and Shacham, Hovav},
    citeulike-article-id = {650282},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s00145-004-0314-9},
    citeulike-linkout-1 = {http://www.ingentaconnect.com/content/klu/145/2004/00000017/00000004/art00005},
    citeulike-linkout-2 = {http://www.springerlink.com/content/4t365jty5jrt6yaj},
    day = {28},
    doi = {10.1007/s00145-004-0314-9},
    issn = {0933-2790},
    journal = {Journal of Cryptology},
    keywords = {cryptography, elliptic\_curve, pairing},
    month = sep,
    number = {4},
    pages = {297--319},
    posted-at = {2010-07-26 13:57:43},
    priority = {1},
    publisher = {Springer},
    title = {Short Signatures from the {W}eil Pairing},
    url = {http://dx.doi.org/10.1007/s00145-004-0314-9},
    volume = {17},
    year = {2004}
}

@incollection{boneh+frankiln01,
    abstract = {We propose a fully functional identity-based encryption scheme ({IBE}). The scheme has chosen ciphertext security in the random oracle model assuming an elliptic curve variant of the computational {Diffie-Hellman} problem. Our system is based on the Weil pairing. We give precise definitions for secure identity based encryption schemes and give several applications for such systems.},
    address = {Berlin, Heidelberg},
    author = {Boneh, Dan and Franklin, Matt},
    booktitle = {Advances in Cryptology — CRYPTO 2001},
    chapter = {13},
    citeulike-article-id = {2754467},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/3-540-44647-8_13},
    citeulike-linkout-1 = {http://www.springerlink.com/content/bf5j8nhdp32pxqgy},
    day = {2},
    doi = {10.1007/3-540-44647-8_13},
    editor = {Kilian, Joe},
    isbn = {978-3-540-42456-7},
    journal = {Advances in Cryptology — CRYPTO 2001},
    keywords = {cryptography, elliptic\_curve, pairing},
    month = aug,
    pages = {213--229},
    posted-at = {2010-07-26 13:56:23},
    priority = {1},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Identity-Based Encryption from the {W}eil Pairing},
    url = {http://dx.doi.org/10.1007/3-540-44647-8_13},
    volume = {2139},
    year = {2001}
}

@inproceedings{lubicz+robert10-2,
    author = {Lubicz, David and Robert, Damein},
    booktitle = {Algorithmic Number Theory: 9th International Symposium, ANTS-IX, Nancy, France, July 19-23, 2010, Proceedings},
    citeulike-article-id = {7539456},
    keywords = {cryp, cryptography, hyperelliptic\_curves, pairing},
    organization = {Springer-Verlag GmbH},
    pages = {251},
    posted-at = {2010-07-26 13:09:33},
    priority = {4},
    title = {Efficient Pairing Computation with Theta Functions},
    year = {2010}
}

@booklet{kenzo,
    abstract = {Standard homological algebra is not constructive, and this is frequently the source of serious problems when algorithms are looked for. In particular the usual exact and spectral sequences of homological algebra frequently are not sufficient to obtain some unknown homology or homotopy group. We will explain it is not difficult to fill in this gap, the main tools being on one hand, from a mathematical point of view, the so-called Homological Perturbation Lemma, and on the other hand, from a computational point of view, Functional Programming.     We will illustrate this area of constructive mathematics by applications in two domains:     a) Commutative Algebra frequently meets homological objects, in particular when resolutions are involved (syzygies). Constructive Homological Algebra produces new methods to process old problems such as homology of Koszul complexes, resolutions, minimal resolutions. The solutions so obtained are constructive and therefore more complete than the usual ones, an important point for their concrete use.     b) Algebraic Topology is the historical origin of Homological Algebra. The usual exact and spectral sequences of Algebraic Topology can be easily transformed into new effective versions, giving algorithms computing for example unknown homology and homotopy groups. In particular the effective version of the {Eilenberg-Moore} spectral sequence gives a very simple solution for the old Adams' problem: what algorithm could compute the homology groups of iterated loop spaces?     Machine programs and machine demonstrations will also concretely illustrate the theoretical results so obtained.},
    author = {Rubio, Julio and Sergeraert, Francis},
    booktitle = {Genova Summer School},
    citeulike-article-id = {7510232},
    citeulike-linkout-0 = {http://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Genova-Lecture-Notes.pdf},
    howpublished = {\url{http://www-fourier.ujf-grenoble.fr/\~sergerar/Papers/Genova-Lecture-Notes.pdf}},
    keywords = {categories, functional\_language, homological\_algebra},
    posted-at = {2010-07-18 20:50:59},
    priority = {2},
    title = {Constructive Homological Algebra},
    url = {http://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Genova-Lecture-Notes.pdf},
    year = {2006}
}

@incollection{coquand+spiwack07,
    abstract = {This paper reports on ongoing work on the project of representing the Kenzo system [15] in type theory [11].},
    address = {Berlin, Heidelberg},
    author = {Coquand, Thierry and Spiwack, Arnaud},
    booktitle = {Towards Mechanized Mathematical Assistants},
    chapter = {4},
    citeulike-article-id = {7510230},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-73086-6_4},
    citeulike-linkout-1 = {http://www.springerlink.com/content/9264606874082662},
    doi = {10.1007/978-3-540-73086-6_4},
    editor = {Kauers, Manuel and Kerber, Manfred and Miner, Robert and Windsteiger, Wolfgang},
    isbn = {978-3-540-73083-5},
    issn = {0302-9743},
    keywords = {categories, functional\_language, types},
    pages = {40--54},
    posted-at = {2010-07-18 20:36:28},
    priority = {4},
    publisher = {Springer Berlin Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Towards Constructive Homological Algebra in Type Theory},
    url = {http://dx.doi.org/10.1007/978-3-540-73086-6_4},
    volume = {4573},
    year = {2007}
}

@article{charles+lauter+goren09,
    abstract = {Abstract   We propose constructing provable collision resistant hash functions from expander graphs in which finding cycles is hard. As examples, we investigate two specific families of optimal expander graphs for provable collision resistant hash function constructions: the families of Ramanujan graphs constructed by {Lubotzky-Phillips}-Sarnak and Pizer respectively. When the hash function is constructed from one of Pizer's Ramanujan graphs, (the set of supersingular elliptic curves over with ℓ-isogenies, ℓ a prime different from p), then collision resistance follows from hardness of computing isogenies between supersingular elliptic curves. For the {LPS} graphs, the underlying hard problem is a representation problem in group theory. Constructing our hash functions from optimal expander graphs implies that the outputs closely approximate the uniform distribution. This property is useful for arguing that the output is indistinguishable from random sequences of bits. We estimate the cost per bit to compute these hash functions, and we implement our hash function for several members of the Pizer and {LPS} graph families and give actual timings.},
    author = {Charles, Denis X. and Lauter, Kristin E. and Goren, Eyal Z.},
    citeulike-article-id = {6466830},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s00145-007-9002-x},
    citeulike-linkout-1 = {http://www.springerlink.com/content/cv4r187833614475},
    day = {1},
    doi = {10.1007/s00145-007-9002-x},
    issn = {0933-2790},
    journal = {Journal of Cryptology},
    keywords = {elliptic\_curve, hash, isogenies},
    month = jan,
    number = {1},
    pages = {93--113},
    posted-at = {2010-07-18 17:03:39},
    priority = {2},
    title = {Cryptographic Hash Functions from Expander Graphs},
    url = {http://dx.doi.org/10.1007/s00145-007-9002-x},
    volume = {22},
    year = {2009}
}

@article{strassen69,
    abstract = {
V. V. Kljuev and N. I. Kokovkin-\v S\v cerbak [\v Z. Vy\v cisl. Mat. i Mat.
Fiz. <strong>5</strong> (1965), 21--33; 
<A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=172453\&loc=fromrevtext">MR0172453 (30 \#2672)</A>; 
English translation, Tech.
Rep. No. CS 24, Comput. Sci. Dept., Stanford Univ., Stanford, Calif., 1965]
have proved that, if one restricts oneself to operations on rows and
columns as a whole, no general system of linear equations can be solved in
fewer multiplications than are required by Gaussian elimination, namely,
\${\textstyle\frac 1{3}}n^3\$ multiplications, where \$n\$ is the order of the
system. In this paper the author describes an ingenious method for
computing the product of two square matrices of order \$n\$ in fewer than
\$4.7n^{2.8}\$ arithmetic operations. Using this method, it is possible to
invert a non-singular matrix of order \$n\$ or compute its determinant in
less than \$6n^{2.8}\$ arithmetic operations.},
    author = {Strassen, Volker},
    citeulike-article-id = {836776},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/BF02165411},
    citeulike-linkout-1 = {http://www.ams.org/mathscinet-getitem?mr=248973},
    citeulike-linkout-2 = {http://www.springerlink.com/content/w71w4445t7m71gm5},
    day = {29},
    doi = {10.1007/BF02165411},
    issn = {0029-599X},
    journal = {Numerische Mathematik},
    keywords = {matrix\_multiplication},
    month = aug,
    mrnumber = {MR248973},
    number = {4},
    pages = {354--356},
    posted-at = {2010-07-17 19:30:42},
    priority = {1},
    publisher = {Springer Berlin / Heidelberg},
    title = {Gaussian elimination is not optimal},
    url = {http://dx.doi.org/10.1007/BF02165411},
    volume = {13},
    year = {1969}
}

@incollection{loebenberger+putzka09,
    abstract = {We use the specific structure of the inputs to the cofactorization step in the general number field sieve ({GNFS}) in order to optimize the runtime for the cofactorization step on a hardware cluster. An optimal distribution of bitlength-specific {ECM} modules is proposed and compared to existing ones. With our optimizations we obtain a speedup between 17\% and 33\% of the cofactorization step of the {GNFS} when compared to the runtime of an unoptimized cluster.},
    address = {Berlin, Heidelberg},
    author = {Loebenberger, Daniel and Putzka, Jens},
    booktitle = {Selected Areas in Cryptography},
    chapter = {11},
    citeulike-article-id = {7505918},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-642-05445-7_11},
    citeulike-linkout-1 = {http://www.springerlink.com/content/t3x7171014248317},
    doi = {10.1007/978-3-642-05445-7_11},
    editor = {Jacobson, Michael J. and Rijmen, Vincent and Safavi-Naini, Reihaneh},
    isbn = {978-3-642-05443-3},
    keywords = {factorization},
    pages = {170--181},
    posted-at = {2010-07-16 21:56:32},
    priority = {1},
    publisher = {Springer Berlin Heidelberg},
    title = {Optimization Strategies for {Hardware-Based} Cofactorization},
    url = {http://dx.doi.org/10.1007/978-3-642-05445-7_11},
    volume = {5867},
    year = {2009}
}

@article{vzgathen+shparlinski09,
    abstract = {doi: {10.1515/JMC}.2009.007 Abstract We show how to accelerate the subset sum pseudorandom number generator with arbitrary weights. Some special choices of weights speed up the naive usage of this generator without losing the property of uniform distribution which has recently been established in the general case. Our results confirm that this generator can be useful for both cryptographic and Quasi Monte Carlo applications.},
    author = {von zur Gathen, Joachim and Shparlinski, Igor E.},
    citeulike-article-id = {7505912},
    citeulike-linkout-0 = {http://www.reference-global.com/doi/abs/10.1515/JMC.2009.007},
    citeulike-linkout-1 = {http://dx.doi.org/10.1515/JMC.2009.007},
    day = {1},
    doi = {10.1515/JMC.2009.007},
    journal = {Journal of Mathematical Cryptology},
    keywords = {cryptography},
    month = aug,
    number = {2},
    pages = {149--163},
    posted-at = {2010-07-16 21:53:38},
    priority = {1},
    publisher = {De Gruyter},
    title = {Subset sum pseudorandom numbers: fast generation and distribution},
    url = {http://dx.doi.org/10.1515/JMC.2009.007},
    volume = {3},
    year = {2009}
}

@unpublished{loebenberger+nusken10-2,
    author = {Loebenberger, Daniel and N\"{u}sken, Michael},
    citeulike-article-id = {7505905},
    citeulike-linkout-0 = {http://cosec.bit.uni-bonn.de/fileadmin/user_upload/publications/pubs/loenus10a.pdf},
    keywords = {factorization},
    posted-at = {2010-07-16 21:45:05},
    priority = {1},
    title = {Notions of {RSA} integers},
    url = {http://cosec.bit.uni-bonn.de/fileadmin/user_upload/publications/pubs/loenus10a.pdf},
    year = {2010}
}

@article{faugere+lubicz+robert10,
    abstract = {The aim of this paper is to give a higher dimensional equivalent of theclassical modular polynomials \$\{Phi\_\ell(X},Y)\$. If \$j\$ is the \$j\$-invariantassociated to an elliptic curve \$E\_k\$ over a field \$k\$ then the roots {of\$\Phi\_}\{ell(j,X})\$ correspond to the \$j\$-invariants of the curves which are\$\ell\$-isogeneous to \$E\_k\$. Denote by \${X\_0(N})\$ the modular curve whichparametrizes the set of elliptic curves together with a \$N\$-torsion {subgroup.It} is possible to interpret \$\{Phi\_\ell(X},Y)\$ as an equation cutting out theimage of a certain modular correspondence \$X\_0(\ell) \to X\_0(1) \times X\_0(1)\$in the product \$X\_0(1) \times X\_0(1)\$. Let \$g\$ be a positive integer and\$\overn \in \N^g\$. We are interested in the moduli space that we denote {by\$\Mn}\$ of abelian varieties of dimension \$g\$ over a field \$k\$ together with anample symmetric line bundle \$\pol\$ and a symmetric theta structure of type\$\overn\$. If \$\ell\$ is a prime and let \$\overl=(\ell, ..., \ell)\$, there existsa modular correspondence \$\Mln \to \Mn \times \Mn\$. We give a system ofalgebraic equations defining the image of this modular correspondence.},
    archivePrefix = {arXiv},
    author = {Faug\`{e}re, Jean-{C}harles and Lubicz, David and Robert, Damien},
    citeulike-article-id = {7500483},
    citeulike-linkout-0 = {http://arxiv.org/abs/0910.4668},
    citeulike-linkout-1 = {http://arxiv.org/pdf/0910.4668},
    day = {24},
    eprint = {0910.4668},
    keywords = {algebraic\_geometry, hyperelliptic\_curves, isogenies},
    month = oct,
    posted-at = {2010-07-16 15:58:54},
    priority = {4},
    title = {Computing modular correspondences for abelian varieties},
    url = {http://arxiv.org/abs/0910.4668},
    year = {2009}
}

@article{lubicz+robert10,
    abstract = {We describe an efficient algorithm for the computation of separable isogenies
between abelian varieties represented in the coordinate system given by
algebraic theta functions. Let \$A\$ be an abelian variety of dimension \$g\$
defined over a field of odd characteristic. Our algorithm decomposes in two
principal steps. First, given a theta null point for \$A\$ and a subgroup \$K\$
isotropic for the Weil pairing, we explain how to compute the theta null point
corresponding to the quotient abelian variety \$A/K\$. Then, from the knowledge
of a theta null point of \$A/K\$, we give an algorithm to obtain a rational
expression for an isogeny from \$A\$ to \$A/K\$. The algorithm resulting as the
combination of these two steps can be viewed as a higher dimensional analog of
the well known algorithm of V\'elu to compute isogenies between elliptic
curves. In the case that \$K\$ is isomorphic to \$(\Z / \ell \Z)^g\$ for \$\ell \in
\N^*\$, the overall time complexity of this algorithm is equivalent to \$O(\log
\ell)\$ additions in \$A\$ and a constant number of \$\ell^{th}\$ root extractions
in the base field of \$A\$. In order to improve the efficiency of our algorithms,
we introduce a compressed representation that allows to encode a point of level
\$4\ell\$ of a \$g\$ dimensional abelian variety using only \$g(g+1)/2\cdot 4^g\$
coordinates. We also give formulas to compute the Weil and commutator pairings
given input points in theta coordinates.},
    archivePrefix = {arXiv},
    author = {Lubicz, David and Robert, Damien},
    citeulike-article-id = {7500481},
    citeulike-linkout-0 = {http://arxiv.org/abs/1001.2016},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1001.2016},
    day = {12},
    eprint = {1001.2016},
    keywords = {algebraic\_geometry, hyperelliptic\_curves, isogenies},
    month = mar,
    posted-at = {2010-07-16 15:57:35},
    priority = {2},
    title = {Computing isogenies between abelian varieties},
    url = {http://arxiv.org/abs/1001.2016},
    year = {2010}
}

@electronic{sutherland10,
    author = {Sutherland, Andrew V.},
    citeulike-article-id = {7499834},
    citeulike-linkout-0 = {#},
    howpublished = {Last accessed July 16, 2010. \url{http://www-math.mit.edu/\~drew/SEArecords.html}},
    keywords = {cryptography, elliptic\_curve, schoof},
    posted-at = {2010-07-16 13:25:40},
    priority = {0},
    title = {Genus 1 point counting over prime fields},
    url = {#},
    year = {2010}
}

@electronic{enge+morain06,
    author = {Enge, Andreas and Morain, Fran\c{c}ois},
    citeulike-article-id = {7499809},
    citeulike-linkout-0 = {http://www.lix.polytechnique.fr/~morain/SEA/d2500x.annonce},
    howpublished = {Last accessed July 16, 2010. \url{http://www.lix.polytechnique.fr/morain/SEA/d2500x.annonce}},
    keywords = {cryptography, elliptic\_curve, schoof},
    posted-at = {2010-07-16 13:19:50},
    priority = {2},
    title = {{SEA} point counting record: 2500 decimal digits},
    url = {http://www.lix.polytechnique.fr/~morain/SEA/d2500x.annonce},
    year = {2006}
}

@electronic{gaudry+schost08,
    author = {Gaudry, Pierrick and Schost, \'{E}ric},
    citeulike-article-id = {7499788},
    citeulike-linkout-0 = {http://webloria.loria.fr/~gaudry/record127/},
    howpublished = {Last accessed July 16, 2010. \url{http://webloria.loria.fr/\~gaudry/record127/}},
    keywords = {cryptography, hyperelliptic\_curves, schoof},
    posted-at = {2010-07-16 13:08:29},
    priority = {0},
    title = {Hyperelliptic curve point counting record: 254 bit {J}acobian},
    url = {http://webloria.loria.fr/~gaudry/record127/},
    year = {2008}
}

@article{lauder+wan06,
    abstract = {We present a deterministic polynomial time algorithm for computing the zetafunction of an arbitrary variety of fixed dimension over a finite field ofsmall characteristic. One consequence of this result is an efficient method forcomputing the order of the group of rational points on the Jacobian of a smoothgeometrically connected projective curve over a finite field of smallcharacteristic.},
    archivePrefix = {arXiv},
    author = {Lauder, Alan G. B. and Wan, Daqing},
    citeulike-article-id = {7493412},
    citeulike-linkout-0 = {http://arxiv.org/abs/math.NT/0612147},
    citeulike-linkout-1 = {http://arxiv.org/pdf/math.NT/0612147},
    day = {6},
    eprint = {math.NT/0612147},
    keywords = {hyperelliptic\_curves, p-adic},
    month = dec,
    posted-at = {2010-07-15 16:21:42},
    priority = {2},
    title = {Counting points on varieties over finite fields of small characteristic},
    url = {http://arxiv.org/abs/math.NT/0612147},
    year = {2006}
}

@article{lauder04,
    abstract = {We describe a method which may be used to compute the zeta function of an arbitrary {Artin-Schreier} cover of the projective line over a finite field. Specifically, for covers defined by equations of the form Z p − Z = f ( X ) we present, and give the complexity analysis of, an algorithm for the case in which f ( X ) is a rational function whose poles all have order 1. However, we only prove the correctness of this algorithm when the field characteristic is at least 5. The algorithm is based upon a cohomological formula for the L -function of an additive character sum. One consequence is a practical method of finding the order of the group of rational points on the Jacobian of a hyperelliptic curve in characteristic 2.},
    author = {Lauder, Alan G. B.},
    citeulike-article-id = {7493406},
    citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.jco.2003.08.009},
    doi = {10.1016/j.jco.2003.08.009},
    issn = {0885064X},
    journal = {Journal of Complexity},
    keywords = {artin-schreier, cryptography, hyperelliptic\_curves, p-adic},
    month = jun,
    number = {2-3},
    pages = {331--349},
    posted-at = {2010-07-15 16:18:27},
    priority = {4},
    title = {Computing zeta functions of Artin-Schreier curves over finite fields {II}},
    url = {http://dx.doi.org/10.1016/j.jco.2003.08.009},
    volume = {20},
    year = {2004}
}

@article{denef+vercauteren06,
    abstract = {Abstract   We present an algorithm to compute the zeta function of an arbitrary hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for odd characteristic. Given a genus g hyperelliptic curve defined over Fqn, the average-case time complexity is O(g4 + ε n3 + ε) and the average-case space complexity is O(g3 n3), whereas the worst-case time and space complexities are O(g5 + ε n3 + ε) and O(g4 n3), respectively.},
    author = {Denef, Jan and Vercauteren, Frederik},
    citeulike-article-id = {7493383},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s00145-004-0231-y},
    citeulike-linkout-1 = {http://www.springerlink.com/content/r79p2q816377j6vn},
    day = {6},
    doi = {10.1007/s00145-004-0231-y},
    issn = {0933-2790},
    journal = {Journal of Cryptology},
    keywords = {cryptography, finite\_field, hyperelliptic\_curves, p-adic},
    month = jan,
    number = {1},
    pages = {1--25},
    posted-at = {2010-07-15 16:09:57},
    priority = {4},
    title = {An Extension of {K}edlaya's Algorithm to Hyperelliptic Curves in Characteristic 2},
    url = {http://dx.doi.org/10.1007/s00145-004-0231-y},
    volume = {19},
    year = {2006}
}

@electronic{mestre02,
    author = {Mestre, Jean-Fran\c{c}ois and Lubicz, David},
    citeulike-article-id = {7493232},
    citeulike-linkout-0 = {http://www.math.jussieu.fr/~mestre/rennescrypto.ps},
    howpublished = {Last accessed July 15, 2010. \url{http://www.math.jussieu.fr/\~mestre/rennescrypto.ps}},
    keywords = {agm, hyperelliptic\_curves, p-adic},
    posted-at = {2010-07-15 15:45:41},
    priority = {2},
    title = {Algorithme pour compter des points de courbes en petite caract\'{e}ristique et petit genre},
    url = {http://www.math.jussieu.fr/~mestre/rennescrypto.ps},
    year = {2002}
}

@electronic{mestre00,
    author = {Mestre, Jean-Fran\c{c}ois},
    citeulike-article-id = {7493226},
    citeulike-linkout-0 = {http://www.math.jussieu.fr/~mestre/lettreGaudryHarley.ps},
    howpublished = {Last accessed July 15, 2010. \url{http://www.math.jussieu.fr/\~mestre/lettreGaudryHarley.ps}},
    keywords = {agm, hyperelliptic\_curves, p-adic},
    posted-at = {2010-07-15 15:43:59},
    priority = {2},
    title = {Lettre adress\'{e}e \`{a} {G}audry et {H}arley},
    url = {http://www.math.jussieu.fr/~mestre/lettreGaudryHarley.ps},
    year = {2000}
}

@incollection{adleman+huang96,
    abstract = {We develop efficient methods for deterministic computations with semi-algebraic sets and apply them to the problem of counting rational points on curves and abelian varieties over finite fields. For abelian varieties of dimension g in projective N space over Fq, we improve Pila's result and show that the problem can be solved in O((log q)\$\$O((\log q)^{O(g^6 )} )\$\$ time.},
    author = {Adleman, Leonard M. and Huang, Ming-Deh},
    booktitle = {Algorithmic Number Theory},
    citeulike-article-id = {7493154},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/3-540-61581-4_36},
    citeulike-linkout-1 = {http://www.springerlink.com/content/f70577004r335173},
    doi = {10.1007/3-540-61581-4_36},
    keywords = {hyperelliptic\_curves, schoof},
    pages = {1--16},
    posted-at = {2010-07-15 15:23:54},
    priority = {2},
    title = {Counting rational points on curves and abelian varieties over finite fields},
    url = {http://dx.doi.org/10.1007/3-540-61581-4_36},
    year = {1996}
}

@article{adleman+huang01,
    abstract = {We develop efficient methods for deterministic computations with semi-algebraic sets and apply them to the problem of counting points on curves and Abelian varieties over finite fields. For Abelian varieties of dimension g in projective N space over Fq, we improve Pila's result and show that the problem can be solved in O((q)) time where is polynomial in g as well as in N. For hyper elliptic curves of genus g over Fq we show that the number of rational points on the curve and the number of rational points on its Jacobian can be computed in ({q)O}(g2g)time.},
    address = {Duluth, MN, USA},
    author = {Adleman, Leonard M. and Huang, Ming D.},
    citeulike-article-id = {7493119},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=500681.500682},
    citeulike-linkout-1 = {http://dx.doi.org/10.1006/jsco.2001.0470},
    doi = {10.1006/jsco.2001.0470},
    issn = {0747-7171},
    journal = {J. Symb. Comput.},
    keywords = {hyperelliptic\_curves, schoof},
    month = sep,
    number = {3},
    pages = {171--189},
    posted-at = {2010-07-15 15:22:16},
    priority = {2},
    publisher = {Academic Press, Inc.},
    title = {Counting points on curves and Abelian varieties over finite fields},
    url = {http://dx.doi.org/10.1006/jsco.2001.0470},
    volume = {32},
    year = {2001}
}

@article{gaudry+schost04-3,
    author = {Gaudry, Pierrick and Schost, \'{E}ric},
    citeulike-article-id = {7492878},
    citeulike-linkout-0 = {http://dx.doi.org/10.1090/S0025-5718-04-01682-5},
    day = {25},
    doi = {10.1090/S0025-5718-04-01682-5},
    issn = {0025-5718},
    journal = {Math. Comp.},
    keywords = {cryptography, hyperelliptic\_curves},
    month = may,
    number = {249},
    pages = {429--455},
    posted-at = {2010-07-15 14:46:19},
    priority = {2},
    title = {Modular equations for hyperelliptic curves},
    url = {http://dx.doi.org/10.1090/S0025-5718-04-01682-5},
    volume = {74},
    year = {2004}
}

@article{lercier+lubicz06,
    abstract = {Abstract\&nbsp;\&nbsp;We describe an algorithm to compute the cardinality of Jacobians of ordinary hyperelliptic curves of small genus over finite fields \$\${\cal F}\_{2^n}\$\$ with cost \$\$O(n^{2+o(1)})\$\$ . This algorithm is derived from ideas due to Mestre. More precisely, we state the mathematical background behind Mestre's algorithm and develop from it a variant with quasi-quadratic time complexity. Among others, we present an algorithm to find roots of a system of generalized Artin-Schreier equations and give results that we obtain with an efficient implementation. Especially, we were able to obtain the cardinality of curves of genus one, two or three in finite fields of huge size.},
    author = {Lercier, Reynald and Lubicz, David},
    citeulike-article-id = {1053038},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s11139-006-0151-6},
    citeulike-linkout-1 = {http://www.ingentaconnect.com/content/klu/rama/2006/00000012/00000003/00000151},
    citeulike-linkout-2 = {http://www.springerlink.com/content/v088708582g28464},
    day = {1},
    doi = {10.1007/s11139-006-0151-6},
    issn = {1382-4090},
    journal = {The Ramanujan Journal},
    keywords = {agm, hyperelliptic\_curves, p-adic},
    month = dec,
    number = {3},
    pages = {399--423},
    posted-at = {2010-07-15 14:26:51},
    priority = {4},
    publisher = {Springer},
    title = {A quasi quadratic time algorithm for hyperelliptic curve point counting},
    url = {http://dx.doi.org/10.1007/s11139-006-0151-6},
    volume = {12},
    year = {2006}
}

@incollection{gaudry+harley00,
    address = {Berlin},
    author = {Gaudry, Pierrick and Harley, Robert},
    booktitle = {Algorithmic number theory (Leiden, 2000)},
    citeulike-article-id = {2145791},
    keywords = {cryptography, hyperelliptic\_curves},
    mrnumber = {MR1850614 (2002f:11072)},
    pages = {313--332},
    posted-at = {2010-07-15 14:12:30},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes in Comput. Sci.},
    title = {Counting points on hyperelliptic curves over finite fields},
    volume = {1838},
    year = {2000}
}

@incollection{gaudry02,
    address = {Berlin},
    author = {Gaudry, Pierrick},
    booktitle = {Advances in cryptology---ASIACRYPT 2002},
    citeulike-article-id = {2146164},
    keywords = {cryptography, hyperelliptic\_curves},
    mrnumber = {MR2087393 (2005h:11124)},
    pages = {311--327},
    posted-at = {2010-07-15 13:54:09},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes in Comput. Sci.},
    title = {A comparison and a combination of {SST} and {AGM} algorithms for counting points of elliptic curves in characteristic 2},
    volume = {2501},
    year = {2002}
}

@incollection{bostan+gaudry+schost04,
    address = {Berlin},
    author = {Bostan, Alin and Gaudry, Pierrick and Schost, \'{E}ric},
    booktitle = {Finite fields and applications},
    citeulike-article-id = {2146165},
    keywords = {cryptography, hyperelliptic\_curves},
    pages = {40--58},
    posted-at = {2010-07-15 13:53:40},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes in Comput. Sci.},
    title = {Linear recurrences with polynomial coefficients and computation of the {C}artier-{M}anin operator on hyperelliptic curves},
    volume = {2948},
    year = {2004}
}

@incollection{gaudry+schost04-2,
    address = {Berlin},
    author = {Gaudry, Pierrick and Schost, \'{E}ric},
    booktitle = {Algorithmic number theory},
    citeulike-article-id = {2146166},
    keywords = {cryptography, hyperelliptic\_curves},
    pages = {208--222},
    posted-at = {2010-07-15 13:53:08},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes in Comput. Sci.},
    title = {A low-memory parallel version of {M}atsuo, {C}hao, and {T}sujii's algorithm},
    volume = {3076},
    year = {2004}
}

@incollection{gaudry+schost04,
    address = {Berlin},
    author = {Gaudry, Pierrick and Schost, \'{E}ric},
    booktitle = {Advances in cryptology---EUROCRYPT 2004},
    citeulike-article-id = {2146168},
    keywords = {cryptography, hyperelliptic\_curves},
    pages = {239--256},
    posted-at = {2010-07-15 13:52:41},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes in Comput. Sci.},
    title = {Construction of secure random curves of genus 2 over prime fields},
    volume = {3027},
    year = {2004}
}

@incollection{kedlaya04,
    address = {Berlin},
    author = {Kedlaya, Kiran S.},
    booktitle = {Algorithmic number theory},
    citeulike-article-id = {2145969},
    keywords = {cryptography, hyperelliptic\_curves, p-adic},
    pages = {1--17},
    posted-at = {2010-07-15 13:27:56},
    priority = {4},
    publisher = {Springer},
    series = {Lecture Notes in Comput. Sci.},
    title = {Computing zeta functions via \(p\)-adic cohomology},
    volume = {3076},
    year = {2004}
}

@article{castryck+denef+vercauteren07,
    abstract = {Abstract. In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using {Monsky-Washnitzer} cohomology. The paper vastly generalizes previous work since in practice all known cases, e.g. hyperelliptic, superelliptic and Cab curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over Fpn, the expected running time {is�O}(n 3 g 6 + n 2 g 6.5), whereas the space complexity amounts {to�O}(n 3 g 4), assuming p is fixed. Keywords: nondegenerate curves, zeta function, {Monsky-Washnitzer} cohomology, Kedlaya's algorithm, Newton polytope, toric geometry, effective Nullstellensatz 1},
    author = {Castryck, W. and Denef, J. and Vercauteren, F.},
    citeulike-article-id = {7491929},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.69.5579},
    journal = {Intl. Math. Res. Notices},
    keywords = {cryptography, hyperelliptic\_curves, p-adic},
    posted-at = {2010-07-15 13:23:01},
    priority = {4},
    title = {Computing zeta functions of nondegenerate curves},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.69.5579},
    volume = {72017},
    year = {2007}
}

@article{harrison10,
    abstract = {In this paper we describe a generalisation and adaptation of Kedlaya's
algorithm for computing the zeta function of a hyperelliptic curve over a
finite field of odd characteristic that the author used for his implementation
of the algorithm in Magma. We generalise the algorithm to the case of an even
degree model. We also analyse the adaptation of working with the \$x^idx/y^3\$
rather than the \$x^idx/y\$ differential basis. This basis has the computational
advantage of always leading to an integral transformation matrix whereas the
latter fails to in small genus cases. There are some theoretical subtleties
that arise in the even degree case where the two differential bases actually
lead to different redundant eigenvalues that must be discarded.},
    archivePrefix = {arXiv},
    author = {Harrison, Michael C.},
    citeulike-article-id = {7491847},
    citeulike-linkout-0 = {http://arxiv.org/abs/1006.4206},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1006.4206},
    day = {22},
    eprint = {1006.4206},
    keywords = {cryptography, hyperelliptic\_curves, p-adic},
    month = jun,
    posted-at = {2010-07-15 12:59:52},
    priority = {4},
    title = {Some notes on Kedlaya's algorithm for hyperelliptic curves},
    url = {http://arxiv.org/abs/1006.4206},
    year = {2010}
}

@article{loebenberger+nusken10,
    abstract = {We count ]{B,C}]-grained (ie. B-rough and C-smooth) integers. Our aim is to
exploit explicit versions of the prime number theorem as much as possible to
get good explicit bounds for the count of ]{B,C}]-grained integers. This analysis
was inspired by certain inner procedures in the general number field sieve. The
result should at least provide some insight in what happens there. We estimate
the given count in terms of some recursively defined functions. Since they are
still difficult to handle, only another approximation step reveals their
orders. Finally, we use the obtained bounds to visually show how good the
desired count can be approximated for the parameters of the general number
field sieve in the mentioned inspiring application.},
    archivePrefix = {arXiv},
    author = {Loebenberger, Daniel and N\"{u}sken, Michael},
    citeulike-article-id = {7479856},
    citeulike-linkout-0 = {http://arxiv.org/abs/1003.2165},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1003.2165},
    day = {10},
    eprint = {1003.2165},
    keywords = {cryptography, factorization},
    month = mar,
    posted-at = {2010-07-14 23:41:04},
    priority = {1},
    title = {Coarse-grained integers Smooth? Rough? Both!},
    url = {http://arxiv.org/abs/1003.2165},
    year = {2010}
}

@article{pila90,
    abstract = {We give a generalization to Abelian varieties over finite fields of the algorithm of Schoof for elliptic curves. Schoof showed that for an elliptic curve \$E\$ over \$\mathbf{F}\_q\$, given by a Weierstrass equation, one can compute the number of \$\mathbf{F}\_q\$-rational points of \$E\$ in time \$O((\log q)^9)\$. Our result is the following. Let \$A\$ be an Abelian variety over \$\mathbf{F}\_q\$. Then one can compute the characteristic polynomial of the Frobenius endomorphism of \$A\$ in time \$O((\log q)^\Delta)\$, where \$\Delta\$ and the implied constant depend only on the dimension of the embedding space of \$A\$, the number of equations defining \$A\$ and the addition law, and their degrees. The method, generalizing that of Schoof, is to use the machinery developed by Weil to prove the Riemann hypothesis for Abelian varieties. By means of this theory, the calculation is reduced to ideal-theoretic computations in a ring of polynomials in several variables over \$\mathbf{F}\_q\$. As applications we show how to count the rational points on the reductions modulo primes \$p\$ of a fixed curve over \$\mathbf {Q}\$ in time polynomial in \$\log p\$; we show also that, for a fixed prime \$l\$, we can compute the \$l\$th roots of unity \$\operatorname{mod} p\$, when they exist, in polynomial time in \$\log p\$. This generalizes Schoof's application of his algorithm to find square roots of a fixed integer \$x \operatorname{mod} p\$.},
    author = {Pila, Jonathan},
    citeulike-article-id = {7478312},
    citeulike-linkout-0 = {http://dx.doi.org/10.2307/2008445},
    citeulike-linkout-1 = {http://www.jstor.org/stable/2008445},
    doi = {10.2307/2008445},
    issn = {00255718},
    journal = {Math. Comp.},
    keywords = {hyperelliptic\_curves, schoof},
    number = {192},
    pages = {745--763},
    posted-at = {2010-07-14 19:34:29},
    priority = {4},
    publisher = {American Mathematical Society},
    title = {Frobenius Maps of {A}belian Varieties and Finding Roots of Unity in Finite Fields},
    url = {http://dx.doi.org/10.2307/2008445},
    volume = {55},
    year = {1990}
}

@article{kedlaya01,
    abstract = {In this paper an algorithm for counting points on hyperelliptic curvesover finite fields of odd characteristic is developed.<P> The approach is to compute in the Monsky-Washnitzer cohomology,described in sections 2 and 3, of an affine curve endowed with anaction of Frobenius which can be \$p\$-adically approximated efficientlyusing certain power series.  The running time of the algorithm is onthe order of \$g^{4+\epsilon}n^{3+\epsilon}\$, where \$n\$ is the degreeof the finite field and \$g\$ is the genus of the curve (assuming thatthe curve has a rational Weierstrass point).},
    author = {Kedlaya, Kiran S.},
    citeulike-article-id = {805320},
    citeulike-linkout-0 = {http://www.ams.org/mathscinet-getitem?mr=1877805},
    journal = {Journal of the Ramanujan Mathematical Society},
    keywords = {hyperelliptic\_curves, p-adic},
    mrnumber = {MR1877805},
    number = {4},
    pages = {323--338},
    posted-at = {2010-07-14 19:30:36},
    priority = {4},
    title = {Counting points on hyperelliptic curves using {M}onsky-{W}ashnitzer cohomology},
    url = {http://www.ams.org/mathscinet-getitem?mr=1877805},
    volume = {16},
    year = {2001}
}

@electronic{nsasuiteb,
    author = {{NSA}},
    citeulike-article-id = {7477942},
    citeulike-linkout-0 = {http://www.nsa.gov/ia/programs/suiteb_cryptography/index.shtml},
    howpublished = {Last accessed 14-Jul-2010. \url{http://www.nsa.gov/ia/programs/suiteb\_cryptography/index.shtml}},
    institution = {National Security Agency},
    keywords = {cryptography},
    organization = {National Security Agency},
    posted-at = {2010-07-14 18:47:44},
    priority = {2},
    publisher = {National Security Agency},
    title = {NSA Suite {B} Cryptography},
    url = {http://www.nsa.gov/ia/programs/suiteb_cryptography/index.shtml},
    year = {2005}
}

@article{gaudry07,
    abstract = {doi: {10.1515/JMC}.2007.012 In 1986, D. V. Chudnovsky and G. V. Chudnovsky proposed to use formulae coming from Theta functions for the arithmetic in Jacobians of genus 2 curves. We follow this idea and derive fast formulae for the scalar multiplication in the Kummer surface associated to a genus 2 curve, using a Montgomery ladder. Our formulae can be used to design very efficient genus 2 cryptosystems that should be faster than elliptic curve cryptosystems in some hardware configurations.},
    author = {Gaudry, P.},
    citeulike-article-id = {7477800},
    citeulike-linkout-0 = {http://www.reference-global.com/doi/abs/10.1515/JMC.2007.012},
    citeulike-linkout-1 = {http://dx.doi.org/10.1515/JMC.2007.012},
    day = {1},
    doi = {10.1515/JMC.2007.012},
    journal = {Journal of Mathematical Cryptology},
    keywords = {cryptography, genus\_2, hyperelliptic\_curves, kummer, montgomery\_curve},
    month = aug,
    number = {3},
    pages = {243--265},
    posted-at = {2010-07-14 18:09:26},
    priority = {2},
    publisher = {De Gruyter},
    title = {Fast genus 2 arithmetic based on Theta functions},
    url = {http://dx.doi.org/10.1515/JMC.2007.012},
    volume = {1},
    year = {2007}
}

@article{edwards07,
    author = {Edwards, Harold M.},
    booktitle = {Bulletin of the American Mathematical Society},
    citeulike-article-id = {3474980},
    journal = {Bulletin of the American Mathematical Society},
    keywords = {edwards, elliptic\_curve},
    month = jul,
    number = {3},
    pages = {393--422},
    posted-at = {2010-07-14 17:12:30},
    priority = {2},
    title = {A Normal Form for Elliptic Curves},
    volume = {44},
    year = {2007}
}

@electronic{bernstein+birkner+lange+peters08-2,
    abstract = {This paper introduces {GMP}-{EECM}, a fast implementation of the elliptic-curve   method of factoring integers.   {GMP}-{EECM} is based on, but faster than, the well-known {GMP}-{ECM} software.   The main changes are as follows:   (1) use Edwards curves instead of Montgomery curves;   (2) use twisted inverted Edwards coordinates;   (3) use signed-sliding-window addition chains;   (4) batch primes to increase the window size;   (5) choose curves with small parameters \${a,d,X\_1},{Y\_1,Z\_1}\$;   (6) choose curves with larger torsion.},
    author = {Bernstein, Daniel J. and Birkner, Peter and Lange, Tanja and Peters, Christiane},
    citeulike-article-id = {3474774},
    citeulike-linkout-0 = {http://eprint.iacr.org/2008/016},
    day = {9},
    howpublished = {Cryptology ePrint Archive, Report 2008/016},
    keywords = {edwards, elliptic\_curve, elliptic\_curve\_method},
    month = jan,
    posted-at = {2010-07-14 17:10:29},
    priority = {2},
    title = {{ECM} using {E}dwards curves},
    url = {http://eprint.iacr.org/2008/016},
    year = {2008}
}

@incollection{bernstein+birkner+lange+peters08,
    abstract = {This paper introduces  ” twisted Edwards curves,” a generalization of the recently introduced Edwards curves; shows that twisted Edwards curves include more curves over finite fields, and in particular every elliptic curve in Montgomery form; shows how to cover even more curves via isogenies; presents fast explicit formulas for twisted Edwards curves in projective and inverted coordinates; and shows that twisted Edwards curves save time for many curves that were already expressible as Edwards curves.},
    author = {Bernstein, Daniel and Birkner, Peter and Joye, Marc and Lange, Tanja and Peters, Christiane},
    citeulike-article-id = {3474768},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-68164-9_26},
    doi = {10.1007/978-3-540-68164-9_26},
    journal = {Progress in Cryptology – AFRICACRYPT 2008},
    keywords = {cryptography, edwards, elliptic\_curve},
    pages = {389--405},
    posted-at = {2010-07-14 17:09:47},
    priority = {2},
    title = {Twisted Edwards Curves},
    url = {http://dx.doi.org/10.1007/978-3-540-68164-9_26},
    year = {2008}
}

@incollection{bernstein+lange08,
    abstract = {Edwards recently introduced a new normal form for elliptic curves. Every elliptic curve over a non-binary field is birationally equivalent to a curve in Edwards form over an extension of the field, and in many cases over the original field. This paper presents fast explicit formulas (and register allocations) for group operations on an Edwards curve. The algorithm for doubling uses only {3M} + {4S}, i.e., 3 field multiplications and 4 field squarings. If curve parameters are chosen to be small then the algorithm for mixed addition uses only {9M} + {1S} and the algorithm for non-mixed addition uses only {10M} + {1S}. Arbitrary Edwards curves can be handled at the cost of just one extra multiplication by a curve parameter. For comparison, the fastest algorithms known for the popular  ” a4 = −3 Jacobian” form use {3M} + {5S} for doubling; use {7M} + {4S} for mixed addition; use {11M} + {5S} for non-mixed addition; and use {10M} + {4S} for non-mixed addition when one input has been added before. The explicit formulas for non-mixed addition on an Edwards curve can be used for doublings at no extra cost, simplifying protection against side-channel attacks. Even better, many elliptic curves (approximately 1/4 of all isomorphism classes of elliptic curves over a non-binary finite field) are birationally equivalent — over the original field — to Edwards curves where this addition algorithm works for all pairs of curve points, including inverses, the neutral element, etc. This paper contains an extensive comparison of different forms of elliptic curves and different coordinate systems for the basic group operations (doubling, mixed addition, non-mixed addition, and unified addition) as well as higher-level operations such as multi-scalar multiplication.},
    author = {Bernstein, Daniel and Lange, Tanja},
    citeulike-article-id = {3474787},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-76900-2_3},
    doi = {10.1007/978-3-540-76900-2_3},
    journal = {Advances in Cryptology – ASIACRYPT 2007},
    keywords = {cryptography, edwards, elliptic\_curve},
    pages = {29--50},
    posted-at = {2010-07-14 16:02:56},
    priority = {2},
    title = {Faster Addition and Doubling on Elliptic Curves},
    url = {http://dx.doi.org/10.1007/978-3-540-76900-2_3},
    year = {2008}
}

@article{citeulike:3890931,
    abstract = {These are lecture notes that arose from a representation theory course given
by the first author to the remaining six authors in March 2004 within the
framework of the Clay Mathematics Institute Research Academy for high school
students, and its extended version given by the first author to {MIT}
undergraduate math students in the Fall of 2008. The notes cover a number of
standard topics in representation theory of groups, Lie algebras, and quivers,
and contain many problems and exercises. They should be accessible to students
with a strong background in linear algebra and a basic knowledge of abstract
algebra, and may be used for an undergraduate or introductory graduate course
in representation theory.},
    archivePrefix = {arXiv},
    author = {Etingof, Pavel and Golberg, Oleg and Hensel, Sebastian and Liu, Tiankai and Schwendner, Alex and Vaintrob, Dmitry and Yudovina, Elena},
    citeulike-article-id = {3890931},
    citeulike-linkout-0 = {http://arxiv.org/abs/0901.0827},
    citeulike-linkout-1 = {http://arxiv.org/pdf/0901.0827},
    day = {1},
    eprint = {0901.0827},
    keywords = {books, representation\_theory},
    month = feb,
    posted-at = {2010-07-14 13:18:14},
    priority = {2},
    title = {Introduction to representation theory},
    url = {http://arxiv.org/abs/0901.0827},
    year = {2011}
}

@article{koblitz89,
    abstract = {In this paper we discuss a source of finite abelian groups suitable for cryptosystems based on the presumed intractability of the discrete logarithm problem for these groups. They are the jacobians of hyperelliptic curves defined over finite fields. Special attention is given to curves defined over the field of two elements. Explicit formulas and examples are given, and the problem of finding groups of almost prime order is discussed.},
    author = {Koblitz, Neal},
    citeulike-article-id = {7471127},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/BF02252872},
    citeulike-linkout-1 = {http://www.springerlink.com/content/f04957m783657u38},
    day = {23},
    doi = {10.1007/BF02252872},
    issn = {0933-2790},
    journal = {Journal of Cryptology},
    keywords = {cryptosystems},
    month = oct,
    number = {3},
    pages = {139--150},
    posted-at = {2010-07-13 18:22:29},
    priority = {2},
    title = {Hyperelliptic cryptosystems},
    url = {http://dx.doi.org/10.1007/BF02252872},
    volume = {1},
    year = {1989}
}

@article{koblitz87,
    author = {Koblitz, Neal},
    citeulike-article-id = {2405523},
    citeulike-linkout-0 = {http://www.jstor.org/stable/2007884},
    journal = {Math. Comp.},
    keywords = {cryptography},
    number = {177},
    pages = {203--209},
    posted-at = {2010-07-13 18:02:46},
    priority = {2},
    title = {Elliptic Curve Cryptosystems},
    url = {http://www.jstor.org/stable/2007884},
    volume = {48},
    year = {1987}
}

@article{elgamal,
    abstract = {A new signature scheme Is proposed together with an implementation of the Diffie - Hellman key distribution scheme that achieves a public key cryptosystem. The security of both systems relies on the difficulty of computing discrete logarithms over finite fields.},
    address = {New York, NY, USA},
    author = {ElGamal, Taher},
    booktitle = {Proceedings of CRYPTO 84 on Advances in cryptology},
    citeulike-article-id = {5722558},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=19480},
    isbn = {0-387-15658-5},
    journal = {IEEE transactions on information theory},
    keywords = {cryptography},
    location = {Santa Barbara, California, United States},
    number = {4},
    pages = {10--18},
    posted-at = {2010-07-13 17:53:41},
    priority = {2},
    publisher = {Institute of Electrical and Electronics Engineers},
    title = {A public key cryptosystem and a signature scheme based on discrete logarithms},
    url = {http://portal.acm.org/citation.cfm?id=19480},
    volume = {31},
    year = {1985}
}

@article{dh,
    abstract = {Two kinds of contemporary developments in cryptography are examined. Widening applications of teleprocessing have given rise to a need for new types of cryptographic systems, which minimize the need for secure key distribution channels and supply the equivalent of a written signature. This paper suggests ways to solve these currently open problems. It also discusses how the theories of communication and computation are beginning to provide the tools to solve cryptographic problems of long...},
    author = {Diffie, Whitfield and Hellman, Martin E.},
    citeulike-article-id = {918759},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.9720},
    journal = {IEEE Transactions on Information Theory},
    keywords = {cryptography},
    number = {6},
    pages = {644--654},
    posted-at = {2010-07-13 17:49:51},
    priority = {2},
    title = {New Directions in Cryptography},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.9720},
    volume = {IT-22},
    year = {1976}
}

@article{rsa,
    abstract = {An encryption method is presented with the novel property that publicly revealing an encryption key does not thereby reveal the corresponding decryption key. This has two important consequences: (1) Couriers or other secure means are not needed to transmit keys, since a message can be enciphered using an encryption key publicly revealed by the intented recipient. Only he can decipher the message, since only he knows the corresponding decryption key. (2) A message can be  ” signed” using a privately held decryption key. Anyone can verify this signature using the corresponding publicly revealed encryption key. Signatures cannot be forged, and a signer cannot later deny the validity of his signature. This has obvious applications in  ” electronic mail” and  ” electronic funds transfer” systems. A message is encrypted by representing it as a number M, raising M to a publicly specified power e, and then taking the remainder when the result is divided by the publicly specified product, n, of two large secret primer numbers p and q. Decryption is similar; only a different, secret, power d is used, where e * d ≡ 1(mod (p - 1) * (q - 1)). The security of the system rests in part on the difficulty of factoring the published divisor, n.},
    address = {New York, NY, USA},
    author = {Rivest, R. L. and Shamir, A. and Adleman, L.},
    citeulike-article-id = {994856},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=359342},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/359340.359342},
    day = {1},
    doi = {10.1145/359340.359342},
    issn = {0001-0782},
    journal = {Commun. ACM},
    keywords = {cryptography},
    month = feb,
    number = {2},
    pages = {120--126},
    posted-at = {2010-07-13 17:46:50},
    priority = {2},
    publisher = {ACM},
    title = {A method for obtaining digital signatures and public-key cryptosystems},
    url = {http://dx.doi.org/10.1145/359340.359342},
    volume = {21},
    year = {1978}
}

@inbook{wadler89,
    abstract = {From the type of a polymorphic function we can derive
a theorem that it satisfies. Every function of the
same type satisfies the same theorem. This provides
a free source of useful theorems, courtesy of Reynolds'
abstraction theorem for the polymorphic lambda calculus.

1 Introduction

Write down the definition of a polymorphic function on
a piece of paper. Tell me its type, but be careful not
to let me see the function's definition. I will tell you a
theorem that the function satisfies.
The...},
    address = {New York},
    author = {Wadler, Philip},
    booktitle = {Proceedings 4th Int.\ Conf.\ on Funct.\ Prog.\ Languages and Computer Arch., FPCA'89, London, UK, 11--13 Sept 1989},
    citeulike-article-id = {70811},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.38.9875},
    keywords = {functional\_language, types},
    pages = {347--359},
    posted-at = {2010-07-13 12:42:36},
    priority = {0},
    publisher = {ACM Press},
    title = {Theorems for Free!},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.38.9875},
    year = {1989}
}

@inproceedings{consel+danvy93,
    author = {Consel, Charles and Danvy, Olivier},
    booktitle = {POPL '93: Proceedings of the 20th ACM SIGPLAN-SIGACT symposium on Principles of programming languages},
    citeulike-article-id = {86986},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=158707},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/158511.158707},
    doi = {10.1145/158511.158707},
    isbn = {0897915607},
    keywords = {partial\_evaluation},
    pages = {493--501},
    posted-at = {2010-07-12 00:14:02},
    priority = {2},
    publisher = {ACM Press},
    title = {Tutorial notes on partial evaluation},
    url = {http://dx.doi.org/10.1145/158511.158707},
    year = {1993}
}

@incollection{riazanov+voronkov04,
    abstract = {Simplification orderings on terms play a crucial role in reducing the search space in paramodulation-based theorem proving. Such a use of orderings requires checking simple ordering constraints on substitutions as an essential part of many operations. Due to their frequency, such checks are costly and are a good target for optimisation. In this paper we present an efficient implementation technique for checking constraints in one of the most widely used simplification orderings, the {Knuth-Bendix} ordering. The technique is based on the idea of run-time algorithm specialisation, which is a close relative of partial evaluation.},
    address = {Berlin, Heidelberg},
    author = {Riazanov, Alexandre and Voronkov, Andrei},
    booktitle = {Automated Reasoning},
    chapter = {3},
    citeulike-article-id = {7461067},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-25984-8_3},
    citeulike-linkout-1 = {http://www.springerlink.com/content/klg9c7rw2k9kn08v},
    doi = {10.1007/978-3-540-25984-8_3},
    editor = {Basin, David and Rusinowitch, Michael},
    isbn = {978-3-540-22345-0},
    keywords = {partial\_evaluation, specialisation},
    pages = {60--74},
    posted-at = {2010-07-11 23:53:39},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes in Computer Science},
    title = {Efficient Checking of Term Ordering Constraints},
    url = {http://dx.doi.org/10.1007/978-3-540-25984-8_3},
    volume = {3097},
    year = {2004}
}

@article{okasaki99,
    abstract = {Square matrices serve as an interesting case study in functional programming. Common representations, such as lists of lists, are both inefficient---at least for access to individual elements---and error-prone, because the compiler cannot enforce "squareness". Switching to a typical balanced-tree representation solves the first problem, but not the second. We develop a representation that solves both problems: it offers logarithmic access to each individual element and it captures the shape invariants in the type, where they can be checked by the compiler. One interesting feature of our solution is that it translates the well-known fast exponentiation algorithm to the level of types. Our implementation also provides a stress test for today's advanced type systems---it uses nested types, polymorphic recursion, higher-order kinds, and rank-2 polymorphism.},
    address = {New York, NY, USA},
    author = {Okasaki, Chris},
    booktitle = {ICFP '99: Proceedings of the fourth ACM SIGPLAN international conference on Functional programming},
    citeulike-article-id = {1857246},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=317781},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/317636.317781},
    doi = {10.1145/317636.317781},
    issn = {0362-1340},
    journal = {SIGPLAN Not.},
    keywords = {functional\_language, haskell, matrix\_multiplication},
    month = sep,
    pages = {28--35},
    posted-at = {2010-07-09 18:12:42},
    priority = {0},
    publisher = {ACM},
    title = {From fast exponentiation to square matrices: an adventure in types},
    url = {http://dx.doi.org/10.1145/317636.317781},
    volume = {34},
    year = {1999}
}

@article{mcbride01,
    abstract = {Dependent types reflect the fact that validity of data is often a relative notion by allowingprior data to affect the types of subsequent data. Not only does this make for a precise typesystem, but also a highly generic one: both the type and the program for each instance ofa family of operations can be computed from the data which codes for that {instance.Recent} experimental extensions to the Haskell type class mechanism give us strong toolsto relativize types to other types. We may...},
    author = {McBride, Connor},
    citeulike-article-id = {4850},
    citeulike-linkout-0 = {http://citeseer.ist.psu.edu/mcbride01faking.html},
    citeulike-linkout-1 = {http://citeseer.lcs.mit.edu/mcbride01faking.html},
    citeulike-linkout-2 = {http://citeseer.ifi.unizh.ch/mcbride01faking.html},
    citeulike-linkout-3 = {http://citeseer.comp.nus.edu.sg/mcbride01faking.html},
    journal = {Journal of functional programming},
    keywords = {dependent\_types, functional\_language, haskell},
    number = {4-5},
    pages = {375--392},
    posted-at = {2010-07-09 18:08:34},
    priority = {0},
    publisher = {Cambridge University Press},
    title = {Faking it---simulating dependent types in {H}askell},
    url = {http://citeseer.ist.psu.edu/mcbride01faking.html},
    volume = {12},
    year = {2003}
}

@inproceedings{kiselyov+lammel+schupke04,
    abstract = {A heterogeneous collection is a datatype that is capable of storing data of different types, while providing operations for look-up, update, iteration, and others. There are various kinds of heterogeneous collections, differing in representation, invariants, and access operations. We describe {HList} --- a Haskell library for strongly typed heterogeneous collections including extensible records. We illustrate {HList}'s benefits in the context of type-safe database access in Haskell. The {HList} library relies on common extensions of Haskell 98. Our exploration raises interesting issues regarding Haskell's type system, in particular, avoidance of overlapping instances, and reification of type equality and type unification.},
    address = {New York, NY, USA},
    author = {Kiselyov, Oleg and L\"{a}mmel, Ralf and Schupke, Keean},
    booktitle = {Haskell '04: Proceedings of the ACM SIGPLAN workshop on Haskell},
    citeulike-article-id = {168497},
    citeulike-linkout-0 = {http://homepages.cwi.nl/~ralf/HList/},
    citeulike-linkout-1 = {http://portal.acm.org/citation.cfm?id=1017488},
    citeulike-linkout-2 = {http://dx.doi.org/10.1145/1017472.1017488},
    doi = {10.1145/1017472.1017488},
    isbn = {1581138504},
    keywords = {depend, dependent\_types, functional\_language, haske, haskell},
    pages = {96--107},
    posted-at = {2010-07-09 16:36:57},
    priority = {0},
    publisher = {ACM Press},
    title = {Strongly typed heterogeneous collections},
    url = {http://homepages.cwi.nl/~ralf/HList/},
    year = {2004}
}

@incollection{carette+kiselyov+shan09,
    abstract = {We have built the first family of tagless interpretations for a higher-order typed object language in a typed metalanguage (Haskell or {ML}) that require no dependent types, generalized algebraic data types, or postprocessing to eliminate tags. The statically type-preserving interpretations include an evaluator, a compiler (or staged evaluator), a partial evaluator, and call-by-name and call-by-value {CPS} transformers. Our main idea is to encode {HOAS} de Bruijn or higher-order abstract syntax using cogen functions rather than data constructors. In other words, we represent object terms not in an initial algebra but using the coalgebraic structure of the λ -calculus. Our representation also simulates inductive maps from types to types, which are required for typed partial evaluation and {CPS} transformations. Our encoding of an object term abstracts over the various ways to interpret it, yet statically assures that the interpreters never get stuck. To achieve self hyp interpretation and show Jones hyp optimality, we relate this exemplar of higher-rank and higher-kind polymorphism (provided by {ML} functors and Haskell 98 constructor classes) to plugging a term into a context of let hyp polymorphic bindings.},
    address = {Berlin, Heidelberg},
    author = {Carette, Jacques and Kiselyov, Oleg and Shan, Chung-chieh and Carette, Jacques and Kiselyov, Oleg and Shan, Chung-chieh},
    booktitle = {Programming Languages and Systems},
    chapter = {15},
    citeulike-article-id = {2703232},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-76637-7_15},
    citeulike-linkout-1 = {http://www.springerlink.com/content/346745075694944p},
    doi = {10.1007/978-3-540-76637-7_15},
    editor = {Shao, Zhong and Shao, Zhong},
    isbn = {978-3-540-76636-0},
    journal = {Programming Languages and Systems},
    keywords = {algebraic\_data\_types, functional\_language, programming\_languages, types},
    pages = {222--238},
    posted-at = {2010-07-09 10:49:09},
    priority = {5},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Finally Tagless, Partially Evaluated Programming Languages and Systems},
    url = {http://dx.doi.org/10.1007/978-3-540-76637-7_15},
    volume = {4807},
    year = {2007}
}

@article{carette+kiselyov+shan09:jfp,
    abstract = {We have built the first family of tagless interpretations for a higher-order typed object language in a typed metalanguage (Haskell or {ML}) that require no dependent types, generalized algebraic data types, or postprocessing to eliminate tags. The statically type-preserving interpretations include an evaluator, a compiler (or staged evaluator), a partial evaluator, and call-by-name and call-by-value continuation-passing style ({CPS}) transformers. Our principal technique is to encode de Bruijn or higher-order abstract syntax using combinator functions rather than data constructors. In other words, we represent object terms not in an initial algebra but using the coalgebraic structure of the \^{I}»-calculus. Our representation also simulates inductive maps from types to types, which are required for typed partial evaluation and {CPS} transformations. Our encoding of an object term abstracts uniformly over the family of ways to interpret it, yet statically assures that the interpreters never get stuck. This family of interpreters thus demonstrates again that it is useful to abstract over higher-kinded types.},
    author = {Carette, Jacques and Kiselyov, Oleg and Shan, Chung C.},
    citeulike-article-id = {7066747},
    citeulike-linkout-0 = {http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6171376},
    citeulike-linkout-1 = {http://dx.doi.org/10.1017/S0956796809007205},
    doi = {10.1017/S0956796809007205},
    journal = {Journal of Functional Programming},
    keywords = {algebraic\_data\_types, functional\_language, partial\_evaluation, types},
    number = {05},
    pages = {509--543},
    posted-at = {2010-07-09 10:47:03},
    priority = {4},
    title = {Finally tagless, partially evaluated: Tagless staged interpreters for simpler typed languages},
    url = {http://dx.doi.org/10.1017/S0956796809007205},
    volume = {19},
    year = {2009}
}

@article{giusti+lecerf+salvy01,
    abstract = {Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial, and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton's iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation of the system. We present our implementation in the Magma system which is called Kronecker in homage to his method for solving systems of polynomial equations. We show that the theoretical complexity of our algorithm is well reflected in practice and we exhibit some cases for which our program is more efficient than the other available software.},
    address = {Orlando, FL, USA},
    author = {Giusti, Marc and Lecerf, Gr\'{e}goire and Salvy, Bruno},
    citeulike-article-id = {7424340},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=374672},
    citeulike-linkout-1 = {http://dx.doi.org/10.1006/jcom.2000.0571},
    doi = {10.1006/jcom.2000.0571},
    issn = {0885-064X},
    journal = {Journal of Complexity},
    keywords = {algebraic\_geometry, algorithm, groebner\_bases, triangular\_set},
    month = mar,
    number = {1},
    pages = {154--211},
    posted-at = {2010-07-06 22:34:51},
    priority = {4},
    publisher = {Academic Press, Inc.},
    title = {A {G}r\"{o}bner free alternative for polynomial system solving},
    url = {http://dx.doi.org/10.1006/jcom.2000.0571},
    volume = {17},
    year = {2001}
}

@incollection{galbraith+hess+vercauteren10,
    abstract = {We survey recent research on pairings on hyperelliptic curves and present a comparison of the performance characteristics of pairings on elliptic curves and hyperelliptic curves. Our analysis indicates that hyperelliptic curves are not more efficient than elliptic curves for general pairing applications.},
    address = {Berlin, Heidelberg},
    author = {Galbraith, Steven and Hess, Florian and Vercauteren, Frederik},
    booktitle = {Pairing-Based Cryptography – Pairing 2007 },
    chapter = {7},
    citeulike-article-id = {7414056},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-73489-5_7},
    citeulike-linkout-1 = {http://www.springerlink.com/content/x52g3734w256r626},
    doi = {10.1007/978-3-540-73489-5_7},
    editor = {Takagi, Tsuyoshi and Okamoto, Tatsuaki and Okamoto, Eiji and Okamoto, Takeshi},
    isbn = {978-3-540-73488-8},
    issn = {0302-9743},
    keywords = {elliptic\_curve, hyperelliptic\_curves, pairing},
    pages = {108--131},
    posted-at = {2010-07-05 17:45:29},
    priority = {3},
    publisher = {Springer Berlin Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Hyperelliptic Pairings},
    url = {http://dx.doi.org/10.1007/978-3-540-73489-5_7},
    volume = {4575},
    year = {2010}
}

@article{dixon+duncan09,
    abstract = {Compact closed categories provide a foundational formalism for a variety of
important domains, including quantum computation. These categories have a
natural visualisation as a form of graphs. We present a formalism for
equational reasoning about such graphs and develop this into a generic proof
system with a fixed logical kernel for equational reasoning about compact
closed categories. Automating this reasoning process is motivated by the slow
and error prone nature of manual graph manipulation. A salient feature of our
system is that it provides a formal and declarative account of derived results
that can include `ellipses'-style notation. We illustrate the framework by
instantiating it for a graphical language of quantum computation and show how
this can be used to perform symbolic computation.},
    archivePrefix = {arXiv},
    author = {Dixon, Lucas and Duncan, Ross},
    citeulike-article-id = {4009848},
    citeulike-linkout-0 = {http://arxiv.org/abs/0902.0514},
    citeulike-linkout-1 = {http://arxiv.org/pdf/0902.0514},
    day = {3},
    eprint = {0902.0514},
    keywords = {graphs, quantum, transposition},
    month = feb,
    posted-at = {2010-07-01 22:33:51},
    priority = {0},
    title = {Graphical Reasoning in Compact Closed Categories for Quantum Computation},
    url = {http://arxiv.org/abs/0902.0514},
    year = {2009}
}

@book{bykov+kytmanov+lazman,
    abstract = {From the Publisher: This book presents a modified method, based on multidimensional residue theory, for the elimination of unknowns from a system of nonlinear algebraic equations. An algorithm is given for constructing the resultant of the system, and a computer implementation making use of formula manipulation software is carried out. Programmes in {MAPLE} are available The algorithms and programmes are then applied to questions from the theory of chemical kinetics, such as the search for all stationary solutions of kinetic equations and the construction of kinetic polynolynomials. The subject of this book is closely connected with a wide range of current problems in the analysis of nonlinear systems.},
    address = {Norwell, MA, USA},
    author = {Bykov, Valery and Kytmanov, Alexander and Lazman, Mark},
    citeulike-article-id = {7369527},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=552711},
    isbn = {0792352408},
    keywords = {change\_of\_order, computer\_algebra, elimination},
    posted-at = {2010-06-30 13:23:34},
    priority = {1},
    publisher = {Kluwer Academic Publishers},
    title = {Elimination Methods in Polynomial Computer Algebra},
    url = {http://portal.acm.org/citation.cfm?id=552711},
    year = {1998}
}

@article{aizenberg+kytmanov81,
    author = {Aizenberg, L. A. and Kytmanov, A. M.},
    citeulike-article-id = {7369523},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/BF00968414},
    citeulike-linkout-1 = {http://www.springerlink.com/content/ux391125249856j5},
    day = {1},
    doi = {10.1007/BF00968414},
    issn = {0037-4466},
    journal = {Siberian Mathematical Journal},
    keywords = {change\_of\_order, newton},
    month = mar,
    number = {2},
    pages = {180--189},
    posted-at = {2010-06-30 13:20:53},
    priority = {4},
    title = {Multidimensional analogues of Newton's formulas for systems of nonlinear algebraic equations and some of their applications},
    url = {http://dx.doi.org/10.1007/BF00968414},
    volume = {22},
    year = {1981}
}

@incollection{diaz+gonzalez01,
    author = {D\'{\i}az-Toca, Gema M. and Gonz\'{a}lez-Vega, Laureano},
    booktitle = {Symbolic computation: solving equations in algebra, geometry and engineering},
    citeulike-article-id = {7369511},
    keywords = {change\_of\_order, trace\_formulas, triangular\_set, zero\_dimensional},
    pages = {21--35},
    posted-at = {2010-06-30 13:05:48},
    priority = {4},
    publisher = {AMS},
    series = {Contemporary Mathematics},
    title = {An explicit description for the triangular decomposition of a zero-dimensional ideal through trace computations},
    volume = {286},
    year = {2001}
}

@article{dahan+jin+moreno+schost08,
    abstract = {We discuss changing the variable order for a regular chain in positive dimension. This quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations. We propose a modular method, reducing the problem to computations in dimension zero and one. The problems raised by the choice of the specialization points and the lack of the (crucial) information of what are the free and algebraic variables for the new order are discussed. Strong (but not unusual) hypotheses for the initial regular chain are required; the main required subroutines are change of order in dimension zero and a formal Newton iteration.},
    author = {Dahan, Xavier and Jin, Xin and Moreno Maza, Marc and Schost, \'{E}ric},
    citeulike-article-id = {7369466},
    citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.tcs.2007.10.003},
    day = {28},
    doi = {10.1016/j.tcs.2007.10.003},
    issn = {03043975},
    journal = {Theoretical Computer Science},
    keywords = {change\_of\_order, triangular\_set},
    month = feb,
    number = {1-3},
    pages = {37--65},
    posted-at = {2010-06-30 12:40:00},
    priority = {4},
    title = {Change of order for regular chains in positive dimension},
    url = {http://dx.doi.org/10.1016/j.tcs.2007.10.003},
    volume = {392},
    year = {2008}
}

@inproceedings{boulier+lemaire+moreno01,
    abstract = {We propose a new algorithm for converting a characteristic set of a prime differential ideal from one ranking into another. This differential algebra algorithm computes characteristic sets by change of ranking (ordering) for prime ideals. It identifies the purely algebraic subproblems which arise during differential computations and solves them algebraically. There are two improvements w.r.t. other approaches: formerly unsolved problems could be carried out; it is conceptually simple. Different variants are implemented.},
    address = {New York, NY, USA},
    author = {Boulier, Fran\c{c}ois and Lemaire, Fran\c{c}ois and Moreno Maza, Marc},
    booktitle = {ISSAC '01: Proceedings of the 2001 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {7369349},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=384101.384108},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/384101.384108},
    doi = {10.1145/384101.384108},
    isbn = {1-58113-417-7},
    keywords = {change\_of\_order, resultant},
    location = {London, Ontario, Canada},
    pages = {38--47},
    posted-at = {2010-06-30 12:26:21},
    priority = {2},
    publisher = {ACM},
    title = {{PARDI!}},
    url = {http://dx.doi.org/10.1145/384101.384108},
    year = {2001}
}

@article{bostan+salvy+schost03,
    abstract = {Many questions concerning a zero-dimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A= k[ X 1,…, X n]/I, where I is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the question of speeding up the linear algebra phase for the computation of minimal polynomials and rational parametrizations in A. We present new formul{\ae} for the rational parametrizations, extending those of Rouillier, and algorithms extending ideas introduced by Shoup in the univariate case. Our approach is based on the A-module structure of the dual space \$\widehat{A}\$ . An important feature of our algorithms is that we do not require \$\widehat{A}\$ to be free and of rank 1. The complexity of our algorithms for computing the minimal polynomial and the rational parametrizations are O(2 nD 5/2) and O( n2 nD 5/2) respectively, where D is the dimension of A. For fixed n, this is better than algorithms based on linear algebra except when the complexity of the available matrix product has exponent less than 5/2.},
    author = {Bostan, Alin and Salvy, Bruno and Schost, \'{E}ric},
    citeulike-article-id = {7367476},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s00200-003-0133-5},
    citeulike-linkout-1 = {http://www.springerlink.com/content/ggfkxwylb3nyqmx2},
    day = {1},
    doi = {10.1007/s00200-003-0133-5},
    issn = {0938-1279},
    journal = {Applicable Algebra in Engineering, Communication and Computing},
    keywords = {computational\_algebraic\_geometry, computer\_algebra, modular\_composition, trace\_formulas, transposed\_algorithms, triangular\_set, zero\_dimensional},
    month = nov,
    number = {4},
    pages = {239--272},
    posted-at = {2010-06-30 00:05:03},
    priority = {0},
    title = {Fast Algorithms for {Zero-Dimensional} Polynomial Systems using Duality},
    url = {http://dx.doi.org/10.1007/s00200-003-0133-5},
    volume = {14},
    year = {2003}
}

@book{asperti+longo,
    author = {Asperti, Andrea and Longo, Giuseppe},
    citeulike-article-id = {590705},
    citeulike-linkout-0 = {http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&amp;path=ASIN/0262011255},
    citeulike-linkout-1 = {http://www.amazon.de/exec/obidos/redirect?tag=citeulike01-21&amp;path=ASIN/0262011255},
    citeulike-linkout-2 = {http://www.amazon.fr/exec/obidos/redirect?tag=citeulike06-21&amp;path=ASIN/0262011255},
    citeulike-linkout-3 = {http://www.amazon.co.uk/exec/obidos/ASIN/0262011255/citeulike00-21},
    citeulike-linkout-4 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0262011255},
    citeulike-linkout-5 = {http://www.worldcat.org/isbn/0262011255},
    citeulike-linkout-6 = {http://books.google.com/books?vid=ISBN0262011255},
    citeulike-linkout-7 = {http://www.amazon.com/gp/search?keywords=0262011255&index=books&linkCode=qs},
    citeulike-linkout-8 = {http://www.librarything.com/isbn/0262011255},
    day = {23},
    howpublished = {Hardcover},
    isbn = {0262011255},
    keywords = {categories, functional\_language, types},
    month = aug,
    posted-at = {2010-06-29 14:12:15},
    priority = {3},
    publisher = {The MIT Press},
    series = {Foundations of Computing},
    title = {Categories, Types, and Structures: An Introduction to Category Theory for the Working Computer Scientist},
    url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0262011255},
    year = {1991}
}

@article{rouiller99,
    abstract = {Abstract.\&nbsp;\&nbsp; This paper is devoted to the resolution of zero-dimensional systems in {K[X} 1, …X n ], where K is a field of characteristic zero (or strictly positive under some conditions). We follow the definition used in {MMM95} and basically due to Kronecker for solving zero-dimensional systems: A system is solved if each root is represented in such way as to allow the performance of any arithmetical operations over the arithmetical expressions of its coordinates. We propose new definitions for solving zero-dimensional systems in this sense by introducing the Univariate Representation of their roots. We show by this way that the solutions of any zero-dimensional system of polynomials can be expressed through a special kind of univariate representation (Rational Univariate Representation): where (f,g,g 1, …,g n ) are polynomials of {K[X} 1, …, X n ]. A special feature of our Rational Univariate Representation is that we dont loose geometrical information contained in the initial system. Moreover we propose different efficient algorithms for the computation of the Rational Univariate Representation, and we make a comparison with standard known tools.},
    author = {Rouillier, Fabrice},
    citeulike-article-id = {7362475},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s002000050114},
    citeulike-linkout-1 = {http://www.springerlink.com/content/xb9f08pv4k8l96jd},
    day = {10},
    doi = {10.1007/s002000050114},
    issn = {0938-1279},
    journal = {Applicable Algebra in Engineering, Communication and Computing},
    keywords = {algebraic\_geometry, algorithm, groebner\_bases, trace\_formulas},
    month = may,
    number = {5},
    pages = {433--461},
    posted-at = {2010-06-27 15:42:40},
    priority = {0},
    title = {Solving {Zero-Dimensional} Systems Through the Rational Univariate Representation},
    url = {http://dx.doi.org/10.1007/s002000050114},
    volume = {9},
    year = {1999}
}

@book{Barendregt:Lambda,
    abstract = {{The revised edition contains a new chapter which provides an elegant description of the semantics. The various classes of lambda calculus models are described in a uniform manner. Some didactical improvements have been made to this edition. An example of a simple model is given and then the general theory (of categorical models) is developed. Indications are given of those parts of the book which can be used to form a coherent course.}},
    author = {Barendregt, H. P.},
    citeulike-article-id = {679666},
    citeulike-linkout-0 = {http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&amp;path=ASIN/0444875085},
    citeulike-linkout-1 = {http://www.amazon.de/exec/obidos/redirect?tag=citeulike01-21&amp;path=ASIN/0444875085},
    citeulike-linkout-2 = {http://www.amazon.fr/exec/obidos/redirect?tag=citeulike06-21&amp;path=ASIN/0444875085},
    citeulike-linkout-3 = {http://www.amazon.jp/exec/obidos/ASIN/0444875085},
    citeulike-linkout-4 = {http://www.amazon.co.uk/exec/obidos/ASIN/0444875085/citeulike00-21},
    citeulike-linkout-5 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0444875085},
    citeulike-linkout-6 = {http://www.worldcat.org/isbn/0444875085},
    citeulike-linkout-7 = {http://books.google.com/books?vid=ISBN0444875085},
    citeulike-linkout-8 = {http://www.amazon.com/gp/search?keywords=0444875085&index=books&linkCode=qs},
    citeulike-linkout-9 = {http://www.librarything.com/isbn/0444875085},
    day = {15},
    edition = {Revised},
    howpublished = {Paperback},
    isbn = {0444875085},
    keywords = {categorical\_models, functional\_language, lambda, logic, types},
    month = nov,
    posted-at = {2010-06-24 13:27:06},
    priority = {2},
    publisher = {North Holland},
    title = {The Lambda Calculus, Its Syntax and Semantics (Studies in Logic and the Foundations of Mathematics, Volume 103). Revised Edition},
    url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0444875085},
    year = {1985}
}

@book{pierce,
    abstract = {{A type system is a syntactic method for automatically checking the absence of certain erroneous behaviors by classifying program phrases according to the kinds of values they compute. The study of type systems--and of programming languages from a type-theoretic perspective-has important applications in software engineering, language design, high-performance compilers, and security.<br /> <br /> This text provides a comprehensive introduction both to type systems in computer science and to the basic theory of programming languages. The approach is pragmatic and operational; each new concept is motivated by programming examples and the more theoretical sections are driven by the needs of implementations. Each chapter is accompanied by numerous exercises and solutions, as well as a running implementation, available via the Web. Dependencies between chapters are explicitly identified, allowing readers to choose a variety of paths through the material.<br /> <br /> The core topics include the untyped lambda-calculus, simple type systems, type reconstruction, universal and existential polymorphism, subtyping, bounded quantification, recursive types, kinds, and type operators. Extended case studies develop a variety of approaches to modeling the features of object-oriented languages.}},
    author = {Pierce, Benjamin C.},
    citeulike-article-id = {105547},
    citeulike-linkout-0 = {http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&amp;path=ASIN/0262162091},
    citeulike-linkout-1 = {http://www.amazon.de/exec/obidos/redirect?tag=citeulike01-21&amp;path=ASIN/0262162091},
    citeulike-linkout-2 = {http://www.amazon.fr/exec/obidos/redirect?tag=citeulike06-21&amp;path=ASIN/0262162091},
    citeulike-linkout-3 = {http://www.amazon.jp/exec/obidos/ASIN/0262162091},
    citeulike-linkout-4 = {http://www.amazon.co.uk/exec/obidos/ASIN/0262162091/citeulike00-21},
    citeulike-linkout-5 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0262162091},
    citeulike-linkout-6 = {http://www.worldcat.org/isbn/0262162091},
    citeulike-linkout-7 = {http://books.google.com/books?vid=ISBN0262162091},
    citeulike-linkout-8 = {http://www.amazon.com/gp/search?keywords=0262162091&index=books&linkCode=qs},
    citeulike-linkout-9 = {http://www.librarything.com/isbn/0262162091},
    day = {01},
    edition = {1},
    howpublished = {Hardcover},
    isbn = {0262162091},
    keywords = {function\_level\_programming, functional\_language, recursive\_types, type\_operators, typechecking},
    month = feb,
    posted-at = {2010-06-24 13:24:56},
    priority = {4},
    publisher = {The MIT Press},
    title = {Types and Programming Languages},
    url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0262162091},
    year = {2002}
}

@book{lidl+niederreiter:2,
    abstract = {{The theory of finite fields is a branch of algebra with diverse applications in such areas as combinatorics, coding theory and the mathematical study of switching circuits. This updated second edition is devoted entirely to the theory of finite fields, and it provides comprehensive coverage of the literature. Bibliographical notes at the end of each chapter give a historical survey of the development of the subject. Worked examples and lists of exercises throughout the book make it useful as a text for advanced level courses for students of algebra.}},
    author = {Lidl, Rudolf and Niederreiter, Harald},
    citeulike-article-id = {523444},
    citeulike-linkout-0 = {http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&amp;path=ASIN/0521392314},
    citeulike-linkout-1 = {http://www.amazon.de/exec/obidos/redirect?tag=citeulike01-21&amp;path=ASIN/0521392314},
    citeulike-linkout-2 = {http://www.amazon.fr/exec/obidos/redirect?tag=citeulike06-21&amp;path=ASIN/0521392314},
    citeulike-linkout-3 = {http://www.amazon.co.uk/exec/obidos/ASIN/0521392314/citeulike00-21},
    citeulike-linkout-4 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0521392314},
    citeulike-linkout-5 = {http://www.worldcat.org/isbn/0521392314},
    citeulike-linkout-6 = {http://books.google.com/books?vid=ISBN0521392314},
    citeulike-linkout-7 = {http://www.amazon.com/gp/search?keywords=0521392314&index=books&linkCode=qs},
    citeulike-linkout-8 = {http://www.librarything.com/isbn/0521392314},
    day = {24},
    howpublished = {Hardcover},
    isbn = {0521392314},
    keywords = {finite\_field},
    month = oct,
    posted-at = {2010-06-20 20:34:48},
    priority = {0},
    publisher = {{Cambridge University Press}},
    title = {Finite Fields (Encyclopedia of Mathematics and its Applications)},
    url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0521392314},
    year = {1996}
}

@book{Cox-Little-OShea:UAG2005,
    abstract = {In recent years, the discovery of new algorithms for dealing with polynomial equations, coupled with their implementation on fast inexpensive computers, has sparked a minor revolution in the study and practice of algebraic geometry. These algorithmic methods have also given rise to some exciting new applications of algebraic geometry. This book illustrates the many uses of algebraic geometry, highlighting some of the more recent applications of Gr\"{o}bner bases and resultants. In order to do this, the authors provide an introduction to some algebraic objects and techniques which are more advanced than one typically encounters in a first course, but nonetheless of great utility. The book is written for nonspecialists and for readers with a diverse range of backgrounds. It assumes knowledge of the material covered in a standard undergraduate course in abstract algebra, and it would help to have some previous exposure to Gr\"{o}bner bases. The book does not assume the reader is familiar with more advanced concepts such as modules. For this new edition the authors added two new sections and a new chapter, updated the references and made numerous minor improvements throughout the text.},
    address = {Berlin-Heidelberg-New York},
    author = {Cox, David A. and Little, John and O'Shea, Donal},
    citeulike-article-id = {549534},
    citeulike-linkout-0 = {http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&amp;path=ASIN/0387207066},
    citeulike-linkout-1 = {http://www.amazon.de/exec/obidos/redirect?tag=citeulike01-21&amp;path=ASIN/0387207066},
    citeulike-linkout-2 = {http://www.amazon.fr/exec/obidos/redirect?tag=citeulike06-21&amp;path=ASIN/0387207066},
    citeulike-linkout-3 = {http://www.amazon.co.uk/exec/obidos/ASIN/0387207066/citeulike00-21},
    citeulike-linkout-4 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0387207066},
    citeulike-linkout-5 = {http://www.worldcat.org/isbn/0387207066},
    citeulike-linkout-6 = {http://books.google.com/books?vid=ISBN0387207066},
    citeulike-linkout-7 = {http://www.amazon.com/gp/search?keywords=0387207066&index=books&linkCode=qs},
    citeulike-linkout-8 = {http://www.librarything.com/isbn/0387207066},
    day = {09},
    howpublished = {Hardcover},
    isbn = {0387207066},
    keywords = {algebraic\_complexity, algebraic\_curves, algebraic\_geometry, algorithm, groebner\_bases, projective\_geometry, zero\_dimensional},
    month = mar,
    posted-at = {2010-06-20 20:02:32},
    priority = {4},
    publisher = {Springer-Verlag},
    series = {Graduate Texts in Mathematics},
    title = {Using Algebraic Geometry},
    url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0387207066},
    year = {2005}
}

@book{cox+little+oshea,
    abstract = {Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory. The algorithms to answer questions such as those posed above are an important part of algebraic geometry. This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered in the 1960's. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with the power of fast computers, have let to some interesting applications, for example in robotics and in geometric theorem proving. In preparing a new edition of Ideals, Varieties and Algorithms the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout's Theorem. Appendix C contains a new section on Axiom and an update about Maple , Mathematica and {REDUCE}.},
    author = {Cox, David and Little, John and O'Shea, Donal},
    citeulike-article-id = {466011},
    citeulike-linkout-0 = {http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&amp;path=ASIN/0387946802},
    citeulike-linkout-1 = {http://www.amazon.de/exec/obidos/redirect?tag=citeulike01-21&amp;path=ASIN/0387946802},
    citeulike-linkout-2 = {http://www.amazon.fr/exec/obidos/redirect?tag=citeulike06-21&amp;path=ASIN/0387946802},
    citeulike-linkout-3 = {http://www.amazon.co.uk/exec/obidos/ASIN/0387946802/citeulike00-21},
    citeulike-linkout-4 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0387946802},
    citeulike-linkout-5 = {http://www.worldcat.org/isbn/0387946802},
    citeulike-linkout-6 = {http://books.google.com/books?vid=ISBN0387946802},
    citeulike-linkout-7 = {http://www.amazon.com/gp/search?keywords=0387946802&index=books&linkCode=qs},
    citeulike-linkout-8 = {http://www.librarything.com/isbn/0387946802},
    day = {22},
    howpublished = {Hardcover},
    isbn = {0387946802},
    keywords = {algebraic\_geometry, computational\_algebraic\_geometry, computer\_algebra, groebner\_bases, projective\_geometry, zero\_dimensional},
    month = jul,
    posted-at = {2010-06-20 20:01:11},
    priority = {4},
    publisher = {Springer},
    series = {(Undergraduate Texts in Mathematics)},
    title = {Ideals, Varieties, and Algorithms : An Introduction to Computational Algebraic Geometry and Commutative Algebra},
    url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0387946802},
    year = {2005}
}

@book{silverman:elliptic,
    author = {Silverman, Joseph H.},
    citeulike-article-id = {7238111},
    keywords = {elliptic\_curve},
    number = {106},
    posted-at = {2010-06-20 19:57:55},
    priority = {4},
    publisher = {Springer},
    series = {Graduate Texts in Mathematics},
    title = {The Arithmetic of Elliptic Curves},
    year = {1986}
}

@book{silverman:advanced,
    abstract = {{This book continues the treatment of the arithmetic theory of elliptic curves begun in the first volume. The book begins with the theory of elliptic and modular functions for the full modular group r(1), including a discussion of Hekcke operators and the L-series associated to cusp forms. This is followed by a detailed study of elliptic curves with complex multiplication, their associated Gr\"{o}ssencharacters and L-series, and applications to the construction of abelian extensions of quadratic imaginary fields. Next comes a treatment of elliptic curves over function fields and elliptic surfaces, including specialization theorems for heights and sections. This material serves as a prelude to the theory of minimal models and N\'{e}ron models of elliptic curves, with a discussion of special fibers, conductors, and Ogg's formula. Next comes a brief description of q-models for elliptic curves over C and R, followed by Tate's theory of q-models for elliptic curves with non-integral j-invariant over p-adic fields. The book concludes with the construction of canonical local height functions on elliptic curves, including explicit formulas for both archimedean and non-archimedean fields.}},
    author = {Silverman, Joseph H.},
    citeulike-article-id = {789887},
    citeulike-linkout-0 = {http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&amp;path=ASIN/0387943285},
    citeulike-linkout-1 = {http://www.amazon.de/exec/obidos/redirect?tag=citeulike01-21&amp;path=ASIN/0387943285},
    citeulike-linkout-2 = {http://www.amazon.fr/exec/obidos/redirect?tag=citeulike06-21&amp;path=ASIN/0387943285},
    citeulike-linkout-3 = {http://www.amazon.co.uk/exec/obidos/ASIN/0387943285/citeulike00-21},
    citeulike-linkout-4 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0387943285},
    citeulike-linkout-5 = {http://www.worldcat.org/isbn/0387943285},
    citeulike-linkout-6 = {http://books.google.com/books?vid=ISBN0387943285},
    citeulike-linkout-7 = {http://www.amazon.com/gp/search?keywords=0387943285&index=books&linkCode=qs},
    citeulike-linkout-8 = {http://www.librarything.com/isbn/0387943285},
    day = {01},
    howpublished = {Paperback},
    isbn = {0387943285},
    keywords = {elliptic\_curve},
    month = jan,
    posted-at = {2010-06-20 19:57:19},
    priority = {4},
    publisher = {Springer},
    title = {Advanced Topics in the Arithmetic of Elliptic Curves (Graduate Texts in Mathematics)},
    url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0387943285},
    year = {1994}
}

@book{lang,
    abstract = {{"Lang's Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books." \&nbsp;NOTICES OF THE AMS "The author has an impressive knack for presenting the important and interesting ideas of algebra in just the right way, and he never gets bogged down in the dry formalism which pervades some parts of algebra."\&nbsp;MATHEMATICAL REVIEWS This book is intended as a basic text for a one-year course in algebra at the graduate level, or as a useful reference for mathematicians and professionals who use higher-level algebra. It successfully addresses the basic concepts of algebra. \&nbsp;For the revised third edition, the author has added exercises and made numerous corrections to the text.}},
    author = {Lang, Serge},
    citeulike-article-id = {117768},
    citeulike-linkout-0 = {http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&amp;path=ASIN/038795385X},
    citeulike-linkout-1 = {http://www.amazon.de/exec/obidos/redirect?tag=citeulike01-21&amp;path=ASIN/038795385X},
    citeulike-linkout-2 = {http://www.amazon.fr/exec/obidos/redirect?tag=citeulike06-21&amp;path=ASIN/038795385X},
    citeulike-linkout-3 = {http://www.amazon.jp/exec/obidos/ASIN/038795385X},
    citeulike-linkout-4 = {http://www.amazon.co.uk/exec/obidos/ASIN/038795385X/citeulike00-21},
    citeulike-linkout-5 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/038795385X},
    citeulike-linkout-6 = {http://www.worldcat.org/isbn/038795385X},
    citeulike-linkout-7 = {http://books.google.com/books?vid=ISBN038795385X},
    citeulike-linkout-8 = {http://www.amazon.com/gp/search?keywords=038795385X&index=books&linkCode=qs},
    citeulike-linkout-9 = {http://www.librarything.com/isbn/038795385X},
    day = {08},
    edition = {3rd},
    howpublished = {Hardcover},
    isbn = {038795385X},
    keywords = {algebra\_books, categories},
    month = jan,
    posted-at = {2010-06-20 19:56:10},
    priority = {3},
    publisher = {Springer},
    title = {Algebra},
    url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/038795385X},
    year = {2002}
}

@article{hanrot+quercia+zimmermann,
    abstract = {We present new algorithms for the inverse, division, and square root of power series. The key trick is a new algorithm – {MiddleProduct} or, for short, {MP} – computing the n middle coefficients of a (2 n-1)× n full product in the same number of multiplications as a full n× n product. This improves previous work of Brent, Mulders, Karp and Markstein, Burnikel and Ziegler. These results apply both to series and polynomials.},
    author = {Hanrot, Guillaume and Quercia, Michel and Zimmermann, Paul},
    citeulike-article-id = {7338006},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s00200-003-0144-2},
    citeulike-linkout-1 = {http://www.springerlink.com/content/57p2fta5k71085wm},
    day = {1},
    doi = {10.1007/s00200-003-0144-2},
    issn = {0938-1279},
    journal = {Applicable Algebra in Engineering, Communication and Computing},
    keywords = {computer\_algebra, polynomial\_multiplication, power\_series, transposed\_algorithms, transposition},
    month = mar,
    number = {6},
    pages = {415--438},
    posted-at = {2010-06-17 19:21:42},
    priority = {2},
    title = {The Middle Product Algorithm {I}},
    url = {http://dx.doi.org/10.1007/s00200-003-0144-2},
    volume = {14},
    year = {2004}
}

@inproceedings{villard+monagan99,
    abstract = {Note: {OCR} errors may be found in this Reference List extracted from the full text article.  {ACM} has opted to expose the complete List rather than only correct and linked references.},
    address = {New York, NY, USA},
    author = {Villard, Dominique and Monagan, Michael B.},
    booktitle = {ISSAC '99: Proceedings of the 1999 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {7333445},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=309941},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/309831.309941},
    doi = {10.1145/309831.309941},
    isbn = {1-58113-073-2},
    keywords = {automatic\_differentiation, transposition},
    location = {Vancouver, British Columbia, Canada},
    pages = {221--228},
    posted-at = {2010-06-17 09:23:26},
    priority = {0},
    publisher = {ACM},
    title = {{ADrien}: an implementation of automatic differentiation in Maple},
    url = {http://dx.doi.org/10.1145/309831.309941},
    year = {1999}
}

@incollection{griewank89,
    author = {Griewank, A. O.},
    booktitle = {Mathematical Programming: Recent Developments and Applications},
    citeulike-article-id = {6904361},
    citeulike-linkout-0 = {http://www.ams.org/mathscinet-getitem?mr=92k:65002},
    editor = {Iri, M. and Tanabe, K.},
    keywords = {automatic\_differentiation, transposition},
    pages = {83--108},
    posted-at = {2010-06-16 15:19:56},
    priority = {0},
    publisher = {Kluwer Academic Publishers},
    title = {On Automatic Differentiation},
    url = {http://www.ams.org/mathscinet-getitem?mr=92k:65002},
    year = {1989}
}

@article{baur+strassen83,
    abstract = {Let L denote the nonscalar complexity in k ( x 1 ,…, x n ). We prove L (ƒ,∂ƒ/∂ x 1 ,…,∂ƒ/∂ x n ) 3 L (ƒ). Using this we determine the complexity of single power sums, single elementary symmetric functions, the resultant and the discriminant as root functions, up to order of magnitude. Also we linearly reduce matrix inversion to computing the determinant.},
    author = {Baur, Walter and Strassen, Volker},
    citeulike-article-id = {7329713},
    citeulike-linkout-0 = {http://dx.doi.org/10.1016/0304-3975(83)90110-X},
    doi = {10.1016/0304-3975(83)90110-X},
    issn = {03043975},
    journal = {Theoretical Computer Science},
    keywords = {automatic\_differentiation, circuit\_complexity, transposition},
    month = feb,
    number = {3},
    pages = {317--330},
    posted-at = {2010-06-16 14:45:38},
    priority = {3},
    title = {The complexity of partial derivatives},
    url = {http://dx.doi.org/10.1016/0304-3975(83)90110-X},
    volume = {22},
    year = {1983}
}

@article{voloch90,
    author = {Voloch, Jos\'{e} F.},
    citeulike-article-id = {7300946},
    journal = {Compositio Mathematica},
    keywords = {elliptic\_curve, p-descent, torsion},
    number = {3},
    pages = {247--258},
    posted-at = {2010-06-14 19:40:17},
    priority = {0},
    title = {Explicit \(p\)-descent for elliptic curves in characteristic \(p\)},
    volume = {74},
    year = {1990}
}

@article{velu71,
    author = {V{\'{e}}lu, Jean},
    citeulike-article-id = {7300942},
    journal = {Comptes Rendus de l'Acad\'{e}mie des Sciences de Paris},
    keywords = {elliptic\_curve, isogenies},
    pages = {238--241},
    posted-at = {2010-06-14 19:38:59},
    priority = {2},
    title = {Isog{\'{e}}nies entre courbes elliptiques},
    volume = {273},
    year = {1971}
}

@article{satoh00,
    author = {Satoh, Takakazu},
    citeulike-article-id = {7300931},
    journal = {Journal of the Ramanujan Mathematical Society},
    keywords = {elliptic\_curve, finite\_field, isogenies, p-adic},
    number = {4},
    pages = {247--270},
    posted-at = {2010-06-14 19:37:29},
    priority = {2},
    publisher = {The Ramanujan Mathematical Society},
    title = {The canonical lift of an ordinary elliptic curve over a finite field and its point counting},
    volume = {15},
    year = {2000}
}

@article{teske06,
    abstract = {Abstract\&nbsp;\&nbsp; We propose an elliptic curve trapdoor system which is of interest in key escrow applications. In this system, a pair (Es, Epb) of elliptic curves over F2161 is constructed with the following properties: (i) the {Gaudry-Hess}-Smart Weil descent attack reduces the elliptic curve discrete logarithm problem ({ECDLP}) in {Es(F2161}) to a hyperelliptic curve {DLP} in the Jacobian of a curve of genus 7 or 8, which is computationally feasible, but by far not trivial; (ii) Es is isogenous to Es; (iii) the best attack on the {ECDLP} in {Es(F2161}) is the parallelized Pollard rho method. The curve Es is used just as usual in elliptic curve cryptosystems. The curve Es is submitted to a trusted authority for the purpose of key escrow. The crucial difference from other key escrow scenarios is that the trusted authority has to invest a considerable amount of computation to compromise a user's private key, which makes applications such as widespread wire-tapping impossible.},
    author = {Teske, Edlyn},
    citeulike-article-id = {7300913},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s00145-004-0328-3},
    citeulike-linkout-1 = {http://www.springerlink.com/content/l489q503440l6v72},
    day = {6},
    doi = {10.1007/s00145-004-0328-3},
    issn = {0933-2790},
    journal = {Journal of Cryptology},
    keywords = {elliptic\_curve, finite\_field, ghs, isogenies, isogeny\_cycles, weil\_descent},
    month = jan,
    number = {1},
    pages = {115--133},
    posted-at = {2010-06-14 19:35:50},
    priority = {2},
    title = {An Elliptic Curve Trapdoor System},
    url = {http://dx.doi.org/10.1007/s00145-004-0328-3},
    volume = {19},
    year = {2006}
}

@article{fouquet+gaudry+harley00,
    abstract = {We describe a fast algorithm for counting points on elliptic curves defined over finite fields of small characteristic, following Satoh. Our main contribution is an extension to characteristics two and three. We give a detailed description with the optimisations necessary for an efficient implementation. Finally we give the number of points we have computed on a "random" curve defined over the field F q with q = 2  8009  .},
    author = {Fouquet, Mireille and Gaudry, Pierrick and Harley, Robert},
    citeulike-article-id = {7300886},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.9478},
    journal = {Journal of the Ramanujan Mathematical Society},
    keywords = {elliptic\_curve, finite\_field, isogenies, p-adic},
    number = {4},
    pages = {281--318},
    posted-at = {2010-06-14 19:32:24},
    priority = {0},
    title = {An extension of {S}atoh's algorithm and its implementation},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.9478},
    volume = {15},
    year = {2000}
}

@article{montgomery87,
    abstract = {Since 1974, several algorithms have been developed that attempt to factor a large number \$N\$ by doing extensive computations modulo \$N\$ and occasionally taking {GCDs} with \$N\$. These began with Pollard's \$p - 1\$ and Monte Carlo methods. More recently, Williams published a \$p + 1\$ method, and Lenstra discovered an elliptic curve method ({ECM}). We present ways to speed all of these. One improvement uses two tables during the second phases of \$p \pm 1\$ and {ECM}, looking for a match. Polynomial preconditioning lets us search a fixed table of size \$n\$ with \$n/2 + o(n)\$ multiplications. A parametrization of elliptic curves lets Step 1 of {ECM} compute the \$x\$-coordinate of \${nP}\$ from that of \$P\$ in about \$9.3 \log\_2 n\$ multiplications for arbitrary \$P\$.},
    author = {Montgomery, Peter L.},
    citeulike-article-id = {7300846},
    citeulike-linkout-0 = {http://dx.doi.org/10.2307/2007888},
    citeulike-linkout-1 = {http://www.jstor.org/stable/2007888},
    doi = {10.2307/2007888},
    issn = {00255718},
    journal = {Math. Comp.},
    keywords = {elliptic\_curve, elliptic\_curve\_method, finite\_field, montgomery\_curve},
    number = {177},
    pages = {243--264},
    posted-at = {2010-06-14 19:27:43},
    priority = {0},
    publisher = {American Mathematical Society},
    title = {Speeding the Pollard and Elliptic Curve Methods of Factorization},
    url = {http://dx.doi.org/10.2307/2007888},
    volume = {48},
    year = {1987}
}

@article{gunji76,
    author = {Gunji, Hiroshi},
    citeulike-article-id = {7300836},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/BF01224654},
    citeulike-linkout-1 = {http://www.springerlink.com/content/jw0333pn20w23534},
    day = {21},
    doi = {10.1007/BF01224654},
    issn = {0003-889X},
    journal = {Archiv der Mathematik},
    keywords = {elliptic\_curve, hasse, invariant, torsion},
    month = dec,
    number = {1},
    pages = {148--158},
    posted-at = {2010-06-14 19:26:03},
    priority = {0},
    title = {The {H}asse invariant and p-division points of an elliptic curve},
    url = {http://dx.doi.org/10.1007/BF01224654},
    volume = {27},
    year = {1976}
}

@electronic{joux+lercier06,
    abstract = {In this paper, we revisit the problem of computing the kernel of a separable isogeny of degree \$\ell\$ between two elliptic curves
 defined over a finite field \$\GF{q}\$ of characteristic \$p\$. We
 describe an algorithm the asymptotic time complexity of which is
 equal to \$\SoftO(\ell^2(1+\ell/p)\log q)\$ bit operations. This
 algorithm is particularly useful when \$\ell \&gt; p\$ and as a
 consequence, we obtain an improvement of the complexity of the SEA
 point counting algorithm for small values of \$p\$. More precisely, we
 obtain a heuristic time complexity \$\SoftO(\log^{4} q)\$ and a space
 complexity \$O(\log^{2} q)\$, in the previously unfavorable case where
 \$p \simeq \log q\$.  Compared to the best previous algorithms, the
 memory requirements of our SEA variation are smaller by a \$\log^2 q\$
 factor.},
    author = {Joux, Antoine and Lercier, Reynald},
    citeulike-article-id = {7300830},
    citeulike-linkout-0 = {http://eprint.iacr.org/2006/176},
    day = {19},
    howpublished = {Cryptology ePrint Archive, Report 2006/176},
    keywords = {elliptic\_curve, finite\_field, isogenies},
    month = may,
    posted-at = {2010-06-14 19:24:29},
    priority = {0},
    title = {Counting points on elliptic curves in medium characteristic},
    url = {http://eprint.iacr.org/2006/176},
    year = {2006}
}

@inproceedings{elkies98,
    abstract = {The paper is concerned with the problem of computingthe cardinality of elliptic curves over finitefields and related questions likecomputing the cardinality of more general curves(varieties), finding simple equations formodular curves \$X\_0(l)\$, etc.<P> The first polynomial time algorithm for computing the cardinality of elliptic curves overa finite field was discovered by R. Schoof \ref[Math. Comp. <strong>44</strong> (1985),no. 170, 483--494; <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=777280\&loc=fromrevtext">MR0777280 (86e:11122)</A>].Schoof's main idea is to compute the residuemodulo \$l\$ of the trace \$t\$ of theFrobenius endomorphism from its actionon \$l\$-torsion points. This is done for enough small primes \$l\$. <P> In a series of unpublished manuscripts, Atkin providedmany improvements that made this algorithm practical.Atkin used the modular equation \$E\_l(X,Y)\$ of \$X\_0(l)\$. The decomposition of thepolynomial \$E\_l(X,j)\in {\bold F}\_p[X]\$ provides some information about \$t\bmod l\$ if\$j\$ is the modular invariant of the elliptic curve.Atkin developed efficient techniquesfor computing nice modular equations (nicer than \$E\_l(X,Y)\$)and was able to efficiently implementthe resulting algorithm.<P> Further improvements were proposed by the author. He showed that for one \$l\$ over two (namely when the Frobeniusaction on the \$l\$-torsion splits) there is a much faster wayto compute the residue modulo \$l\$ of the trace \$t\$.<P> A nice survey paper on these questions is that of Schoof\ref[J. Th\'{e}or. Nombres Bordeaux <strong>7</strong> (1995), no. 1, 219--254; <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=1413578\&loc=fromrevtext">MR1413578 (97i:11070)</A>]. <P> The present  paper recalls these various contributionsand provides several remarks, generalizations and improvements.Several ideas are published here for the first time.<P> One sectionis devotedto the problem of computing the cardinality of curves of smallgenus (up to \$9\$)and varieties of higher dimensions.The problem of computing modular equationswith smallest possible coefficients is discussedboth in general and for some special values of the level \$l\$ for which various geometric andnumber-theoretic observations are given.<P> \{See also the following review[<A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=1486832\&loc=fromrevtext">MR1486832 (99a:11079)</A>].\}<P>{For the entire collection see <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=1486828\&loc=fromrevtext">MR1486828 (98g:11001)</A>.}},
    address = {Providence, RI},
    author = {Elkies, Noam D.},
    booktitle = {Computational perspectives on number theory (Chicago, IL, 1995)},
    citeulike-article-id = {805332},
    citeulike-linkout-0 = {http://www.ams.org/mathscinet-getitem?mr=1486831},
    keywords = {elliptic\_curve, finite\_field, isogenies, modular\_curves},
    mrnumber = {MR1486831},
    pages = {21--76},
    posted-at = {2010-06-14 19:22:40},
    priority = {2},
    publisher = {AMS International Press},
    series = {Studies in Advanced Mathematics},
    title = {Elliptic and modular curves over finite fields and related computational issues},
    url = {http://www.ams.org/mathscinet-getitem?mr=1486831},
    volume = {7},
    year = {1998}
}

@phdthesis{couveignes94,
    author = {Couveignes, Jean-Marc},
    citeulike-article-id = {7300816},
    keywords = {isogenies},
    posted-at = {2010-06-14 19:21:22},
    priority = {2},
    school = {Universit\'{e} de Bordeaux},
    title = {{Quelques calculs en th{\'{e}}orie des nombres}},
    year = {1994}
}

@article{yorgey:typeclassopedia,
    author = {Yorgey, Brent},
    citeulike-article-id = {7300811},
    day = {12},
    editor = {Swierstra, Wouter},
    journal = {The Monad.Reader},
    keywords = {haskell, type\_classes},
    month = mar,
    number = {13},
    pages = {17--68},
    posted-at = {2010-06-14 19:17:11},
    priority = {0},
    title = {Typeclassopedia},
    year = {2009}
}

@book{vollmer,
    address = {Secaucus, NJ, USA},
    author = {Vollmer, Heribert},
    citeulike-article-id = {2906263},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=520668},
    isbn = {3540643109},
    keywords = {algebraic\_complexity, circuit\_complexity, uniformity},
    posted-at = {2010-06-14 19:15:26},
    priority = {2},
    publisher = {Springer-Verlag New York, Inc.},
    title = {Introduction to Circuit Complexity: A Uniform Approach},
    url = {http://portal.acm.org/citation.cfm?id=520668},
    year = {1999}
}

@inproceedings{villard00,
    abstract = {. We probabilistically determine the Frobenius form and thus the characteristic polynomial of a matrix A 2 F  nn  by O(n log(n)) multiplications of A by vectors and O n  2  log  2  (n) log log(n)    arithmetic operations in the eld F. The parameter  is the number of distinct invariant factors of A, it is less than 3  p  n=2 in the worst case. The method requires O(n) storage space in addition to that needed for the matrix A. 1 Introduction  The known complexity estimates of the computation of the characteristic polynomial and a fortiori, of the Frobenius normal form of special { sparse or black box { square matrices A over a eld F, seem to not be satisfactory. We refer to Kaltofen [8, Open Problem 3] and to Pan et al. [16, 15] for discussions on this subject and survey of current solutions. We denote by M(n) the number of operations in F required for nn matrix multiplications. The characteristic polynomial of a general matrix A can be computed at cost of O(n  3  ) or O(M(n) log...}}},
    author = {Villard, Gilles},
    booktitle = {CASC 2000 Proc. the Third International Workshop on Computer Algebra in Scientific Computing},
    citeulike-article-id = {7300798},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.2012},
    keywords = {black\_box, characteristic\_polynomial, frobenius, minimal\_polynomial, sparse\_linear\_equations},
    pages = {395--407},
    posted-at = {2010-06-14 19:14:40},
    priority = {2},
    title = {Computing the Frobenius Normal Form of a Sparse Matrix},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.2012},
    year = {2000}
}

@article{strachey00,
    abstract = {This paper forms the substance of a course of lectures given at the International Summer School in Computer Programming at Copenhagen in August, 1967. The lectures were originally given from notes and the paper was written after the course was finished. In spite of this, and only partly because of the shortage of time, the paper still retains many of the shortcomings of a lecture course. The chief of these are an uncertainty of aim—it is never quite clear what sort of audience there will be for such lectures—and an associated switching from formal to informal modes of presentation which may well be less acceptable in print than it is natural in the lecture room. For these (and other) faults, I apologise to the reader.},
    author = {Strachey, Christopher},
    citeulike-article-id = {1454140},
    citeulike-linkout-0 = {http://dx.doi.org/10.1023/A:1010000313106},
    citeulike-linkout-1 = {http://www.springerlink.com/content/x878vn91l4763752},
    day = {1},
    doi = {10.1023/A:1010000313106},
    issn = {13883690},
    journal = {Higher-Order and Symbolic Computation},
    keywords = {function\_level\_programming, functional\_language, polymorphic, programming\_languages, programming\_style},
    month = apr,
    number = {1},
    pages = {11--49},
    posted-at = {2010-06-14 19:13:06},
    priority = {0},
    publisher = {Springer Netherlands},
    title = {Fundamental Concepts in Programming Languages},
    url = {http://dx.doi.org/10.1023/A:1010000313106},
    volume = {13},
    year = {2000}
}

@article{sergeev08,
    abstract = {Abstract  The {Baur-Strassen} method implies L(∇f) ⩽ {4L}(f), where L(f) is the complexity of computing a rational function f by arithmetic circuits, and ∇f is the gradient of f. We show that L(∇ f) ⩽ {3L}(f) + n, where n is the number of variables in f. In addition, the depth of a circuit for the gradient is estimated.},
    author = {Sergeev, Igor},
    citeulike-article-id = {3614797},
    citeulike-linkout-0 = {http://dx.doi.org/10.1134/S1990478908030095},
    citeulike-linkout-1 = {http://www.springerlink.com/content/n732u510264874l6},
    day = {1},
    doi = {10.1134/S1990478908030095},
    journal = {Journal of Applied and Industrial Mathematics},
    keywords = {algebraic\_complexity, automatic\_differentiation, gradients, transposition},
    number = {3},
    pages = {385--396},
    posted-at = {2010-06-14 19:11:05},
    priority = {0},
    title = {On the complexity of the gradient of a rational function},
    url = {http://dx.doi.org/10.1134/S1990478908030095},
    volume = {2},
    year = {2008}
}

@inproceedings{kaltofen+saunders91,
    address = {London, UK},
    author = {Kaltofen, Erich and Saunders, B. David},
    booktitle = {AAECC-9: Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes},
    citeulike-article-id = {7300776},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=646027.676885},
    isbn = {3-540-54522-0},
    keywords = {black\_box, sparse\_linear\_equations, sparse\_linear\_systems},
    pages = {29--38},
    posted-at = {2010-06-14 19:09:49},
    priority = {0},
    publisher = {Springer-Verlag},
    title = {On Wiedemann's Method of Solving Sparse Linear Systems},
    url = {http://portal.acm.org/citation.cfm?id=646027.676885},
    year = {1991}
}

@inproceedings{hopcroft+musinski73,
    abstract = {The paper considers the complexity of bilinear forms in a noncommutative ring. The dual of a computation is defined and applied to matrix multiplication and other bilinear forms. It is shown that the dual of an optimal computation gives an optimal computation for a dual problem. An nxm by mxp matrix product is shown to be the dual of an nxp by pxm or an mxn by nxp matrix product implying that each of the matrix products requires the same number of multiplications to compute. Finally an algorithm for computing a single bilinear form over a noncommutative ring with a minimum number of multiplications is derived by considering a dual problem.},
    address = {New York, NY, USA},
    author = {Hopcroft, John E. and Musinski, Jean},
    booktitle = {STOC '73: Proceedings of the fifth annual ACM symposium on Theory of computing},
    citeulike-article-id = {7300772},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=800125.804038},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/800125.804038},
    doi = {10.1145/800125.804038},
    keywords = {duality, matrix\_multiplication, transposition},
    location = {Austin, Texas, United States},
    pages = {73--87},
    posted-at = {2010-06-14 19:08:08},
    priority = {0},
    publisher = {ACM},
    title = {Duality applied to the complexity of matrix multiplications and other bilinear forms},
    url = {http://dx.doi.org/10.1145/800125.804038},
    year = {1973}
}

@article{griewank92,
    abstract = {this paper we assume that the simplicity and convenience of taking snapshots of the whole state space outweigh the savings in storage that might be achieved by a more localized approach. As we will briefly discuss in the final section, the operating system may automatically provide this localization through its virtual memory manager.  {LOGARITHMIC} {REVERSE} {DIFFERENTIATION} 11},
    author = {Griewank, Andreas},
    citeulike-article-id = {7300766},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.1450},
    journal = {Optimization Methods and software},
    keywords = {automatic\_differentiation, spatial\_complexity},
    pages = {35--54},
    posted-at = {2010-06-14 19:04:13},
    priority = {1},
    publisher = {Taylor \& Francis},
    title = {Achieving Logarithmic Growth Of Temporal And Spatial Complexity In Reverse Automatic Differentiation},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.1450},
    volume = {1},
    year = {1992}
}

@article{gashkov+gashkov05,
    author = {Gashkov, Sergey B. and Gashkov, Igor B.},
    citeulike-article-id = {350534},
    citeulike-linkout-0 = {http://dx.doi.org/10.1163/156939205774464936},
    citeulike-linkout-1 = {http://www.ingentaconnect.com/content/vsp/dma/2005/00000015/00000004/art00001},
    doi = {10.1163/156939205774464936},
    issn = {0924-9265},
    journal = {Discrete Mathematics and Applications},
    keywords = {automatic\_differentiation, differentials, gradients, transposition},
    month = may,
    number = {4},
    pages = {327--350},
    posted-at = {2010-06-14 18:57:32},
    priority = {3},
    publisher = {VSP},
    title = {On the complexity of calculation of differentials and gradients},
    url = {http://dx.doi.org/10.1163/156939205774464936},
    volume = {15},
    year = {2005}
}

@article{massey69,
    abstract = {It is shown in this paper that the iterative algorithm introduced by Berlekamp for decoding {BCH} codes actually provides a general solution to the problem of synthesizing the shortest linear feedback shift register capable of generating a prescribed finite sequence of digits. The shift-register approach leads to a simple proof of the validity of the algorithm as well as providing additional insight into its properties. The equivalence of the decoding problem for {BCH} codes to a shift-register synthesis problem is demonstrated, and other applications for the algorithm are suggested.},
    author = {Massey, Jim},
    citeulike-article-id = {581085},
    citeulike-linkout-0 = {http://dx.doi.org/10.1109/TIT.1969.1054260},
    citeulike-linkout-1 = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1054260},
    day = {06},
    doi = {10.1109/TIT.1969.1054260},
    issn = {0018-9448},
    journal = {IEEE Transactions on Information Theory},
    keywords = {bch\_codes, berlekamp, coding\_theory, linear\_feedback\_shift, linear\_recurring\_sequence},
    month = jan,
    number = {1},
    pages = {122--127},
    posted-at = {2010-06-14 18:55:55},
    priority = {0},
    title = {Shift-register synthesis and {BCH} decoding},
    url = {http://dx.doi.org/10.1109/TIT.1969.1054260},
    volume = {15},
    year = {1969}
}

@inproceedings{eberly00,
    abstract = {A new randomized algorithm is presented for computation of the Frobenius form and transition matrix for an  n  \&times;  n  matrix over a field. Using standard matrix and polynomial arithmetic, the algorithm has an asymptotic expected complexity that matches the worst case complexity of the best known deterministic algorithmic for this problem, recently given by Storjohann and Villard [16]. The new algorithm is based on the evaluation of Krylov spaces, rather than an climination technique, and may therefore be superior when applied to sparse or structured matrices with a small number of invariant factors.},
    address = {New York, NY, USA},
    author = {Eberly, Wayne},
    booktitle = {ISSAC '00: Proceedings of the 2000 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {7300737},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=345542.345596},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/345542.345596},
    doi = {10.1145/345542.345596},
    isbn = {1-58113-218-2},
    keywords = {algebraic\_complexity, black\_box, finite\_field, frobenius},
    location = {St. Andrews, Scotland},
    pages = {106--113},
    posted-at = {2010-06-14 18:53:49},
    priority = {2},
    publisher = {ACM},
    title = {Black box Frobenius decompositions over small fields},
    url = {http://dx.doi.org/10.1145/345542.345596},
    year = {2000}
}

@article{bordewijk57,
    abstract = {{Summary\&nbsp;\&nbsp;As} an extension of the concept of reciprocity a new concept calledinterreciprocity is introduced. The latter concept applies to both reciprocal and non-reciprocal structures. On behalf of the use of inter-reciprocity for electrical network investigations a unique topological network operation, calledtransposition, has been devised in such a way that a network and its transpose are inter-reciprocal. This transposition is applied in a number of direct calculations as well as in considerations of a more general character concerning transmission and noise properties of gyrator, triode and transistor networks.},
    author = {Bordewijk, J.},
    citeulike-article-id = {6396833},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/BF02920362},
    citeulike-linkout-1 = {http://www.springerlink.com/content/5871843321111834},
    day = {1},
    doi = {10.1007/BF02920362},
    issn = {0365-7140},
    journal = {Applied Scientific Research, Section B},
    keywords = {transposition},
    month = dec,
    number = {1},
    pages = {1--74},
    posted-at = {2010-06-14 18:51:48},
    priority = {0},
    title = {Inter-reciprocity applied to electrical networks},
    url = {http://dx.doi.org/10.1007/BF02920362},
    volume = {6},
    year = {1957}
}

@article{schoof85,
    abstract = {In this paper we present a deterministic algorithm to compute the number of \$\mathbf{F}\_q\$-points of an elliptic curve that is defined over a finite field \$\mathbf{F}\_q\$ and which is given by a Weierstrass equation. The algorithm takes \$O(\log^9 q)\$ elementary operations. As an application we give an algorithm to compute square roots \$\operatorname{mod} p\$. For fixed \$x \in \mathbf{Z}\$, it takes \$O(\log^9 p)\$ elementary operations to compute \$\sqrt x \operatorname{mod} p\$.},
    author = {Schoof, Ren\'{e}},
    citeulike-article-id = {7300735},
    citeulike-linkout-0 = {http://dx.doi.org/10.2307/2007968},
    citeulike-linkout-1 = {http://www.jstor.org/stable/2007968},
    doi = {10.2307/2007968},
    issn = {00255718},
    journal = {Math. Comp.},
    keywords = {elliptic\_curve, finite\_field, schoof},
    number = {170},
    pages = {483--494},
    posted-at = {2010-06-14 18:50:35},
    priority = {0},
    publisher = {American Mathematical Society},
    title = {Elliptic Curves Over Finite Fields and the Computation of Square Roots mod \(p\)},
    url = {http://dx.doi.org/10.2307/2007968},
    volume = {44},
    year = {1985}
}

@article{schoof95,
    abstract = {Les Dix-huiti\`{e}mes Journ\'{e}es Arithm\'{e}tiques (Bordeaux, 1993)},
    author = {Schoof, Ren\'{e}},
    citeulike-article-id = {805313},
    citeulike-linkout-0 = {http://www.ams.org/mathscinet-getitem?mr=1413578},
    journal = {Journal de Th\'{e}orie des Nombres de Bordeaux},
    keywords = {elliptic\_curve, finite\_field, isogenies, schoof},
    mrnumber = {MR1413578},
    number = {1},
    pages = {219--254},
    posted-at = {2010-06-14 18:49:20},
    priority = {0},
    title = {Counting points on elliptic curves over finite fields},
    url = {http://www.ams.org/mathscinet-getitem?mr=1413578},
    volume = {7},
    year = {1995}
}

@unpublished{elkies92,
    author = {Elkies, Noam D.},
    citeulike-article-id = {7300729},
    howpublished = {manuscript, Boston MA},
    keywords = {elliptic\_curve, isogenies, schoof},
    posted-at = {2010-06-14 18:45:22},
    priority = {2},
    title = {Explicit isogenies},
    year = {1992}
}

@article{bostan+morain+salvy+schost08,
    abstract = {We survey algorithms for computing isogenies between elliptic curvesdefined over a field of characteristic either 0 or a large prime. Weintroduce a new algorithm that computes an isogeny of degree  ell (  elldifferent from the characteristic) in time quasi-linear with respect to ell E This is based in particular on fast algorithms for power seriesexpansion of the Weierstrass  wp -function and related functions.},
    author = {Bostan, Alin and Morain, Fran\c{c}ois and Salvy, Bruno and Schost, \'{E}ric},
    citeulike-article-id = {7300722},
    citeulike-linkout-0 = {http://dx.doi.org/10.1090/S0025-5718-08-02066-8},
    citeulike-linkout-1 = {http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2008MaCom..77.1755B},
    doi = {10.1090/S0025-5718-08-02066-8},
    journal = {Math. Comp.},
    keywords = {algorithm, differential\_equations, elliptic\_curve, isogenies, weierstrass},
    month = sep,
    pages = {1755--1778},
    posted-at = {2010-06-14 18:42:41},
    priority = {0},
    title = {Fast algorithms for computing isogenies between elliptic curves},
    url = {http://dx.doi.org/10.1090/S0025-5718-08-02066-8},
    volume = {77},
    year = {2008}
}

@unpublished{atkin88,
    author = {Atkin, A. O. L.},
    citeulike-article-id = {7300712},
    howpublished = {manuscript, Chicago IL},
    keywords = {elliptic\_curve, isogenies, schoof},
    posted-at = {2010-06-14 18:38:52},
    priority = {2},
    title = {The number of points on an elliptic curve modulo a prime},
    year = {1988}
}

@article{atkin+morain93,
    abstract = {Using the parametrizations of Kubert, we show how to produce infinite families of elliptic curves which have prescribed nontrivial torsion over Q and rank at least one. These curves can be used to speed up the {ECM} factorization algorithm of Lenstra. We also briefly discuss curves with complex multiplication},
    author = {Atkin, A. O. L. and Morain, Fran\c{c}ois},
    citeulike-article-id = {3474702},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.2624},
    journal = {Math. Comp.},
    keywords = {ecm, elliptic\_curve, elliptic\_curve\_method, factorization},
    pages = {399--405},
    posted-at = {2010-06-14 18:37:36},
    priority = {2},
    title = {Finding suitable curves for the elliptic curve method of factorization},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.2624},
    volume = {60},
    year = {1993}
}

@article{wang+zhu88,
    abstract = {The Fourier transform over finite fields is mainly required in theencoding and decoding of {Reed-Solomon} and {BCH} codes. An algorithm forcomputing the Fourier transform over any finite {fieldGF}( p m ) is introduced. It requires {onlyO}( n (log  n ) 2 /4) additions and the samenumber of multiplications for an  n -point transform and allowsin some fields a further reduction of the number of multiplications {toO}( n  log  n ). Because of its highly regular structure,this algorithm can be easily implementation by {VLSI} technology},
    author = {Wang, Yao and Zhu, Xuelong},
    citeulike-article-id = {840894},
    citeulike-linkout-0 = {http://dx.doi.org/10.1109/49.1926},
    citeulike-linkout-1 = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1926},
    doi = {10.1109/49.1926},
    issn = {07338716},
    journal = {IEEE Journal on Selected Areas in Communications},
    keywords = {artin-schreier, finite\_field, polynomial\_multiplication, vlsi\_implementation},
    month = apr,
    number = {3},
    pages = {572--577},
    posted-at = {2010-06-14 18:35:55},
    priority = {0},
    title = {A fast algorithm for the {F}ourier transform over finite fields and its {VLSI} implementation},
    url = {http://dx.doi.org/10.1109/49.1926},
    volume = {6},
    year = {1988}
}

@misc{shoup2003ntl,
    author = {Shoup, Victor},
    citeulike-article-id = {7300697},
    howpublished = {\url{http://www.shoup.net/ntl}},
    keywords = {ntl},
    posted-at = {2010-06-14 18:33:18},
    priority = {2},
    title = {{NTL}: A library for doing number theory},
    year = {2003}
}

@article{lercier+sirvent08,
    author = {Lercier, Reynald and Sirvent, Thomas},
    citeulike-article-id = {7300675},
    citeulike-linkout-0 = {http://perso.univ-rennes1.fr/reynald.lercier/file/LS08.pdf},
    citeulike-linkout-1 = {http://jtnb.cedram.org/jtnb-bin/item?id=JTNB_2008__20_3_783_0},
    comment = {Eq. A.1 has a minor error: it should be modulo x^{d+1} instead of modulo x^d},
    journal = {Journal de th\'{e}orie des nombres de Bordeaux},
    keywords = {elliptic\_curve, isogenies},
    number = {3},
    pages = {783--797},
    posted-at = {2010-06-14 18:26:31},
    priority = {0},
    title = {On {E}lkies subgroups of \(\ell\)-torsion points in elliptic curves defined over a finite field},
    url = {http://perso.univ-rennes1.fr/reynald.lercier/file/LS08.pdf},
    volume = {20},
    year = {2008}
}

@phdthesis{lercier-algorithmique,
    author = {Lercier, Reynald},
    citeulike-article-id = {7300668},
    keywords = {lercier},
    month = jun,
    posted-at = {2010-06-14 18:24:56},
    priority = {0},
    school = {LIX -- CNRS},
    title = {Algorithmique des courbes elliptiques dans les corps finis},
    year = {1997}
}

@book{burgisser+clausen-shokrollahi,
    abstract = {This is the first book to present an up-to-date and self-contained account of Algebraic Complexity Theory that is both comprehensive and unified. Requiring of the reader only some basic algebra and offering over 350 exercises, it is well-suited as a textbook for beginners at graduate level. With its extensive bibliography covering about 500 research papers, this text is also an ideal reference book for the professional researcher. The subdivision of the contents into 21 more or less independent chapters enables readers to familiarize themselves quickly with a specific topic, and facilitates the use of this book as a basis for complementary courses in other areas such as computer algebra.},
    author = {B\"{u}rgisser, Peter and Clausen, Michael and Shokrollahi, Mohammad A.},
    citeulike-article-id = {494273},
    citeulike-linkout-0 = {http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&amp;path=ASIN/3540605827},
    citeulike-linkout-1 = {http://www.amazon.de/exec/obidos/redirect?tag=citeulike01-21&amp;path=ASIN/3540605827},
    citeulike-linkout-2 = {http://www.amazon.fr/exec/obidos/redirect?tag=citeulike06-21&amp;path=ASIN/3540605827},
    citeulike-linkout-3 = {http://www.amazon.co.uk/exec/obidos/ASIN/3540605827/citeulike00-21},
    citeulike-linkout-4 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/3540605827},
    citeulike-linkout-5 = {http://www.worldcat.org/isbn/3540605827},
    citeulike-linkout-6 = {http://books.google.com/books?vid=ISBN3540605827},
    citeulike-linkout-7 = {http://www.amazon.com/gp/search?keywords=3540605827&index=books&linkCode=qs},
    citeulike-linkout-8 = {http://www.librarything.com/isbn/3540605827},
    day = {14},
    howpublished = {Hardcover},
    isbn = {3540605827},
    keywords = {algebraic\_complexity, transposition},
    month = feb,
    posted-at = {2010-06-14 18:22:07},
    priority = {0},
    publisher = {Springer},
    series = {A Series of Comprehensive Studies in Mathematics},
    title = {Algebraic Complexity Theory},
    url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/3540605827},
    volume = {315},
    year = {1997}
}

@article{brent93,
    abstract = {For odd square-free <latex>\$n > 1\$</latex> the cyclotomic polynomial Φ<sub>n</sub>(x) satisfies the identity of Gauss, 4Φ<sub>n</sub>(x) = A<sup>2</sup><sub>n</sub> - (-1)<sup>(n - {1)/2</sup>nB}<sup>2</sup><sub>n</sub>. A similar identity of Aurifeuille, Le Lasseur, and Lucas is Φ<sub>n</sub>((-1)<sup>(n - 1)/2</sup>x) = C<sup>2</sup><sub>n</sub> - {nxD}<sup>2</sup><sub>n</sub> or, in the case that n is even and square-free, ±Φ<sub>n/2</sub>(-x<sup>2</sup>) = C<sup>2</sup><sub>n</sub> - {nxD}<sup>2</sup><sub>n</sub>. Here, A<sub>n</sub>(x), ..., D<sub>n</sub>(x) are polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require O(n<sup>2</sup>) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for A<sub>n</sub>(x), ..., D<sub>n</sub>(x), and illustrate the application to integer factorization with some numerical examples.},
    author = {Brent, Richard P.},
    citeulike-article-id = {7300624},
    citeulike-linkout-0 = {http://dx.doi.org/10.2307/2152941},
    citeulike-linkout-1 = {http://www.jstor.org/stable/2152941},
    doi = {10.2307/2152941},
    issn = {00255718},
    journal = {Math. Comp.},
    keywords = {cyclotomic\_polynomial, factoring\_polynomials, finite\_field},
    number = {203},
    pages = {131--149},
    posted-at = {2010-06-14 18:17:20},
    priority = {0},
    publisher = {American Mathematical Society},
    title = {On Computing Factors of Cyclotomic Polynomials},
    url = {http://dx.doi.org/10.2307/2152941},
    volume = {61},
    year = {1993}
}

@article{garcia+stichtenoth96,
    author = {Garcia, A. and Stichtenoth, H.},
    citeulike-article-id = {7108052},
    citeulike-linkout-0 = {http://www.ingentaconnect.com/content/ap/nt/1996/00000061/00000002/art00147},
    issn = {0022-314X},
    journal = {Journal of Number Theory},
    keywords = {algebraic\_curves, algebraic\_geometry, artin-schreier, coding\_theory},
    month = dec,
    pages = {248--273},
    posted-at = {2010-06-14 18:11:28},
    priority = {3},
    publisher = {Academic Press},
    title = {On the Asymptotic Behaviour of Some Towers of Function Fields over Finite Fields},
    url = {http://www.ingentaconnect.com/content/ap/nt/1996/00000061/00000002/art00147},
    year = {1996}
}

@inproceedings{guruswami+sudan98,
    abstract = {Given an error-correcting code of block length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding Reed-Solomon codes. The list decoding problem for Reed-Solomon codes reduces to the following ``curve-fitting'' problem over a field F: Given n points (x[i].y[i]), 1 <= i <= n (where x[i],y[i] are elements of the field F), a degree parameter k and error parameter e, find all univariate polynomials P over the field F of degree at most k such that y[i]=P(x[i]) for all but at most e values of i in the range 1 <= i <= n. We give an algorithm that solves this problem for e < n - sqrt{kn}, which improves over the previous best result due to Sudan [Journal of Complexity, Vol. 13, 1997], for every choice of k and n. Of particular interest is the case of k/n > 1/3, where the result yields the first asymptotic improvement since Peterson's original algorithm nearly four decades ago.The algorithm generalizes to solve the list decoding problem for other algebraic codes, specifically alternant codes (a class of codes including BCH codes) and algebraic-geometric codes. In both cases, we obtain a list decoding algorithm that corrects up to n - sqrt{n(n-d')} errors, where n is the block length and d' is the designed distance of the code. The improvement for the case of algebraic-geometric codes extends the methods of Shokrollahi and Wasserman [STOC '98] and improves upon their bound for every choice of n and d'. We also present some other consequences of our algorithm including a solution to a weighted curve fitting problem, which is of use in soft-decision decoding algorithms for Reed-Solomon codes.},
    address = {Washington, DC, USA},
    author = {Guruswami, Venkatesan and Sudan, Madhu},
    booktitle = {FOCS '98: Proceedings of the 39th Annual Symposium on Foundations of Computer Science},
    citeulike-article-id = {7300529},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=796393},
    isbn = {0-8186-9172-7},
    keywords = {algebraic\_curves, algebraic\_geometry, coding\_theory, list\_decoding, reed\_solomon},
    pages = {28+},
    posted-at = {2010-06-14 18:06:49},
    priority = {2},
    publisher = {IEEE Computer Society},
    title = {Improved Decoding of {Reed-Solomon} and {Algebraic-Geometric} Codes},
    url = {http://portal.acm.org/citation.cfm?id=796393},
    year = {1998}
}

@inproceedings{miller86,
    address = {New York, NY, USA},
    author = {Miller, Victor S.},
    booktitle = {Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85},
    citeulike-article-id = {5722624},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=25413},
    isbn = {0-387-16463-4},
    keywords = {cryptography, elliptic\_curve},
    location = {Santa Barbara, California, United States},
    pages = {417--426},
    posted-at = {2010-06-14 18:05:04},
    priority = {2},
    publisher = {Springer-Verlag New York, Inc.},
    title = {Use of elliptic curves in cryptography},
    url = {http://portal.acm.org/citation.cfm?id=25413},
    year = {1986}
}

@inproceedings{MOV,
    abstract = {Note: {OCR} errors may be found in this Reference List extracted from the full text article.  {ACM} has opted to expose the complete List rather than only correct and linked references.},
    address = {New York, NY, USA},
    author = {Menezes, Alfred and Vanstone, Scott and Okamoto, Tatsuaki},
    booktitle = {STOC '91: Proceedings of the twenty-third annual ACM symposium on Theory of computing},
    citeulike-article-id = {7300505},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=103434},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/103418.103434},
    doi = {10.1145/103418.103434},
    isbn = {0-89791-397-3},
    keywords = {cryptography, elliptic\_curve, finite\_field, mov, supersingular},
    location = {New Orleans, Louisiana, United States},
    pages = {80--89},
    posted-at = {2010-06-14 18:03:48},
    priority = {2},
    publisher = {ACM},
    title = {Reducing elliptic curve logarithms to logarithms in a finite field},
    url = {http://dx.doi.org/10.1145/103418.103434},
    year = {1991}
}

@inproceedings{mauer+menezes+teske01,
    address = {London, UK},
    author = {Maurer, Markus and Menezes, Alfred and Teske, Edlyn},
    booktitle = {INDOCRYPT '01: Proceedings of the Second International Conference on Cryptology in India},
    citeulike-article-id = {7300494},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=716020},
    isbn = {3-540-43010-5},
    keywords = {elliptic\_curve, ghs, isogenies, weil\_descent},
    pages = {195--213},
    posted-at = {2010-06-14 18:02:45},
    priority = {2},
    publisher = {Springer-Verlag},
    title = {Analysis of the {GHS} {W}eil Descent Attack on the {ECDLP} over Characteristic Two Finite Fields of Composite Degree},
    url = {http://portal.acm.org/citation.cfm?id=716020},
    year = {2001}
}

@incollection{GHS,
    address = {Berlin},
    author = {Galbraith, Steven D. and Hess, Florian and Smart, Nigel P.},
    booktitle = {Advances in cryptology---{EUROCRYPT} 2002 ({A}msterdam)},
    citeulike-article-id = {7300483},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=715831},
    isbn = {3-540-43553-0},
    keywords = {cryptography, elliptic\_curve, ghs, isogenies, weil\_descent},
    mrnumber = {MR1975526 (2004f:94060)},
    pages = {29--44},
    posted-at = {2010-06-14 18:01:45},
    priority = {2},
    publisher = {Springer},
    series = {Lecture Notes in Comput. Sci.},
    title = {Extending the {GHS} {W}eil descent attack},
    url = {http://portal.acm.org/citation.cfm?id=715831},
    volume = {2332},
    year = {2002}
}

@inproceedings{guruswami+sudan00,
    abstract = {Note: {OCR} errors may be found in this Reference List extracted from the full text article.  {ACM} has opted to expose the complete List rather than only correct and linked references.},
    address = {New York, NY, USA},
    author = {Guruswami, Venkatesan and Sudan, Madhu},
    booktitle = {STOC '00: Proceedings of the thirty-second annual ACM symposium on Theory of computing},
    citeulike-article-id = {754970},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=335305.335327},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/335305.335327},
    doi = {10.1145/335305.335327},
    isbn = {1-58113-184-4},
    keywords = {coding\_theory, list\_decoding, reed\_solomon},
    location = {Portland, Oregon, United States},
    pages = {181--190},
    posted-at = {2010-06-14 18:00:33},
    priority = {2},
    publisher = {ACM},
    title = {List decoding algorithms for certain concatenated codes},
    url = {http://dx.doi.org/10.1145/335305.335327},
    year = {2000}
}

@inproceedings{lercier+morain95,
    abstract = {Cryptographic schemes using elliptic curves over finite fields require the computation of the cardinality of the curves. Dramatic progress have been achieved recently in that field by various authors. The aim of this article is to highlight part of these improvements and to describe an efficient implementation of them in the particular case of the fields  {GF} (2 n ), for  n  \&\#8804; 600.},
    address = {Berlin, Heidelberg},
    author = {Lercier, Reynald and Morain, Fran\c{c}ois},
    booktitle = {EUROCRYPT'95: Proceedings of the 14th annual international conference on Theory and application of cryptographic techniques},
    citeulike-article-id = {7300454},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1755019},
    isbn = {3-540-59409-4},
    keywords = {elliptic\_curve, schoof},
    location = {Saint-Malo, France},
    pages = {79--94},
    posted-at = {2010-06-14 17:58:34},
    priority = {0},
    publisher = {Springer-Verlag},
    title = {Counting the number of points on elliptic curves over finite fields: strategies and performances},
    url = {http://portal.acm.org/citation.cfm?id=1755019},
    year = {1995}
}

@inproceedings{hess03,
    abstract = {We generalize the Weil descent construction of the {GHS} attack to arbitrary {Artin-Schreier} extensions. We give a formula for the characteristic polynomial of Frobenius of the obtained curves and prove that the large cyclic factor of the input elliptic curve is not contained in the kernel of the composition of the conorm and norm maps. As an application we almost square the number of elliptic curves which succumb to the basic {GHS} attack, thereby weakening curves over F 2 155  further. We also discuss other possible extensions or variations of the {GHS} attack and conclude that they are not likely to yield further improvements.},
    address = {Berlin, Heidelberg},
    author = {Hess, Florian},
    booktitle = {EUROCRYPT'03: Proceedings of the 22nd international conference on Theory and applications of cryptographic techniques},
    citeulike-article-id = {7300422},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1766203},
    isbn = {3-540-14039-5},
    keywords = {cryptography, elliptic\_curve, ghs, isogenies, weil\_descent},
    location = {Warsaw, Poland},
    pages = {374--387},
    posted-at = {2010-06-14 17:56:30},
    priority = {2},
    publisher = {Springer-Verlag},
    title = {The {GHS} attack revisited},
    url = {http://portal.acm.org/citation.cfm?id=1766203},
    year = {2003}
}

@article{sudan97,
    abstract = {We present a randomized algorithm which takes as inputndistinct points {(xi,yi)}i= 1nfromF×F(whereFis a field) and integer parameterstanddand returns a list of all univariate polynomialsfoverFin the variablexof degree at mostdwhich agree with the given set of points in at leasttplaces (i.e.,yi=f(xi) for at leasttvalues ofi), providedt= \^{I}{\copyright}(). The running time is bounded by a polynomial inn. This immediately provides a maximum likelihood decoding algorithm for Reed Solomon Codes, which works in a setting with a larger number of errors than any previously known algorithm. To the best of our knowledge, this is the first efficient (i.e., polynomial time bounded) algorithm which provides error recovery capability beyond the error-correction bound of a code for any efficient (i.e., constant or even polynomial rate) code.},
    address = {Orlando, FL, USA},
    author = {Sudan, Madhu},
    citeulike-article-id = {495137},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=270513},
    citeulike-linkout-1 = {http://dx.doi.org/10.1006/jcom.1997.0439},
    citeulike-linkout-2 = {http://www.sciencedirect.com/science/article/B6WHX-45SJBN4-P/2/1efc9cb68d2c21c9a032da7d5cd1b714},
    doi = {10.1006/jcom.1997.0439},
    issn = {0885064X},
    journal = {Journal of Complexity},
    keywords = {coding\_theory, list\_decoding, reed\_solomon},
    month = mar,
    number = {1},
    pages = {180--193},
    posted-at = {2010-06-14 17:53:54},
    priority = {2},
    publisher = {Academic Press, Inc.},
    title = {Decoding of Reed Solomon Codes beyond the {Error-Correction} Bound},
    url = {http://dx.doi.org/10.1006/jcom.1997.0439},
    volume = {13},
    year = {1997}
}

@book{guruswami,
    address = {Secaucus, NJ, USA},
    author = {Guruswami, Venkatesan},
    citeulike-article-id = {7300390},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1076300},
    isbn = {3540240519},
    keywords = {coding\_theory, list\_decoding, reed\_solomon},
    posted-at = {2010-06-14 17:52:55},
    priority = {2},
    publisher = {Springer-Verlag New York, Inc.},
    title = {List Decoding of {Error-Correcting} Codes: Winning Thesis of the 2002 {ACM} Doctoral Dissertation Competition (Lecture Notes in Computer Science)},
    url = {http://portal.acm.org/citation.cfm?id=1076300},
    year = {2005}
}

@inproceedings{gf2x,
    abstract = {In this paper, we discuss an implementation of various algorithmsfor multiplying polynomials in {GF}(2)[ x ]: variants of the windowmethods, Karatsuba's, {Toom-Cook}'s, Sch\"{o}nhage's and Cantor's {algorithms.For} most of them, we propose improvements that lead to practicalspeedups.},
    address = {Berlin, Heidelberg},
    author = {Brent, Richard P. and Gaudry, Pierrick and Thom\'{e}, Emmanuel and Zimmermann, Paul},
    booktitle = {ANTS-VIII'08: Proceedings of the 8th international conference on Algorithmic number theory},
    citeulike-article-id = {7300384},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1789728},
    isbn = {3-540-79455-7, 978-3-540-79455-4},
    keywords = {field\_extensions, finite\_field, implementation, polynomial\_multiplication},
    location = {Banff, Canada},
    pages = {153--166},
    posted-at = {2010-06-14 17:51:32},
    priority = {0},
    publisher = {Springer-Verlag},
    title = {Faster multiplication in {GF(2)[x]}},
    url = {http://portal.acm.org/citation.cfm?id=1789728},
    year = {2008}
}

@inproceedings{couveignes+morain94,
    address = {London, UK},
    author = {Couveignes, Jean M. and Morain, Fran\c{c}ois},
    booktitle = {ANTS-I: Proceedings of the First International Symposium on Algorithmic Number Theory},
    citeulike-article-id = {7300377},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=749555},
    isbn = {3-540-58691-1},
    keywords = {eigenvalue, elliptic\_curve, frobenius, isogenies, isogeny\_cycles, schoof},
    pages = {43--58},
    posted-at = {2010-06-14 17:49:55},
    priority = {0},
    publisher = {Springer-Verlag},
    title = {Schoof's algorithm and isogeny cycles},
    url = {http://portal.acm.org/citation.cfm?id=749555},
    year = {1994}
}

@article{coppersmith+winograd,
    abstract = {We present a new method for accelerating matrix multiplication asymptotically. Thiswork builds on recent ideas of Volker Strassen, by using a basic trilinear form which is not a matrix product. We make novel use of the {Salem-Spencer} Theorem, which gives a fairly dense set of integers with no three-term arithmetic progression. Our resulting matrix exponent is 2.376.},
    address = {Duluth, MN, USA},
    author = {Coppersmith, Don and Winograd, Shmuel},
    citeulike-article-id = {7300360},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=77766},
    citeulike-linkout-1 = {http://dx.doi.org/10.1016/S0747-7171(08)80013-2},
    doi = {10.1016/S0747-7171(08)80013-2},
    issn = {07477171},
    journal = {J. Symbolic Comput.},
    keywords = {matrix\_multiplication},
    month = mar,
    number = {3},
    pages = {251--280},
    posted-at = {2010-06-14 17:47:39},
    priority = {2},
    publisher = {Academic Press, Inc.},
    title = {Matrix multiplication via arithmetic progressions},
    url = {http://dx.doi.org/10.1016/S0747-7171(08)80013-2},
    volume = {9},
    year = {1990}
}

@book{blake+seroussi+smart,
    address = {New York, NY, USA},
    author = {Blake, Ian F. and Seroussi, Gadiel and Smart, Niegel P.},
    citeulike-article-id = {7300302},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=345194},
    isbn = {0-521-65374-6},
    keywords = {cryptography, elliptic\_curve},
    posted-at = {2010-06-14 17:44:49},
    priority = {0},
    publisher = {Cambridge University Press},
    title = {Elliptic curves in cryptography},
    url = {http://portal.acm.org/citation.cfm?id=345194},
    year = {1999}
}

@inproceedings{guruswami+rudra06,
    abstract = {For every 0 \&lt; R \&lt; 1 and \&\#949; \&gt; 0, we present an explicit construction of error-correcting codes of rate R that can be list decoded in polynomial time up to a fraction ({1-R}-\&\#949;) of errors. These codes achieve the "capacity" for decoding from  adversarial  errors, i.e., achieve the  optimal  trade-off between rate and error-correction radius. At least theoretically, this meets one of the central challenges in coding {theory.Prior} to this work, explicit codes achieving capacity were not known for  any  rate R. In fact, our codes are the first to beat the error-correction radius of {1-\&\#8730;R}, that was achieved for {Reed-Solomon} ({RS}) codes in [9], for all rates R. (For rates R \&lt; 1/16, Parvaresh and Vardy [12] had recently improved upon the {1-\&\#8730;R} bound; for R \&\#8594; 0, their algorithm can decode a fraction {1-O}(R {log(1/R})) of {errors.)Our} codes are simple to describe --- they are certain  folded {Reed-Solomon} codes , which are in fact  exactly  {RS} codes, but viewed as a code over a larger alphabet by careful bundling of codeword symbols. Given the ubiquity of {RS} codes, this is an appealing feature of our result, since the codes we propose are not too far from the ones in actual {use.The} main insight in our work is that some carefully chosen folded {RS} codes are "compressed" versions of a related family of {Parvaresh-Vardy} codes. Further, the decoding of the folded {RS} codes can be reduced to list decoding the related {Parvaresh-Vardy} codes. The alphabet size of these folded {RS} codes is polynomial in the block length. This can be reduced to a constant that depends on the distance \&\#949; to capacity using ideas concerning "list recovering" and expander-based codes from [7, 8]. Concatenating the folded {RS} codes with suitable inner codes also gives us polytime constructible binary codes that can be efficiently list decoded up to the Zyablov bound.},
    address = {New York, NY, USA},
    author = {Guruswami, Venkatesan and Rudra, Atri},
    booktitle = {STOC '06: Proceedings of the thirty-eighth annual ACM symposium on Theory of computing},
    citeulike-article-id = {891139},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1132516.1132518},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1132516.1132518},
    doi = {10.1145/1132516.1132518},
    isbn = {1-59593-134-1},
    keywords = {coding\_theory, list\_decoding, reed\_solomon},
    location = {Seattle, WA, USA},
    pages = {1--10},
    posted-at = {2010-06-14 17:43:09},
    priority = {2},
    publisher = {ACM},
    title = {Explicit capacity-achieving list-decodable codes},
    url = {http://dx.doi.org/10.1145/1132516.1132518},
    year = {2006}
}

@article{kaltofen+shoup98,
    address = {Boston, MA, USA},
    author = {Kaltofen, Erich and Shoup, Victor},
    citeulike-article-id = {7300278},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=294748},
    citeulike-linkout-1 = {http://dx.doi.org/10.1090/S0025-5718-98-00944-2},
    doi = {10.1090/S0025-5718-98-00944-2},
    issn = {0025-5718},
    journal = {Math. Comp.},
    keywords = {computer\_algebra, finite\_field, polynomial\_factorization, transposed\_algorithms},
    number = {223},
    pages = {1179--1197},
    posted-at = {2010-06-14 17:41:01},
    priority = {0},
    publisher = {American Mathematical Society},
    title = {Subquadratic-time factoring of polynomials over finite fields},
    url = {http://dx.doi.org/10.1090/S0025-5718-98-00944-2},
    volume = {67},
    year = {1998}
}

@phdthesis{mateer08,
    abstract = {This manuscript describes a number of algorithms that can be used to quickly evaluate a polynomial over a collection of points and interpolate these evaluations back into a polynomial. Engineers define the  ” Fast Fourier Transform” as a method of solving the interpolation problem where the coefficient ring used to construct the polynomials has a special multiplicative structure. Mathematicians define the  ” Fast Fourier Transform” as a method of solving the evaluation problem. One purpose of the document is to provide a mathematical treatment of the topic of the  ” Fast Fourier Transform” that can also be understood by someone who has an understanding of the topic from the engineering perspective.},
    address = {Clemson, SC, USA},
    author = {Mateer, Todd},
    citeulike-article-id = {7300268},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1559083},
    isbn = {978-0-549-69375-8},
    keywords = {artin-schreier, polynomial\_evaluation, polynomial\_interpolation, polynomial\_multiplication},
    posted-at = {2010-06-14 17:39:54},
    priority = {1},
    school = {Clemson University},
    title = {Fast {F}ourier transform algorithms with applications},
    url = {http://portal.acm.org/citation.cfm?id=1559083},
    year = {2008}
}

@inproceedings{li+moreno+schost07,
    abstract = {We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, we reach quasi-linear complexity. The main outcome we have in mind is the acceleration of higher-level algorithms, by interfacing our low-level implementation with languages such as {AXIOM} or Maple We show the potential for huge speed-ups, by comparing two {AXIOM} implementations of van Hoeij and Monagan's modular {GCD} algorithm.},
    address = {New York, NY, USA},
    author = {Li, Xin and Moreno Maza, Marc and Schost, \'{E}ric},
    booktitle = {ISSAC '07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {7300246},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1277585},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1277548.1277585},
    doi = {10.1145/1277548.1277585},
    isbn = {978-1-59593-743-8},
    keywords = {computer\_algebra, ideal, multivariate\_polynomial, polynomial\_multiplication, triangular\_set, zero\_dimensional},
    location = {Waterloo, Ontario, Canada},
    pages = {269--276},
    posted-at = {2010-06-14 17:36:12},
    priority = {3},
    publisher = {ACM},
    title = {Fast arithmetic for triangular sets: from theory to practice},
    url = {http://dx.doi.org/10.1145/1277548.1277585},
    year = {2007}
}

@inproceedings{lercier96,
    address = {London, UK},
    author = {Lercier, Reynald},
    booktitle = {ANTS-II: Proceedings of the Second International Symposium on Algorithmic Number Theory},
    citeulike-article-id = {7300245},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=749582},
    isbn = {3-540-61581-4},
    keywords = {elliptic\_curve, finite\_field, isogenies, systems},
    pages = {197--212},
    posted-at = {2010-06-14 17:34:08},
    priority = {0},
    publisher = {Springer-Verlag},
    title = {Computing Isogenies in {GF(2,n)}},
    url = {http://portal.acm.org/citation.cfm?id=749582},
    year = {1996}
}

@article{couveignes00,
    address = {Boston, MA, USA},
    author = {Couveignes, Jean-Marc},
    citeulike-article-id = {7300242},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=360598},
    citeulike-linkout-1 = {http://dx.doi.org/10.1090/S0025-5718-00-01193-5},
    doi = {10.1090/S0025-5718-00-01193-5},
    issn = {0025-5718},
    journal = {Math. Comp.},
    keywords = {artin-schreier, computer\_algebra, field\_extensions, finite\_field},
    number = {232},
    pages = {1625--1631},
    posted-at = {2010-06-14 17:32:32},
    priority = {0},
    publisher = {American Mathematical Society},
    title = {Isomorphisms between {A}rtin-{S}chreier towers},
    url = {http://dx.doi.org/10.1090/S0025-5718-00-01193-5},
    volume = {69},
    year = {2000}
}

@inproceedings{couveignes96,
    address = {London, UK},
    author = {Couveignes, Jean-Marc},
    booktitle = {ANTS-II: Proceedings of the Second International Symposium on Algorithmic Number Theory},
    citeulike-article-id = {7300238},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=749581},
    isbn = {3-540-61581-4},
    keywords = {artin-schreier, elliptic\_curve, euclid, field\_extensions, finite\_field, isogenies, polynomial\_interpolation, torsion},
    pages = {59--65},
    posted-at = {2010-06-14 17:31:15},
    priority = {0},
    publisher = {Springer-Verlag},
    title = {Computing {l-Isogenies} Using the {p-Torsion}},
    url = {http://portal.acm.org/citation.cfm?id=749581},
    year = {1996}
}

@article{cantor89,
    address = {Orlando, FL, USA},
    author = {Cantor, David G.},
    citeulike-article-id = {7300235},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=65178},
    citeulike-linkout-1 = {http://dx.doi.org/10.1016/0097-3165(89)90020-4},
    doi = {10.1016/0097-3165(89)90020-4},
    issn = {0097-3165},
    journal = {J.  Combin. Theory Ser. A},
    keywords = {artin-schreier, field\_extensions, finite\_field, polynomial\_multiplication, polynomials\_over\_finite\_fields},
    number = {2},
    pages = {285--300},
    posted-at = {2010-06-14 17:29:24},
    priority = {0},
    publisher = {Academic Press, Inc.},
    title = {On arithmetical algorithms over finite fields},
    url = {http://dx.doi.org/10.1016/0097-3165(89)90020-4},
    volume = {50},
    year = {1989}
}

@article{MAGMA,
    address = {Duluth, MN, USA},
    author = {Bosma, Wieb and Cannon, John and Playoust, Catherine},
    citeulike-article-id = {7300233},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=268922},
    citeulike-linkout-1 = {http://dx.doi.org/10.1006/jsco.1996.0125},
    doi = {10.1006/jsco.1996.0125},
    issn = {0747-7171},
    journal = {J. Symbolic Comput.},
    keywords = {algebra\_system, computer\_algebra, magma},
    number = {3-4},
    pages = {235--265},
    posted-at = {2010-06-14 17:27:53},
    priority = {0},
    publisher = {Academic Press, Inc.},
    title = {The {MAGMA} algebra system {I}: the user language},
    url = {http://dx.doi.org/10.1006/jsco.1996.0125},
    volume = {24},
    year = {1997}
}

@article{FGLM,
    abstract = {We present an efficient algorithm for the transformation of a Gr\"{o}bner basis of a zero-dimensional ideal with respect to any given ordering into a Gr\"{o}bner basis with respect to any other ordering. This algorithm is polynomial in the degree of the ideal. In particular the lexicographical Gr\"{o}bner basis can be obtained by applying this algorithm after a total degree Gr\"{o}bner basis computation: it is usually much faster to compute the basis this way than with a direct application of Buchberger's algorithm.},
    address = {Duluth, MN, USA},
    author = {Faug\`{e}re, Jean-Charles and Gianni, Patricia and Lazard, Daniel and Mora, Teo},
    citeulike-article-id = {707494},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=181785},
    citeulike-linkout-1 = {http://dx.doi.org/10.1006/jsco.1993.1051},
    citeulike-linkout-2 = {http://www.ams.org/mathscinet-getitem?mr=1263871},
    doi = {10.1006/jsco.1993.1051},
    issn = {07477171},
    journal = {J. Symbolic Comput.},
    keywords = {change\_of\_order, computer\_algebra, groebner\_bases, ideal, zero\_dimensional},
    month = oct,
    mrnumber = {MR1263871},
    number = {4},
    pages = {329--344},
    posted-at = {2010-06-14 17:26:02},
    priority = {3},
    publisher = {Academic Press, Inc.},
    title = {Efficient computation of zero-dimensional {G}r\"{o}bner bases by change of ordering},
    url = {http://dx.doi.org/10.1006/jsco.1993.1051},
    volume = {16},
    year = {1993}
}

@article{bernstein98,
    address = {Duluth, MN, USA},
    author = {Bernstein, Daniel J.},
    citeulike-article-id = {7300220},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=291737},
    citeulike-linkout-1 = {http://dx.doi.org/10.1006/jsco.1998.0216},
    doi = {10.1006/jsco.1998.0216},
    issn = {0747-7171},
    journal = {J. Symbolic Comput.},
    keywords = {algorithm, finite\_field, formal\_power\_series, frobenius, modular\_composition, power\_series},
    number = {3},
    pages = {339--341},
    posted-at = {2010-06-14 17:24:30},
    priority = {0},
    publisher = {Academic Press, Inc.},
    title = {Composing power series over a finite ring in essentially linear time},
    url = {http://dx.doi.org/10.1006/jsco.1998.0216},
    volume = {26},
    year = {1998}
}

@inproceedings{enge+morain03,
    abstract = {Let  {f(X})  be a separable polynomial with coefficients in a field  K , generating a field extension  {M/K} . If this extension is Galois with a solvable automorphism group, then the equation  {f(X})  = 0 can be solved by radicals. The first step of the solution consists of splitting the extension  {M/K}  into intermediate fields. Such computations are classical, and we explain how fast polynomial arithmetic can be used to speed up the process. Moreover, we extend the algorithms to a more general case of extensions that are no longer Galois. Numerical examples are provided, including results obtained with our implementation for Hilbert class fields of imaginary quadratic fields.},
    address = {Berlin, Heidelberg},
    author = {Enge, Andreas and Morain, Fran\c{c}ois},
    booktitle = {AAECC'03: Proceedings of the 15th international conference on Applied algebra, algebraic algorithms and error-correcting codes},
    citeulike-article-id = {7300206},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1759867},
    isbn = {3-540-40111-3},
    keywords = {field\_extensions, galois\_group, polynomial\_factorization, polynomial\_interpolation},
    location = {Toulouse, France},
    pages = {254--264},
    posted-at = {2010-06-14 17:22:49},
    priority = {0},
    publisher = {Springer-Verlag},
    title = {Fast decomposition of polynomials with known Galois group},
    url = {http://portal.acm.org/citation.cfm?id=1759867},
    year = {2003}
}

@inproceedings{gaudry+morain06,
    abstract = {The {Schoof-Elkies}-Atkin algorithm is the best known algorithm for counting the number of points of an elliptic curve defined over a finite field of large characteristic. Several practical and asymptotical improvements for the phase called eigenvalue computation are proposed.},
    address = {New York, NY, USA},
    author = {Gaudry, Pierrick and Morain, Fran\c{c}ois},
    booktitle = {ISSAC '06: Proceedings of the 2006 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {7300197},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1145791},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1145768.1145791},
    doi = {10.1145/1145768.1145791},
    isbn = {1-59593-276-3},
    keywords = {eigenvalue, elliptic\_curve, field\_extensions, finite\_field, frobenius, isogenies, schoof},
    location = {Genoa, Italy},
    pages = {109--115},
    posted-at = {2010-06-14 17:20:48},
    priority = {2},
    publisher = {ACM},
    title = {Fast algorithms for computing the eigenvalue in the {S}choof-{E}lkies-{A}tkin algorithm},
    url = {http://dx.doi.org/10.1145/1145768.1145791},
    year = {2006}
}

@article{vzgatehn+gerhard02,
    abstract = {We describe algorithms for polynomial factorization over the binary field F 2 , and their implementation. They allow polynomials of degree up to 250 000 to be factored in about one day of {CPU} time, distributing the work on two processors.},
    address = {Boston, MA, USA},
    author = {Joachim von zur Gathen and Gerhard, J\"{u}rgen},
    citeulike-article-id = {7300183},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=774048},
    citeulike-linkout-1 = {http://dx.doi.org/10.1090/S0025-5718-02-01421-7},
    doi = {10.1090/S0025-5718-02-01421-7},
    issn = {0025-5718},
    journal = {Math. Comp.},
    keywords = {artin-schreier, computer\_algebra, cpu\_time, field\_extensions, finite\_field, implementation, polynomial\_factorization, polynomials\_over\_finite\_fields},
    number = {240},
    pages = {1677--1698},
    posted-at = {2010-06-14 17:18:21},
    priority = {0},
    publisher = {American Mathematical Society},
    title = {Polynomial factorization over {GF(2)}},
    url = {http://dx.doi.org/10.1090/S0025-5718-02-01421-7},
    volume = {71},
    year = {2002}
}

@inproceedings{mihailescu+morain+schost07,
    abstract = {The {Schoof-Elkies}-Atkin algorithm is the best known method for counting the number of points of an elliptic curve defined over a finite field of large characteristic. We use Abelian properties of division polynomials to design a fast theoretical and practical algorithm for nding the eigenvalue.},
    address = {New York, NY, USA},
    author = {Mihailescu, Preda and Morain, Fran\c{c}ois and Schost, \'{E}ric},
    booktitle = {ISSAC '07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {7300175},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1277587},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1277548.1277587},
    doi = {10.1145/1277548.1277587},
    isbn = {978-1-59593-743-8},
    keywords = {abelian\_lift, eigenvalue, elliptic\_curve, finite\_field, frobenius, schoof},
    location = {Waterloo, Ontario, Canada},
    pages = {285--292},
    posted-at = {2010-06-14 17:15:30},
    priority = {5},
    publisher = {ACM},
    title = {Computing the eigenvalue in the {S}choof-{E}lkies-{A}tkin algorithm using abelian lifts},
    url = {http://dx.doi.org/10.1145/1277548.1277587},
    year = {2007}
}

@inproceedings{shoup93,
    abstract = {Note: {OCR} errors may be found in this Reference List extracted from the full text article.  {ACM} has opted to expose the complete List rather than only correct and linked references.},
    address = {Philadelphia, PA, USA},
    author = {Shoup, Victor},
    booktitle = {SODA '93: Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms},
    citeulike-article-id = {7300159},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=313865},
    isbn = {0-89871-313-7},
    keywords = {computer\_algebra, finite\_field, irreducible\_polynomials, transposed\_algorithms},
    location = {Austin, Texas, United States},
    pages = {484--492},
    posted-at = {2010-06-14 17:13:42},
    priority = {0},
    publisher = {Society for Industrial and Applied Mathematics},
    title = {Fast construction of irreducible polynomials over finite fields},
    url = {http://portal.acm.org/citation.cfm?id=313865},
    year = {1993}
}

@inproceedings{kedlaya+umans08,
    abstract = {We give an algorithm for modular composition of degree n univariate polynomials over a finite field F\_q requiring n^{1 + o(1)}log^{1 + o(1)}q bit operations; this had earlier been achieved in characteristic n^{o(1)} by Umans (2008). As an application, we obtain a randomized algorithm for factoring degree n polynomials over F\_q requiring (n^{1.5 + o(1)} + n^{1 + o(1)}log q)log^{1 + o(1)}q bit operations, improving upon the methods of von zur Gathen \& Shoup (1992) and Kaltofen \& Shoup (1998). Our results also imply algorithms for irreducibility testing and computing minimal polynomials whose running times are best-possible, up to lower order terms.As in Umans (2008), we reduce modular composition to certain instances of multipoint evaluation of multivariate polynomials. We then give an algorithm that solves this problem optimally (up to lower order terms), in arbitrary characteristic. The main idea is to lift to characteristic 0, apply a small number of rounds of multimodular reduction, and finish with a small number of multidimensional FFTs. The final evaluations are then reconstructed using the Chinese Remainder Theorem. As a bonus, we obtain a very efficient data structure supporting polynomial evaluation queries, which is of independent interest.Our algorithm uses techniques which are commonly employed in practice, so it may be competitive for real problem sizes. This contrasts with previous asymptotically fast methods relying on fast matrix multiplication.},
    address = {Washington, DC, USA},
    author = {Kedlaya, Kiran S. and Umans, Christopher},
    booktitle = {FOCS '08: Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science},
    citeulike-article-id = {7300127},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1470582.1470655},
    citeulike-linkout-1 = {http://dx.doi.org/10.1109/FOCS.2008.13},
    doi = {10.1109/FOCS.2008.13},
    isbn = {978-0-7695-3436-7},
    keywords = {chinese\_remainder\_theorem, modular\_composition, multivariate\_polynomial, polynomial\_evaluation, polynomial\_interpolation, randomized\_algorithm, transposed\_algorithms},
    pages = {146--155},
    posted-at = {2010-06-14 17:12:01},
    priority = {5},
    publisher = {IEEE Computer Society},
    title = {Fast Modular Composition in any Characteristic},
    url = {http://dx.doi.org/10.1109/FOCS.2008.13},
    year = {2008}
}

@inproceedings{umans:08,
    abstract = {We obtain randomized algorithms for factoring degree n univariate polynomials over F\_q that use O(n 1.5 + o(1)  + n 1 + o(1) log q) field operations, when the characteristic is at most n o(1) . When log q < n, this is asymptotically faster than the best previous algorithms (von zur Gathen \& Shoup (1992) and Kaltofen \& Shoup (1998)); for log q \&\#8805; n, it matches the asymptotic running time of the best known algorithms.},
    address = {New York, NY, USA},
    author = {Umans, Christopher},
    booktitle = {STOC '08: Proceedings of the 40th annual ACM symposium on Theory of computing},
    citeulike-article-id = {7300116},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1374445},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1374376.1374445},
    doi = {10.1145/1374376.1374445},
    isbn = {978-1-60558-047-0},
    keywords = {modular\_composition, multivariate\_polynomial, polynomial\_factorization, polynomial\_interpolation, transposed\_algorithms},
    location = {Victoria, British Columbia, Canada},
    pages = {481--490},
    posted-at = {2010-06-14 17:08:11},
    priority = {0},
    publisher = {ACM},
    title = {Fast polynomial factorization and modular composition in small characteristic},
    url = {http://dx.doi.org/10.1145/1374376.1374445},
    year = {2008}
}

@inproceedings{vzgathen+shoup92,
    abstract = {A new probabilistic algorithm for factoring univariate polynomials over finite fields is presented whose asymptotic running time improves upon previous results. To factor a polynomial of degree  n  over  F q , the algorithm uses  O (( n 2  +  n  log  q )•(log  n ) 2  log log  n ) arithmetic operations in  F q . The main technical innovation is a new way to compute Frobenius and trace maps in the ring of polynomials modulo the polynomial to be factored.},
    address = {New York, NY, USA},
    author = {Joachim von zur Gathen and Shoup, Victor},
    booktitle = {STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of computing},
    citeulike-article-id = {7300092},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=129722},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/129712.129722},
    doi = {10.1145/129712.129722},
    isbn = {0-89791-511-9},
    keywords = {computer\_algebra, finite\_field, polynomial\_factorization, polynomials\_over\_finite\_fields, probabilistic\_algorithm, transposed\_algorithms, transposition},
    location = {Victoria, British Columbia, Canada},
    pages = {97--105},
    posted-at = {2010-06-14 17:05:52},
    priority = {0},
    publisher = {ACM},
    title = {Computing {F}robenius maps and factoring polynomials},
    url = {http://dx.doi.org/10.1145/129712.129722},
    year = {1992}
}

@inproceedings{kaltofen+shoup97,
    abstract = {Note: {OCR} errors may be found in this Reference List extracted from the full text article.  {ACM} has opted to expose the complete List rather than only correct and linked references.},
    address = {New York, NY, USA},
    author = {Kaltofen, Erich and Shoup, Victor},
    booktitle = {ISSAC '97: Proceedings of the 1997 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {7300069},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=258777},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/258726.258777},
    doi = {10.1145/258726.258777},
    isbn = {0-89791-875-4},
    keywords = {computer\_algebra, field\_extensions, finite\_field, modular\_composition, polynomial\_factorization, transposed\_algorithms},
    location = {Kihei, Maui, Hawaii, United States},
    pages = {184--188},
    posted-at = {2010-06-14 17:03:22},
    priority = {0},
    publisher = {ACM},
    title = {Fast polynomial factorization over high algebraic extensions of finite fields},
    url = {http://dx.doi.org/10.1145/258726.258777},
    year = {1997}
}

@article{dornstetter87,
    address = {Piscataway, NJ, USA},
    author = {Dornstetter, Jean-Louis},
    citeulike-article-id = {7300055},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=32509},
    citeulike-linkout-1 = {http://dx.doi.org/10.1109/TIT.1987.1057299},
    doi = {10.1109/TIT.1987.1057299},
    issn = {0018-9448},
    journal = {IEEE Transactions on Information Theory},
    keywords = {algorithm, berlekamp, euclid},
    number = {3},
    pages = {428--431},
    posted-at = {2010-06-14 17:00:58},
    priority = {0},
    publisher = {IEEE Press},
    title = {On the equivalence between {B}erlekamp's and {E}uclid's algorithms},
    url = {http://dx.doi.org/10.1109/TIT.1987.1057299},
    volume = {33},
    year = {1987}
}

@inproceedings{df+schost09,
    abstract = {An {Artin-Schreier} tower over the finite field  F p  is a tower of field extensions generated by polynomials of the form  X p - X -α. Following Cantor and Couveignes, we give algorithms with quasi-linear time complexity for arithmetic operations in such towers. As an application, we present an implementation of Couveignes' algorithm for computing isogenies between elliptic curves using the  p -torsion.},
    address = {New York, NY, USA},
    author = {De Feo, Luca and Schost, \'{E}ric},
    booktitle = {ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {7300031},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1576722},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1576702.1576722},
    doi = {10.1145/1576702.1576722},
    isbn = {978-1-60558-609-0},
    keywords = {algorithm, artin-schreier, computer\_algebra, field\_extensions, finite\_field, isogenies, transposed\_algorithms},
    location = {Seoul, Republic of Korea},
    pages = {127--134},
    posted-at = {2010-06-14 16:59:28},
    priority = {0},
    publisher = {ACM},
    title = {Fast arithmetics in {A}rtin-{S}chreier towers over finite fields},
    url = {http://dx.doi.org/10.1145/1576702.1576722},
    year = {2009}
}

@inproceedings{pascal+schost06,
    abstract = {Changing the order of variables in bivariate triangular sets has applications in Trager's factorization algorithm, or in rational function integration. We discuss the complexity of this question, using baby steps / giant steps techniques and trace formulas, obtaining subquadratic estimates.},
    address = {New York, NY, USA},
    author = {Pascal, Cyril and Schost, \'{E}ric},
    booktitle = {ISSAC '06: Proceedings of the 2006 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {7300011},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1145814},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1145768.1145814},
    doi = {10.1145/1145768.1145814},
    isbn = {1-59593-276-3},
    keywords = {change\_of\_order, computer\_algebra, trace\_formulas, transposed\_algorithms, triangular\_set},
    location = {Genoa, Italy},
    pages = {277--284},
    posted-at = {2010-06-14 16:56:59},
    priority = {0},
    publisher = {ACM},
    title = {Change of order for bivariate triangular sets},
    url = {http://dx.doi.org/10.1145/1145768.1145814},
    year = {2006}
}

@article{Ka2K,
    abstract = {The success of the symbolic mathematical computation discipline is striking. The theoretical advances have been continuous and significant: Gr\"{o}bner bases, the Risch integration algorithm, integer lattice basis reduction, hypergeometric summation algorithms, etc. From the beginning in the early 1960s, it has been the tradition of our discipline to create software that makes our ideas readily available to scientists, engineers, and educators: {SAC}-1, Reduce, Macsyma, etc. The commercial viability of our system products is proven by Maple and Mathematica. Today's user communities of symbolic computation systems are diverse: educators, engineers, stock market analysts, etc. The mathematics and computer science in the design and implementation of our algorithms are sophisticated. The research challenges in symbolic computation at the close of the twentieth century are formidable. I state my favorite eight open problems in symbolic computation. They range from problems in symbolic/numeric computing, symbolic algorithm synthesis, to system component construction. I have worked on seven of my problems and borrowed one from George Collins. I present background to each of my problems and a clear-cut test that evaluates whether a proposed attack has solved one of my problems. An additional ninth open problem by Rob Corless and David Jeffrey on complex function semantics is given in an appendix.},
    address = {Duluth, MN, USA},
    author = {Kaltofen, Erich},
    citeulike-article-id = {7300004},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=342625},
    citeulike-linkout-1 = {http://dx.doi.org/10.1006/jsco.2000.0370},
    doi = {10.1006/jsco.2000.0370},
    issn = {07477171},
    journal = {J. Symbolic Comput.},
    keywords = {challenges, computer\_algebra, symbolic\_computation},
    month = jun,
    number = {6},
    pages = {891--919},
    posted-at = {2010-06-14 16:55:30},
    priority = {0},
    publisher = {Academic Press, Inc.},
    title = {Challenges of Symbolic Computation: My Favorite Open Problems},
    url = {http://dx.doi.org/10.1006/jsco.2000.0370},
    volume = {29},
    year = {2000}
}

@article{vdHoven02,
    abstract = {Assume that we wish to expand the product  h = fg  of two formal power series  f  and  g . Classically, there are two types of algorithms to do this:  zealous  algorithms first expand  f  and  g  up to order  n , multiply the results and truncate at order  n .  Lazy  algorithms on the contrary compute the coefficients of  f, g  and  h  gradually and they perform no more computations than strictly necessary at each stage. In particular, at the moment we compute the coefficient  h i  of  z i  in  h , only  f 0 ,...,  f i  and  g 0 ,...,  g i  are {known.Lazy} algorithms have the advantage that the coefficients of  f  and  g  may actually depend on "previous" coefficients of  h , as long as they are computed before they are needed in the multiplication, i.e. the coefficients  f i  and  g i  may depend on  h 0 ,...,  h i -1 . For this reason, lazy algorithms are extremely useful when solving functional equations in rings of formal power series. However, lazy algorithms have the disadvantage that the classical asymptotically fast multiplication algorithms on polynomials--such as the divide and conquer algorithm and fast Fourier multiplication--cannot be {used.In} a previous paper, we therefore introduced  relaxed  algorithms, which share the property concerning the resolution of functional equations with lazy algorithms, but perform slightly more computations than lazy algorithms during the computation of a given coefficient of  h . These extra computations anticipate the computations of the next coefficients of  h  and dramatically improve the asymptotic time complexities of such {algorithms.In} this paper, we survey several classical and new zealous algorithms for manipulating formal power series, including algorithms for multiplication, division, resolution of differential equations, composition and reversion. Next, we give various relaxed algorithms for these operations. All algorithms are specified in great detail and we prove theoretical time and space complexity bounds. Most algorithms have been experimentally implemented in C++ and we provide benchmarks. We conclude by some suggestions for future developments and a discussion of the fitness of the lazy and relaxed approaches for specific {applications.This} paper is intended both for those who are interested in the most recent algorithms for the manipulation of formal power series and for those who want to actually implement a power series library into a computer algebra system.},
    address = {Duluth, MN, USA},
    author = {Van der Hoeven, Joris},
    citeulike-article-id = {29388},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=766453},
    citeulike-linkout-1 = {http://dx.doi.org/10.1006/jsco.2002.0562},
    citeulike-linkout-2 = {http://www.ingentaconnect.com/content/ap/sy/2002/00000034/00000006/art00562},
    doi = {10.1006/jsco.2002.0562},
    issn = {0747-7171},
    journal = {J. Symb. Comput.},
    keywords = {composition, computer\_algebra, divide\_and\_conquer\_algorithm, formal\_power\_series, lazy\_evaluation, multiplication, power\_series},
    month = dec,
    number = {6},
    pages = {479--542},
    posted-at = {2010-06-14 16:53:44},
    priority = {2},
    publisher = {Academic Press, Inc.},
    title = {Relax, but don't be too lazy},
    url = {http://dx.doi.org/10.1006/jsco.2002.0562},
    volume = {34},
    year = {2002}
}

@book{vzGG,
    address = {New York, NY, USA},
    author = {Joachim von zur Gathen and Gerhard, J\"{u}rgen},
    citeulike-article-id = {7299960},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=304952},
    isbn = {0-521-64176-4},
    keywords = {computer\_algebra, modern\_computer\_algebra},
    posted-at = {2010-06-14 16:49:37},
    priority = {0},
    publisher = {Cambridge University Press},
    title = {Modern computer algebra},
    url = {http://portal.acm.org/citation.cfm?id=304952},
    year = {1999}
}

@article{shoup95,
    address = {Duluth, MN, USA},
    author = {Shoup, Victor},
    citeulike-article-id = {7299933},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=226280},
    citeulike-linkout-1 = {http://dx.doi.org/10.1006/jsco.1995.1055},
    doi = {10.1006/jsco.1995.1055},
    issn = {0747-7171},
    journal = {J. Symbolic Comput.},
    keywords = {computer\_algebra, finite\_field, polynomial\_factorization, transposed\_algorithms, transposition},
    number = {4},
    pages = {363--397},
    posted-at = {2010-06-14 16:47:34},
    priority = {0},
    publisher = {Academic Press, Inc.},
    title = {A new polynomial factorization algorithm and its implementation},
    url = {http://dx.doi.org/10.1006/jsco.1995.1055},
    volume = {20},
    year = {1995}
}

@article{shoup94,
    address = {Duluth, MN, USA},
    author = {Shoup, Victor},
    citeulike-article-id = {7299924},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=187280},
    citeulike-linkout-1 = {http://dx.doi.org/10.1006/jsco.1994.1025},
    doi = {10.1006/jsco.1994.1025},
    issn = {0747-7171},
    journal = {J. Symbolic Comput.},
    keywords = {computer\_algebra, finite\_field, irreducible\_polynomials, transposed\_algorithms, transposition},
    number = {5},
    pages = {371--391},
    posted-at = {2010-06-14 16:46:25},
    priority = {0},
    publisher = {Academic Press, Inc.},
    title = {Fast construction of irreducible polynomials over finite fields},
    url = {http://dx.doi.org/10.1006/jsco.1994.1025},
    volume = {17},
    year = {1994}
}

@inproceedings{shoup91,
    abstract = {Note: {OCR} errors may be found in this Reference List extracted from the full text article.  {ACM} has opted to expose the complete List rather than only correct and linked references.},
    address = {New York, NY, USA},
    author = {Shoup, Victor},
    booktitle = {ISSAC '91: Proceedings of the 1991 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {7299904},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=120697},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/120694.120697},
    doi = {10.1145/120694.120697},
    isbn = {0-89791-437-6},
    keywords = {deterministic\_algorithm, factoring\_polynomials, finite\_field, transposed\_algorithms},
    location = {Bonn, West Germany},
    pages = {14--21},
    posted-at = {2010-06-14 16:44:55},
    priority = {5},
    publisher = {ACM},
    title = {A fast deterministic algorithm for factoring polynomials over finite fields of small characteristic},
    url = {http://dx.doi.org/10.1145/120694.120697},
    year = {1991}
}

@article{kaminski+kirkpatrick+bshouty88,
    address = {Duluth, MN, USA},
    author = {Kaminski, Michael and Kirkpatrick, David G. and Bshouty, Nader H.},
    citeulike-article-id = {7299884},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=51373},
    citeulike-linkout-1 = {http://dx.doi.org/10.1016/0196-6774(88)90026-0},
    doi = {10.1016/0196-6774(88)90026-0},
    issn = {0196-6774},
    journal = {J. Algorithms},
    keywords = {duality, matrix\_multiplication, transposition},
    number = {3},
    pages = {354--364},
    posted-at = {2010-06-14 16:42:13},
    priority = {4},
    publisher = {Academic Press, Inc.},
    title = {Addition requirements for matrix and transposed matrix products},
    url = {http://dx.doi.org/10.1016/0196-6774(88)90026-0},
    volume = {9},
    year = {1988}
}

@article{brent+kung,
    abstract = {Note: {OCR} errors may be found in this Reference List extracted from the full text article.  {ACM} has opted to expose the complete List rather than only correct and linked references.},
    address = {New York, NY, USA},
    author = {Brent, Richard P. and Kung, Hsiang-Tsung},
    citeulike-article-id = {7299837},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=322099},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/322092.322099},
    doi = {10.1145/322092.322099},
    issn = {0004-5411},
    journal = {Journal of the ACM},
    keywords = {computer\_algebra, modular\_composition, power\_series},
    number = {4},
    pages = {581--595},
    posted-at = {2010-06-14 16:39:21},
    priority = {0},
    publisher = {ACM},
    title = {Fast Algorithms for Manipulating Formal Power Series},
    url = {http://dx.doi.org/10.1145/322092.322099},
    volume = {25},
    year = {1978}
}

@inproceedings{dumas+pernet+saunders09,
    abstract = {We present algorithms and heuristics to compute the characteristic polynomial of a matrix given its minimal polynomial. The matrix is represented as a black-box, i.e., by a function to compute its matrix-vector product. The methods apply to matrices either over the integers or over a large enough finite field. Experiments show that these methods perform efficiently in practice. Combined in an adaptive strategy, these algorithms reach significant speedups in practice for some integer matrices arising in an application from graph theory.},
    address = {New York, NY, USA},
    author = {Dumas, Jean G. and Pernet, Cl\'{e}ment and Saunders, B. David},
    booktitle = {ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {7299815},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1576723},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1576702.1576723},
    doi = {10.1145/1576702.1576723},
    isbn = {978-1-60558-609-0},
    keywords = {black\_box, characteristic\_polynomial, minimal\_polynomial, sparse\_linear\_equations},
    location = {Seoul, Republic of Korea},
    pages = {135--142},
    posted-at = {2010-06-14 16:37:12},
    priority = {0},
    publisher = {ACM},
    title = {On finding multiplicities of characteristic polynomial factors of black-box matrices},
    url = {http://dx.doi.org/10.1145/1576702.1576723},
    year = {2009}
}

@inproceedings{kaltofen+shoup95,
    abstract = {Note: {OCR} errors may be found in this Reference List extracted from the full text article.  {ACM} has opted to expose the complete List rather than only correct and linked references.},
    address = {New York, NY, USA},
    author = {Kaltofen, Erich and Shoup, Victor},
    booktitle = {STOC '95: Proceedings of the twenty-seventh annual ACM symposium on Theory of computing},
    citeulike-article-id = {7299777},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=225166},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/225058.225166},
    doi = {10.1145/225058.225166},
    isbn = {0-89791-718-9},
    keywords = {computer\_algebra, factoring\_of\_polynomials, finite\_field, polynomials\_over\_finite\_fields, transposed\_algorithms},
    location = {Las Vegas, Nevada, United States},
    pages = {398--406},
    posted-at = {2010-06-14 16:33:56},
    priority = {0},
    publisher = {ACM},
    title = {Subquadratic-time factoring of polynomials over finite fields},
    url = {http://dx.doi.org/10.1145/225058.225166},
    year = {1995}
}

@inproceedings{shoup99,
    abstract = {Note: {OCR} errors may be found in this Reference List extracted from the full text article.  {ACM} has opted to expose the complete List rather than only correct and linked references.},
    address = {New York, NY, USA},
    author = {Shoup, Victor},
    booktitle = {ISSAC '99: Proceedings of the 1999 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {7299740},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=309859},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/309831.309859},
    doi = {10.1145/309831.309859},
    isbn = {1-58113-073-2},
    keywords = {computer\_algebra, finite\_field, minimal\_polynomials, transposed\_algorithms, transposition},
    location = {Vancouver, British Columbia, Canada},
    pages = {53--58},
    posted-at = {2010-06-14 16:28:47},
    priority = {0},
    publisher = {ACM},
    title = {Efficient computation of minimal polynomials in algebraic extensions of finite fields},
    url = {http://dx.doi.org/10.1145/309831.309859},
    year = {1999}
}

@inproceedings{kaltofen+lakshman89,
    address = {London, UK},
    author = {Kaltofen, Erich and Yagati, Lakshman N.},
    booktitle = {ISSAC '88: Proceedings of the International Symposium ISSAC'88 on Symbolic and Algebraic Computation},
    citeulike-article-id = {7299718},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=690740},
    isbn = {3-540-51084-2},
    keywords = {black\_box, interpolation\_algorithms, multivariate\_polynomial, polynomial\_interpolation},
    pages = {467--474},
    posted-at = {2010-06-14 16:25:05},
    priority = {0},
    publisher = {Springer-Verlag},
    title = {Improved Sparse Multivariate Polynomial Interpolation Algorithms},
    url = {http://portal.acm.org/citation.cfm?id=690740},
    year = {1989}
}

@article{wiedemann:sparse,
    abstract = {A "coordinate recurrence" method for solving sparse systems of linear equations over finite fields is described. The algorithms discussed all require O(n\_{1}(omega + n\_{1})log^{k}n\_{1}) field operations, where n\_{1} is the maximum dimension of the coefficient matrix, omega is approximately the number of field operations required to apply the matrix to a test vector, and the value of k depends on the algorithm. A probabilistic algorithm is shown to exist for finding the determinant of a square matrix. Also, probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error.},
    address = {Piscataway, NJ, USA},
    author = {Wiedemann, Douglas H.},
    citeulike-article-id = {7299705},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=13744},
    citeulike-linkout-1 = {http://dx.doi.org/10.1109/TIT.1986.1057137},
    citeulike-linkout-2 = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1057137},
    doi = {10.1109/TIT.1986.1057137},
    issn = {0018-9448},
    journal = {IEEE Transactions on Information Theory},
    keywords = {black\_box, computer\_algebra, sparse\_linear\_equations},
    month = jan,
    number = {1},
    pages = {54--62},
    posted-at = {2010-06-14 16:23:58},
    priority = {0},
    publisher = {IEEE Press},
    title = {Solving sparse linear equations over finite fields},
    url = {http://dx.doi.org/10.1109/TIT.1986.1057137},
    volume = {32},
    year = {1986}
}

@inproceedings{canny+kaltofen+yagati89,
    abstract = {Note: {OCR} errors may be found in this Reference List extracted from the full text article.  {ACM} has opted to expose the complete List rather than only correct and linked references.},
    address = {New York, NY, USA},
    author = {Canny, John F. and Kaltofen, Eric and Yagati, Lakshman N.},
    booktitle = {ISSAC '89: Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {1421483},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=74556},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/74540.74556},
    doi = {10.1145/74540.74556},
    isbn = {0-89791-325-6},
    keywords = {computer\_algebra, polynomial\_equations, systems},
    location = {Portland, Oregon, United States},
    pages = {121--128},
    posted-at = {2010-06-14 16:22:35},
    priority = {0},
    publisher = {ACM},
    title = {Solving systems of nonlinear polynomial equations faster},
    url = {http://dx.doi.org/10.1145/74540.74556},
    year = {1989}
}

@phdthesis{fiduccia:phd,
    address = {Providence, RI, USA},
    author = {Fiduccia, Charles M.},
    citeulike-article-id = {7299574},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=906618},
    citeulike-linkout-1 = {http://portal.acm.org/citation.cfm?id=906618},
    keywords = {algebraic\_complexity, matrix\_multiplication, transposition},
    posted-at = {2010-06-14 16:06:24},
    priority = {2},
    school = {Brown University},
    title = {{On the algebraic complexity of matrix multiplication.}},
    url = {http://portal.acm.org/citation.cfm?id=906618},
    year = {1973}
}

@inproceedings{bostan+lecerf+schost:tellegen,
    abstract = {The transposition principle, also called Tellegen's principle, is a set of transformation rules for linear programs. Yet, though well known, it is not used systematically, and few practical implementations rely on it. In this article, we propose explicit transposed versions of polynomial multiplication and division but also new faster algorithms for multipoint evaluation, interpolation and their transposes. We report on their implementation in Shoup's {NTL} C++ library.},
    address = {New York, NY, USA},
    author = {Bostan, Alin and Lecerf, Gr\'{e}goire and Schost, \'{E}ric},
    booktitle = {ISSAC '03: Proceedings of the 2003 international symposium on Symbolic and algebraic computation},
    citeulike-article-id = {1432771},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=860854.860870},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/860854.860870},
    doi = {10.1145/860854.860870},
    isbn = {1-58113-641-2},
    keywords = {duality, polynomial\_multiplication, tellegen, transposition},
    location = {Philadelphia, PA, USA},
    pages = {37--44},
    posted-at = {2010-06-14 16:04:15},
    priority = {0},
    publisher = {ACM},
    title = {Tellegen's principle into practice},
    url = {http://dx.doi.org/10.1145/860854.860870},
    year = {2003}
}

@article{BSS,
    author = {Blum, Lenore and Shub, Mike and Smale, Steve},
    citeulike-article-id = {7299552},
    citeulike-linkout-0 = {http://dx.doi.org/10.1090/S0273-0979-1989-15750-9},
    day = {01},
    doi = {10.1090/S0273-0979-1989-15750-9},
    issn = {0273-0979},
    journal = {Bulletin of the American Mathematical Society},
    keywords = {bss, complexity, real\_numbers, theory\_of\_computation, universal\_machines},
    month = jul,
    number = {1},
    pages = {1--47},
    posted-at = {2010-06-14 16:02:04},
    priority = {0},
    title = {On a theory of computation and complexity over the real numbers: {NP}-completeness, recursive functions and universal machines},
    url = {http://dx.doi.org/10.1090/S0273-0979-1989-15750-9},
    volume = {21},
    year = {1989}
}

@incollection{backus:turing+lecture,
    abstract = {Conventional programming languages are growing ever more enormous, but not stronger. Inherent defects at the most basic level cause them to be both fat and weak: their primitive word-at-a-time style of programming inherited from their common ancestor--the yon Neumann computer, their close coupling of semantics to state transitions, their division of programming into a world of expressions and a world of statements, their inability to effectively use powerful combining forms for building new programs from existing ones, and their lack of useful mathematical properties for reasoning about programs. An alternative functional style of programming is founded on the use of combining forms for creating programs. Functional programs deal with structured data, are often nonrepetitive and nonrecursive, are hierarchically constructed, do not name their arguments, and do not require the complex machinery of procedure declarations to become generally applicable. Combining forms can use high-level programs to build still higher level ones in a style not possible in conventional languages. Associated with the functional style of programming is an algebra of programs whose variables range over programs and whose operations are combining forms. This algebra can be used to transform programs and to solve equations whose "unknowns" are programs in much the same way one tranforms equations in high school algebra. These transformations are given by algebraic laws and are carried out in the same language in which programs are written. Combining forms are chosen not only for their programming power but also the power of their associated algebraic laws. General theorems of the algebra give the detailed behavior and termination conditions for large classes of programs. A new class of computing systems uses the functional programming style both in its programming language and in its stage transition rules. Unlike von Neumann languages, these systems have semantics loosely coupled to states --- only one state transition occurs per major computation.},
    address = {New York, NY, USA},
    author = {Backus, John},
    chapter = {Can programming be liberated from the von Neumann style?: a functional style and its algebra of programs},
    citeulike-article-id = {7299551},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1283933},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1283920.1283933},
    doi = {10.1145/1283920.1283933},
    isbn = {978-1-4503-1049-9},
    keywords = {function\_level\_programming, functional\_language},
    pages = {1977+},
    posted-at = {2010-06-14 15:59:52},
    priority = {0},
    publisher = {ACM},
    title = {{ACM} Turing award lectures},
    url = {http://dx.doi.org/10.1145/1283920.1283933},
    year = {2007}
}

@inproceedings{liu+cheng+hudak09,
    abstract = {Arrows are a popular form of abstract computation. Being more general than monads, they are more broadly applicable, and in particular are a good abstraction for signal processing and dataflow computations. Most notably, arrows form the basis for a domain specific language called Yampa, which has been used in a variety of concrete applications, including animation, robotics, sound synthesis, control systems, and graphical user interfaces. Our primary interest is in better understanding the class of abstract computations captured by Yampa. Unfortunately, arrows are not concrete enough to do this with precision. To remedy this situation we introduce the concept of commutative arrows that capture a kind of non-interference property of concurrent computations. We also add an init operator, and identify a crucial law that captures the causal nature of arrow effects. We call the resulting computational model causal commutative arrows. To study this class of computations in more detail, we define an extension to the simply typed lambda calculus called causal commutative arrows ({CCA}), and study its properties. Our key contribution is the identification of a normal form for {CCA} called causal commutative normal form ({CCNF}). By defining a normalization procedure we have developed an optimization strategy that yields dramatic improvements in performance over conventional implementations of arrows. We have implemented this technique in Haskell, and conducted benchmarks that validate the effectiveness of our approach. When combined with stream fusion, the overall methodology can result in speed-ups of greater than two orders of magnitude.},
    address = {New York, NY, USA},
    author = {Liu, Hai and Cheng, Eric and Hudak, Paul},
    booktitle = {Proceedings of the 14th ACM SIGPLAN international conference on Functional programming},
    citeulike-article-id = {6663203},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=1596550.1596559},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/1596550.1596559},
    doi = {10.1145/1596550.1596559},
    isbn = {978-1-60558-332-7},
    keywords = {arrows, functional\_language, haskell},
    location = {Edinburgh, Scotland},
    pages = {35--46},
    posted-at = {2010-06-14 15:33:33},
    priority = {2},
    publisher = {ACM},
    series = {ICFP '09},
    title = {Causal commutative arrows and their optimization},
    url = {http://dx.doi.org/10.1145/1596550.1596559},
    year = {2009}
}

@inproceedings{Walder+Blott-ad-hoc-polymorphism,
    abstract = {This paper presents type classes, a new approach to ad-hoc polymorphism. Type classes permit overloading of arithmetic operators such as multiplication, and generalise the "eqtype variables" of Standard {ML}. Type classes extend the {Hindley/Milner} polymorphic type system, and provide a new approach to issues that arise in object-oriented programming, bounded type quantification, and abstract data types. This paper provides an informal introduction to type classes, and defines them formally by...},
    author = {Wadler, Philip and Blott, Stephen},
    booktitle = {Conference Record of the 16th Annual ACM Symposium on Principles of Programming Languages},
    citeulike-article-id = {549400},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.1059},
    keywords = {functional\_language, haskell, polymorphic, programming\_languages, typechecking},
    month = jan,
    pages = {60--76},
    posted-at = {2010-06-14 15:27:50},
    priority = {0},
    publisher = {ACM},
    title = {How to make ad-hoc polymorphism less ad-hoc},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.1059},
    year = {1989}
}

@inproceedings{Damas+Milner,
    abstract = {An abstract is not available.},
    address = {New York, NY, USA},
    author = {Damas, Luis and Milner, Robin},
    booktitle = {Proceedings of the 9th ACM SIGPLAN-SIGACT symposium on Principles of programming languages},
    citeulike-article-id = {370442},
    citeulike-linkout-0 = {http://portal.acm.org/citation.cfm?id=582176},
    citeulike-linkout-1 = {http://dx.doi.org/10.1145/582153.582176},
    doi = {10.1145/582153.582176},
    isbn = {0-89791-065-6},
    keywords = {functional\_language, polymorphic, programming\_languages, typechecking},
    location = {Albuquerque, New Mexico},
    pages = {207--212},
    posted-at = {2010-06-14 15:25:09},
    priority = {0},
    publisher = {ACM},
    series = {POPL '82},
    title = {Principal type-schemes for functional programs},
    url = {http://dx.doi.org/10.1145/582153.582176},
    year = {1982}
}

@article{Cardelli:Typechecking,
    abstract = {Polymorphic means to have many forms. As related to programming languages, it refers to data or programs which have many types, or which operate on many types. There are several arbitrary ways in which programs can have many types; we are mostly interested in a},
    author = {Cardelli, Luca},
    citeulike-article-id = {3425283},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.7951},
    journal = {Science of Computer Programming},
    keywords = {functional\_language, polymorphic, programming\_languages, typechecking},
    pages = {147--172},
    posted-at = {2010-06-14 15:22:07},
    priority = {0},
    title = {Basic polymorphic typechecking},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.7951},
    volume = {8},
    year = {1987}
}

@article{hughes98,
    abstract = {this paper. Pleasingly, the arrow interface turned out to be applicable to other kinds of non-monadic library also, for example the fudgets library for graphical user interfaces [{CH93}], and a new library for programming active web pages. These applications will be described in sections 6 and 9. While arrows are a little less convenient to use than monads, they have significantly wider applicability. They can therefore be used to bring the benefits of monad-like programming to a much wider class of applications. 2 Background: Library Design Using Monads},
    author = {Hughes, John},
    citeulike-article-id = {3425308},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.4575},
    journal = {Science of Computer Programming},
    keywords = {arrows, haskell, monad, syntax},
    pages = {67--111},
    posted-at = {2010-06-14 15:05:58},
    priority = {0},
    title = {Generalising Monads to Arrows},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.4575},
    volume = {37},
    year = {1998}
}

@inproceedings{paterson01,
    abstract = {The categorical notion of monad, used by Moggi to structure denotational descriptions, has proved to be a powerful tool for structuring combinator libraries. Moreover, the monadic programming style provides a convenient syntax for many kinds of computation, so that each library defines a new sublanguage. Recently, several workers have proposed a generalization of monads, called variously  ” arrows ” or Freyd-categories. The extra generality promises to increase the power, expressiveness and efficiency of the embedded approach, but does not mesh as well with the native abstraction and application. Definitions are typically given in a point-free style, which is useful for proving general properties, but can be awkward for programming specific instances. In this paper we define a simple extension to the functional language Haskell that makes these new notions of computation more convenient to use. Our language is similar to the monadic style, and has similar reasoning properties. Moreover, it is extensible, in the sense that new combining forms can be defined as expressions in the host language. 1.},
    author = {Paterson, Ross},
    booktitle = {ICFP '01: International Conference on Functional Programming},
    citeulike-article-id = {7299325},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.161.6678},
    keywords = {haskell, monad, programming\_style, syntax},
    pages = {229--240},
    posted-at = {2010-06-14 15:03:59},
    priority = {0},
    title = {A new notation for arrows},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.161.6678},
    year = {2001}
}

@inproceedings{alimarine+al:invertible-arrows,
    abstract = {Invertible programming occurs in the area of data conversion where it is required that the conversion in one direction is the inverse of the other. For that purpose, we introduce bidirectional arrows (bi- arrows). The bi-arrow class is an extension of Haskell's arrow class with an extra combinator that changes the direction of computation. The advantage of the use of bi-arrows for invertible programming is the preservation of invertibility properties using the biarrow combinators. Programming with bi-arrows in a polytypic or generic way exploits this the most. Besides bidirectional polytypic examples, including invertible serialization, we give the definition of a monadic bi-arrow transformer, which we use to construct a bidirectional parser/pretty printer.},
    author = {Alimarine, Artem and Smetsers, Sjaak and van Weelden, Arjen and van Eekelen, Marko and Plasmeijer, Rinus},
    booktitle = {Proceedings of the 2005 ACM SIGPLAN workshop on Haskell},
    citeulike-article-id = {5624151},
    citeulike-linkout-0 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.60.7278},
    keywords = {arrows, combinator, haskell, invertible\_computation},
    organization = {ACM},
    pages = {97+},
    posted-at = {2010-06-14 15:01:23},
    priority = {2},
    title = {There and Back Again: Arrows for Invertible Programming},
    url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.60.7278},
    year = {2005}
}

@book{mclane,
    abstract = {{Categories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields.  Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality.  The book then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits.  These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors.  The categories of algebraic systems are constructed from certain adjoint-like data and characterized by Beck's theorem.  After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions.   This second edition includes a number of revisions and additions, including two new chapters on topics of active interest.  One is on symmetric monoidal categories and braided monoidal categories and the coherence theorems for them.  The second describes 2-categories and the higher dimensional categories which have recently come into prominence.  The bibliography has also been expanded to cover some of the many other recent advances concerning categories.}},
    author = {Mac Lane, Saunders},
    citeulike-article-id = {423557},
    citeulike-linkout-0 = {http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&amp;path=ASIN/0387984038},
    citeulike-linkout-1 = {http://www.amazon.de/exec/obidos/redirect?tag=citeulike01-21&amp;path=ASIN/0387984038},
    citeulike-linkout-2 = {http://www.amazon.fr/exec/obidos/redirect?tag=citeulike06-21&amp;path=ASIN/0387984038},
    citeulike-linkout-3 = {http://www.amazon.jp/exec/obidos/ASIN/0387984038},
    citeulike-linkout-4 = {http://www.amazon.co.uk/exec/obidos/ASIN/0387984038/citeulike00-21},
    citeulike-linkout-5 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0387984038},
    citeulike-linkout-6 = {http://www.worldcat.org/isbn/0387984038},
    citeulike-linkout-7 = {http://books.google.com/books?vid=ISBN0387984038},
    citeulike-linkout-8 = {http://www.amazon.com/gp/search?keywords=0387984038&index=books&linkCode=qs},
    citeulike-linkout-9 = {http://www.librarything.com/isbn/0387984038},
    day = {25},
    edition = {2nd},
    howpublished = {Hardcover},
    isbn = {0387984038},
    keywords = {categories, functor, natural\_transformation},
    month = sep,
    posted-at = {2010-06-14 15:00:19},
    priority = {0},
    publisher = {Springer},
    series = {Graduate Texts in Mathematics},
    title = {Categories for the Working Mathematician},
    url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0387984038},
    year = {1998}
}

@book{Conway:ONAG2000,
    abstract = {{ONAG, as the book is known, is one of those rare publications that sprang to life in a  moment of creative energy and has remained influential for over a quarter of a century. Still in high  demand, it is being republished with some adjustments and corrections. The original motivation for  writing the book was an attempt to understand the relation between the theories of transfinite numbers and  mathematical games. By defining numbers as the strengths of positions in certain games, the author  arrives at a new class, the surreal numbers (so named by Donald Knuth) that includes at the same time the  real numbers and the ordinal numbers.   <P>This new edition ends with an epilogue that sets the stage for further research on surreal numbers. The  book is a must-have for all readers with a serious interest in the mathematical foundations of game  strategies.}},
    author = {Conway, John H.},
    citeulike-article-id = {1223195},
    citeulike-linkout-0 = {http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&amp;path=ASIN/1568811276},
    citeulike-linkout-1 = {http://www.amazon.de/exec/obidos/redirect?tag=citeulike01-21&amp;path=ASIN/1568811276},
    citeulike-linkout-2 = {http://www.amazon.fr/exec/obidos/redirect?tag=citeulike06-21&amp;path=ASIN/1568811276},
    citeulike-linkout-3 = {http://www.amazon.jp/exec/obidos/ASIN/1568811276},
    citeulike-linkout-4 = {http://www.amazon.co.uk/exec/obidos/ASIN/1568811276/citeulike00-21},
    citeulike-linkout-5 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/1568811276},
    citeulike-linkout-6 = {http://www.worldcat.org/isbn/1568811276},
    citeulike-linkout-7 = {http://books.google.com/books?vid=ISBN1568811276},
    citeulike-linkout-8 = {http://www.amazon.com/gp/search?keywords=1568811276&index=books&linkCode=qs},
    citeulike-linkout-9 = {http://www.librarything.com/isbn/1568811276},
    day = {01},
    edition = {2nd},
    howpublished = {Hardcover},
    isbn = {1568811276},
    keywords = {finite\_field, games, mathematical\_foundations, ordinal\_numbers, transfinite\_numbers},
    month = dec,
    posted-at = {2010-06-14 14:57:52},
    priority = {0},
    publisher = {AK Peters, Ltd.},
    title = {On Numbers and Games},
    url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/1568811276},
    year = {2000}
}

@unpublished{goren+lauter10,
    abstract = {Genus 2 curves are useful in cryptography for both discrete-log based and
pairing-based systems, but a method is required to compute genus 2 curves such
that the Jacobian has a given number of points. Currently, all known methods
involve constructing genus 2 curves with complex multiplication via computing
their three Igusa class polynomials. These polynomials have rational
coefficients and require extensive computation and precision to compute. Both
the computation and the complexity analysis of these algorithms can be improved
by a more precise understanding of the denominators of the coefficients of the
polynomials. The main goal of this paper is to give a bound on the denominators
of Igusa class polynomials of genus 2 curves with {CM} by a primitive quartic {CM}
field. We give an overview of Igusa's results on the moduli space of genus two
curves and the method to construct genus 2 curves via their Igusa invariants.
We also give a complete characterization of the reduction type of a {CM} abelian
surface, for biquadratic, cyclic, and {non-Galois} quartic {CM} fields, and for any
type of prime decomposition of the prime, including ramified primes. The
methods of the proof of the main result involve studying the embedding problem
of the quartic {CM} field into certain matrix algebras over quaternions and
invoking techniques from crystalline deformation theory.},
    archivePrefix = {arXiv},
    author = {Goren, Eyal Z. and Lauter, Kristin E.},
    citeulike-article-id = {7062176},
    citeulike-linkout-0 = {http://arxiv.org/abs/1003.4759},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1003.4759},
    day = {24},
    eprint = {1003.4759},
    keywords = {complex\_multiplication, elliptic\_curve, genus\_2, igusa},
    month = mar,
    posted-at = {2010-04-22 16:53:42},
    priority = {2},
    title = {Genus 2 Curves with Complex Multiplication},
    url = {http://arxiv.org/abs/1003.4759},
    year = {2010}
}

@article{citeulike:7062171,
    abstract = {We study the collection of group structures that can be realized as a group
of rational points on an elliptic curve over a finite field (such groups are
well known to be of rank at most two). We also study various subsets of this
collection which correspond to curves over prime fields or to curves with a
prescribed torsion. Some of our results are rigorous and are based on recent
advances in analytic number theory, some are conditional under certain widely
believed conjectures, and others are purely heuristic in nature.},
    archivePrefix = {arXiv},
    author = {Banks, William D. and Pappalardi, Francesco and Shparlinski, Igor E.},
    citeulike-article-id = {7062171},
    citeulike-linkout-0 = {http://arxiv.org/abs/1003.3004},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1003.3004},
    day = {15},
    eprint = {1003.3004},
    keywords = {elliptic\_curve, estimates, finite\_field, group\_structure},
    month = mar,
    posted-at = {2010-04-22 16:52:30},
    priority = {3},
    title = {On group structures realized by elliptic curves over arbitrary finite fields},
    url = {http://arxiv.org/abs/1003.3004},
    year = {2010}
}

@article{citeulike:7062163,
    abstract = {We obtain explicit formulas for the number of non-isomorphic elliptic curves
with a given group structure (considered as an abstract abelian group).
Moreover, we give explicit formulas for the number of distinct group structures
of all elliptic curves over a finite field. We use these formulas to derive
some asymptotic estimates and tight upper and lower bounds for various counting
functions related to classification of elliptic curves accordingly to their
group structure. Finally, we present results of some numerical tests which
exhibit several interesting phenomena in the distribution of group structures.
We pose getting an explanation to these as an open problem.},
    archivePrefix = {arXiv},
    author = {Farashahi, Reza R. and Shparlinski, Igor E.},
    citeulike-article-id = {7062163},
    citeulike-linkout-0 = {http://arxiv.org/abs/1003.3000},
    citeulike-linkout-1 = {http://arxiv.org/pdf/1003.3000},
    day = {15},
    eprint = {1003.3000},
    keywords = {elliptic\_curve, estimates, finite\_field, group\_structure},
    month = mar,
    posted-at = {2010-04-22 16:51:28},
    priority = {3},
    title = {On group structures realized by elliptic curves over a fixed finite field},
    url = {http://arxiv.org/abs/1003.3000},
    year = {2010}
}

@incollection{Chen-Ling-Padro-Wang-Xing:CC2009,
    abstract = {Key predistribution schemes ({KPSs}) and one-time broadcast encryption schemes ({OTBESs}) are unconditionally secure protocols for key distribution in networks. The efficiency of these schemes has been measured in previous works in terms of their information rate, that is, the ratio between the length of the secret keys and the length of the secret information that must be stored by every user. Several constructions with optimal information rate have been proposed, but in them the secret keys are taken from a finite field with at least as many elements as the number of users in the network. This can be an important drawback in very large networks in which the nodes have limited computational resources as, for instance, wireless sensor networks. Actually, key predistribution schemes have been applied recently in the design of key distribution protocols for such networks.  In this paper we present a method to construct key predistribution schemes from linear codes that provide new families of {KPSs} and {OTBESs} for an arbitrarily large number of users and with secret keys of constant size. As a consequence of the {Gilbert-Varshamov} bound, we can prove that our {KPSs} are asymptotically more efficient than previous constructions, specially if we consider {KPSs} that are secure against coalitions formed by a constant fraction of the users. We analyze as well the {KPSs} that are obtained from families of algebraic geometry linear codes that are above the {Gilbert-Varshamov} bound, as the ones constructed from the curves of Garcia and Stichtenoth. Finally, we discuss how the use of {KPSs} based on algebraic geometry codes can provide more efficient {OTBESs}.},
    address = {Berlin, Heidelberg},
    author = {Chen, Hao and Ling, San and Padr\'{o}, Carles and Wang, Huaxiong and Xing, Chaoping},
    booktitle = {Cryptography and Coding},
    chapter = {16},
    citeulike-article-id = {6543583},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-642-10868-6_16},
    citeulike-linkout-1 = {http://www.springerlink.com/content/0161h875u6690670},
    doi = {10.1007/978-3-642-10868-6_16},
    editor = {Parker, Matthew},
    isbn = {978-3-642-10867-9},
    keywords = {coding\_theory, daniel, secret\_keys},
    pages = {263--277},
    posted-at = {2010-04-21 09:17:04},
    priority = {2},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Key Predistribution Schemes and {One-Time} Broadcast Encryption Schemes from Algebraic Geometry Codes},
    url = {http://dx.doi.org/10.1007/978-3-642-10868-6_16},
    volume = {5921},
    year = {2009}
}

@incollection{Chen-Cramer-Goldwasser-Haan-Vaikuntanathan:EURCOCRYPT2007,
    abstract = {Secure computation consists of protocols for secure arithmetic: secret values are added and multiplied securely by networked processors. The striking feature of secure computation is that security is maintained even in the presence of an adversary who corrupts a quorum of the processors and who exercises full, malicious control over them. One of the fundamental primitives at the heart of secure computation is secret-sharing. Typically, the required secret-sharing techniques build on Shamir's scheme, which can be viewed as a cryptographic twist on the {Reed-Solomon} error correcting code. In this work we further the connections between secure computation and error correcting codes. We demonstrate that threshold secure computation in the secure channels model can be based on arbitrary codes. For a network of size n, we then show a reduction in communication for secure computation amounting to a multiplicative logarithmic factor (in n) compared to classical methods for small, e.g., constant size fields, while tolerating  players to be corrupted, where ε\&gt; 0 can be arbitrarily small. For large networks this implies considerable savings in communication. Our results hold in the broadcast/negligible error model of Rabin and {Ben-Or}, and complement results from {CRYPTO} 2006 for the zero-error model of {Ben-Or}, Goldwasser and Wigderson ({BGW}). Our general theory can be extended so as to encompass those results from {CRYPTO} 2006 as well. We also present a new method for constructing high information rate ramp schemes based on arbitrary codes, and in particular we give a new construction based on algebraic geometry codes.},
    address = {Berlin, Heidelberg},
    author = {Chen, Hao and Cramer, Ronald and Goldwasser, Shafi and de Haan, Robbert and Vaikuntanathan, Vinod},
    booktitle = {Advances in Cryptology - EUROCRYPT 2007},
    chapter = {17},
    citeulike-article-id = {5832902},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-540-72540-4_17},
    citeulike-linkout-1 = {http://www.springerlink.com/content/37570763475553wn},
    doi = {10.1007/978-3-540-72540-4_17},
    editor = {Naor, Moni},
    isbn = {978-3-540-72539-8},
    journal = {Advances in Cryptology - EUROCRYPT 2007},
    keywords = {coding\_theory, daniel, secure\_computation},
    pages = {291--310},
    posted-at = {2010-04-21 09:16:27},
    priority = {2},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Secure Computation from Random Error Correcting Codes},
    url = {http://dx.doi.org/10.1007/978-3-540-72540-4_17},
    volume = {4515},
    year = {2007}
}

@incollection{Cascudo-Chen-Cramer-Xing:CRYPTO2009,
    abstract = {This work deals with  {MPC}-friendly  linear secret sharing schemes ({LSSS}), a mathematical primitive upon which secure multi-party computation ({MPC}) can be based and which was introduced by Cramer, Damgaard and Maurer ({EUROCRYPT} 2000). Chen and Cramer proposed a special class of such schemes that is constructed from algebraic geometry and that enables efficient secure multi-party computation over fixed finite fields ({CRYPTO} 2006). We extend this in four ways. First, we propose an abstract coding-theoretic framework in which this class of schemes and its (asymptotic) properties can be cast and analyzed. Second, we show that for every finite field , there exists an infinite family of {LSSS} over  that is asymptotically good in the following sense: the schemes are  ideal,  i.e., each share consists of a single -element, and the schemes have t-strong multiplication on n players, where the corruption tolerance  tends to a constant ν(q) with 0 \&lt; ν(q) \&lt; 1 when n tends to infinity. Moreover, when  tends to infinity, ν(q) tends to 1, which is optimal. This leads to explicit lower bounds on , our measure of asymptotic optimal corruption tolerance. We achieve this by combining the results of Chen and Cramer with a dedicated field-descent method. In particular, in the -case there exists a family of binary t-strongly multiplicative ideal {LSSS} with  when n tends to infinity, a one-bit secret and just a one-bit share for every player. Previously, such results were shown for  with q ≥ 49 a square. Third, we present an infinite family of ideal schemes with t-strong multiplication that does not rely on algebraic geometry and that works over every finite field . Its corruption tolerance vanishes, yet still . Fourth and finally, we give an improved non-asymptotic upper bound on corruption tolerance.},
    address = {Berlin, Heidelberg},
    author = {Cascudo, Ignacio and Chen, Hao and Cramer, Ronald and Xing, Chaoping},
    booktitle = {Advances in Cryptology - CRYPTO 2009},
    chapter = {28},
    citeulike-article-id = {5832899},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-3-642-03356-8_28},
    citeulike-linkout-1 = {http://www.springerlink.com/content/bgj52w775h64h851},
    doi = {10.1007/978-3-642-03356-8_28},
    editor = {Halevi, Shai},
    isbn = {978-3-642-03355-1},
    journal = {Advances in Cryptology - CRYPTO 2009},
    keywords = {coding\_theory, daniel, secret\_sharing},
    pages = {466--486},
    posted-at = {2010-04-21 09:15:52},
    priority = {2},
    publisher = {Springer Berlin / Heidelberg},
    series = {Lecture Notes in Computer Science},
    title = {Asymptotically Good Ideal Linear Secret Sharing with Strong Multiplication over \&{lt;i\&gt;Any}\&lt;/i\&gt; Fixed Finite Field},
    url = {http://dx.doi.org/10.1007/978-3-642-03356-8_28},
    volume = {5677},
    year = {2009}
}

@article{Liu-Shen:DCC2006,
    author = {Liu, Lihua and Shen, Hao},
    citeulike-article-id = {955139},
    citeulike-linkout-0 = {http://dx.doi.org/10.1007/s10623-006-9004-y},
    doi = {10.1007/s10623-006-9004-y},
    journal = {Designs, Codes and Cryptography},
    keywords = {algebraic\_curves, coding\_theory, daniel, hash},
    month = nov,
    number = {2},
    pages = {221--233},
    posted-at = {2010-04-21 09:14:57},
    priority = {2},
    title = {Explicit constructions of separating hash families from algebraic curves over finite fields},
    url = {http://dx.doi.org/10.1007/s10623-006-9004-y},
    volume = {V41},
    year = {2006}
}

@article{Fernandez-Soriano:IEEE_SP:2004,
    abstract = {In a fingerprinting scheme, a distributor places marks in each copy of a digital object. Placing different marks in different copies uniquely identifies the recipient of each copy and therefore allows the tracing of the source of an unauthorized redistribution. A widely used approach to the fingerprinting problem is the use of error-correcting codes with a suitable minimum distance. With this approach, the set of embedded marks in a given copy is precisely a codeword of the error correcting code. The focus of this paper is in the identification of traitors when the error-correcting code is an algebraic-geometric ({AG}) code. The authors present a tracing algorithm that employs the {Guruswami-Sudan} soft-decision list decoding algorithm to find all provably identifiable dishonest users.},
    author = {Fernandez, M. and Soriano, M.},
    citeulike-article-id = {754465},
    citeulike-linkout-0 = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1337283},
    journal = {IEEE Transactions on Signal Processing},
    keywords = {algebraic\_geometry, coding\_theory, daniel},
    number = {10},
    pages = {3073--3077},
    posted-at = {2010-04-21 09:14:39},
    priority = {2},
    title = {Identification of traitors in algebraic-geometric traceability codes},
    url = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1337283},
    volume = {52},
    year = {2004}
}

@electronic{Chen-Ma-Li:ARXIV2006,
    abstract = {Propagation criterion of degree \$l\$ and order \$k\$ (\${PC}(l)\$ of order \$k\$) for Boolean functions is important for the design of block ciphers. In [1-2] Kurosawa, Stoh and Carlet give several constructions of Boolean functions satisfying \${PC}(l)\$ of order \$k\$ from binary linear or nonlinear codes. Matsumoto, Kurosawa and Itoh get some lower bounds and an non-constructive upper bound on the input length the Boolean functions in {Kurosawa-Satoh} construction. In this paper, algebraic-geometric codes over \${GF}(2^m)\$ are inserted into {Kurosawa-Satoh} construction for giving explicit constructions of Boolean functions satisfying \${PC}(l)\$ of order \$k\$ by this {AG}-construction. This method give some constructive upper bound on the minimum input length of Boolean functions satisfying \${PC}(l)\$ of order \$k\$ in {Kurosawa-Satoh} construction.},
    archivePrefix = {arXiv},
    author = {Chen, Hao and Ma, Liang and Li, Jianhua},
    citeulike-article-id = {684069},
    citeulike-linkout-0 = {http://arxiv.org/abs/cs.CR/0606011},
    citeulike-linkout-1 = {http://arxiv.org/pdf/cs.CR/0606011},
    day = {2},
    eprint = {cs.CR/0606011},
    keywords = {boolean\_functions, coding\_theory, daniel},
    month = jun,
    posted-at = {2010-04-21 09:14:21},
    priority = {2},
    title = {Constructing \({PC}(l)\) of order \(k\) Boolean Functions from {Algebraic-Geometric} Codes},
    url = {http://arxiv.org/abs/cs.CR/0606011},
    year = {2006}
}

@article{Feng-Ling-Xing:IEEE_IT2006,
    abstract = {We generalize a characterization of p-ary (p is a prime) quantum codes given by Feng and Xing to q-ary (q is a prime power) quantum codes. This characterization makes it possible to convert an asymptotic bound of Stichtenoth and Xing for nonlinear algebraic geometry codes to a quantum asymptotic bound. Besides, we also investigate the asymptotic behavior of quantum codes.},
    author = {Feng, Keqin and Ling, San and Xing, Chaoping},
    citeulike-article-id = {674960},
    citeulike-linkout-0 = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1603766},
    journal = {IEEE Transactions on Information Theory},
    keywords = {algebraic\_geometry, coding\_theory, daniel, quantum},
    number = {3},
    pages = {986--991},
    posted-at = {2010-04-21 09:13:47},
    priority = {2},
    title = {Asymptotic bounds on quantum codes from algebraic geometry codes},
    url = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1603766},
    volume = {52},
    year = {2006}
}

@article{Xing-Ding:IEEE_IT2009,
    abstract = {In the present paper, by making use of some special properties of the Hermitian function fields, we construct multisequences with both large linear complexity and k-error linear complexity. Moreover, these sequences can be explicitly constructed.},
    author = {Xing, Chaoping and Ding, Yang},
    booktitle = {Information Theory, IEEE Transactions on},
    citeulike-article-id = {5237585},
    citeulike-linkout-0 = {http://dx.doi.org/10.1109/TIT.2009.2023714},
    citeulike-linkout-1 = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5165170},
    day = {14},
    doi = {10.1109/TIT.2009.2023714},
    journal = {Information Theory, IEEE Transactions on},
    keywords = {coding\_theory, daniel, linear\_complexity, sequences},
    month = jul,
    number = {8},
    pages = {3858--3863},
    posted-at = {2010-04-21 09:11:10},
    priority = {2},
    title = {Multisequences With Large Linear and \({k\)-Error} Linear Complexity From Hermitian Function Fields},
    url = {http://dx.doi.org/10.1109/TIT.2009.2023714},
    volume = {55},
    year = {2009}
}

@article{Maharaj-Matthews-Pirsic:JPAA2005,
    abstract = {This paper is concerned with two applications of bases of {Riemann-Roch} spaces. In the first application, we define the floor of a divisor and obtain improved bounds on the parameters of algebraic geometry codes. These bounds apply to a larger class of codes than that of Homma and Kim (J. Pure Appl. Algebra 162 (2001) 273). Then we determine explicit bases for large classes of {Riemann-Roch} spaces of the Hermitian function field. These bases give better estimates on the parameters of a large class of m-point Hermitian codes. In the second application, these bases are used for fast implementation of Xing and Niederreiter's method (Acta. Arith. 72 (1995) 281) for the construction of low-discrepancy sequences.},
    author = {Maharaj, Hiren and Matthews, Gretchen L. and Pirsic, Gottlieb},
    citeulike-article-id = {494259},
    citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.jpaa.2004.06.010},
    citeulike-linkout-1 = {http://www.sciencedirect.com/science/article/B6V0K-4D0Y5RT-1/2/c7af6d8d01a2c3def3da19a087b2acd4},
    day = {1},
    doi = {10.1016/j.jpaa.2004.06.010},
    journal = {Journal of Pure and Applied Algebra},
    keywords = {coding\_theory, daniel, riemann-roch},
    month = feb,
    number = {3},
    pages = {261--280},
    posted-at = {2010-04-21 09:10:08},
    priority = {2},
    title = {{R}iemann-{R}och spaces of the {H}ermitian function field with applications to algebraic geometry codes and low-discrepancy sequences},
    url = {http://dx.doi.org/10.1016/j.jpaa.2004.06.010},
    volume = {195},
    year = {2005}
}

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