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Lattice algorithms using floating-point arithmetic

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fplll

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fplll contains implementations of several lattice algorithms. The implementation relies on floating-point orthogonalization, and LLL [LLL82] is central to the code, hence the name.

It includes implementations of floating-point LLL reduction algorithms [NS09,MSV09], offering different speed/guarantees ratios. It contains a 'wrapper' choosing the estimated best sequence of variants in order to provide a guaranteed output as fast as possible [S10]. In the case of the wrapper, the succession of variants is oblivious to the user.

It includes an implementation of the BKZ reduction algorithm [SE94], including the BKZ-2.0 [CN11] improvements (extreme enumeration pruning, pre-processing of blocks, early termination). Additionally, Slide reduction [GN08] and self dual BKZ [MW16] are supported.

It also includes a floating-point implementation of the Kannan-Fincke-Pohst algorithm [K83,FP85] that finds a shortest non-zero lattice vector. For the same task, the GaussSieve algorithm [MV10] is also available in fplll. Finally, it contains a variant of the enumeration algorithm that computes a lattice vector closest to a given vector belonging to the real span of the lattice.

fplll is distributed under the GNU Lesser General Public License (either version 2.1 of the License, or, at your option, any later version) as published by the Free Software Foundation.

How to cite

@unpublished{fplll,
    author = {The {FPLLL} development team},
    title = {{fplll}, a lattice reduction library},
    year = 2016,
    note = {Available at \url{https://github.com/fplll/fplll}},
    url = {https://github.com/fplll/fplll}
}

Table of contents

Compilation

Dependencies

Required

Optional

Installation

Linux

You should downloaded the source code from github and then run

./autogen.sh

which generates the ./configure script used to configure fplll by calling the appropriate autotools command.

Then, to compile and install type

./configure
make
make install			# (as root)

If GMP, MPFR and/or MPIR are not in the $LD_LIBRARY_PATH, you have to point to the directories where the libraries are, with

./configure --with-gmp=path/to/gmp

or

./configure --with-mpfr=path/to/mpfr

The same philosophy applies to the (optional) QD library. If you want to use mpir instead of gmp, use --enable-mpir and --with-mpir=path/to/mpir.

You can remove the program binaries and object files from the source code directory by typing make clean. To also remove the files that ./configure created (so you can compile the package for a different kind of computer), type make distclean. By default, make install installs the package commands under /usr/local/bin, include files under /usr/local/include, etc. You can specify an installation directory name other than /usr/local by giving ./configure the option --prefix=dirname. Run ./configure --help for further details.

Windows 10

Windows 10 has a "Windows Subsystem for Linux", which essentially allows you to use Linux features in Windows without the need for a dual-boot system or a virtual machine. To activate this, first go to Settings -> Update and security -> For developers and enable developer mode. (This may take a while.) Afterwards, open command prompt and run

lxrun /install

This will install the WSL, and afterwards this system can be accessed e.g. by opening command prompt and typing bash. With this Linux-like subsystem, installing fplll is then similar to above, except that most likely the package repository is not up to date, and various additional packages need to be installed first. To make sure you only install the most recent software, run:

sudo apt-get update

Then run sudo apt-get install <packages> for the (indirectly) required packages, such as make, autoconf, libtool, gcc, g++, libgmp-dev, and libmpfr-dev. Finally, download the fplll source code, extract the contents, navigate to this folder in Bash (commonly found under /mnt/c/<local path> when stored somewhere on the C:\ drive), and run:

./autogen.sh
./configure
make

The same comments as before apply for using e.g. make install or make distclean instead of make.

Check

Type

make check

Optimization

The default compilation flag is -O3. One may use the -march=native -O3 flag to optimize the binaries. See "this issue" for its impact on the enumeration speed.

How to use

Executable files fplll and latticegen are installed in the directory /usr/local/bin. (Note that the programs generated by make in the fplll/ directory are only wrappers to the programs in fplll/.libs/).

If you type make check, it will also generate the executable file llldiff, in fplll/.libs/.

latticegen

latticegen is a utility for generating matrices (rows form input lattice basis vectors).

The options are:

  • r d b : generates a knapsack like matrix of dimension d x (d+1) and b bits (see, e.g., [S09]): the i-th vector starts with a random integer of bit-length <=b and the rest is the i-th canonical unit vector.
  • s d b b2 : generates a d x d matrix of a form similar to that is involved when trying to find rational approximations to reals with the same small denominator (see, e.g., [LLL82]): the first vector starts with a random integer of bit-length <=b2 and continues with d-1 independent integers of bit-lengths <=b; the i-th vector for i>1 is the i-th canonical unit vector scaled by a factor 2^b.
  • u d b : generates a d x d matrix whose entries are independent integers of bit-lengths <=b.
  • n d b c : generates an ntru-like matrix. If char is 'b', then it first samples an integer q of bit-length <=b, whereas if char is 'q', then it sets q to the provided value. Then it samples a uniform h in the ring Z_q[x]/(x^n-1). It finally returns the 2 x 2 block matrix [[I, Rot(h)], [0, q*I]], where each block is d x d, the first row of Rot(h) is the coefficient vector of h, and the i-th row of Rot(h) is the shift of the (i-1)-th (with last entry put back in first position), for all i>1. Warning: this does not produce a genuine ntru lattice with h a genuine public key (see [HPS98]).
  • N d b c : as the previous option, except that the contructed matrix is [[q*I, 0], [Rot(h), I]].
  • q d k b c : generates a q-ary matrix. If char is 'b', then it first samples an integer q of bit-length <=b; if char is 'p', it does the same and updates q to the smallest (probabilistic) prime that is greater; if char is 'q', then it sets q to the provided value. It returns a 2 x 2 block matrix [[I, H], [0, q*I]], where H is (d-k) x k and uniformly random modulo q. These bases correspond to the SIS/LWE q-ary lattices (see [MR09]). Goldstein-Mayer lattices correspond to k=1 and q prime (see [GM03]).
  • t d f : generates a d x d lower-triangular matrix B with B_ii = 2^(d-i+1)^f for all i, and B_ij is uniform between -B_jj/2 and B_jj/2 for all j<i.
  • T d : also takes as input a d-dimensional vector vec read from a file. It generates a d x d lower-triangular matrix B with B_ii = vec[i] for all i and B_ij is uniform between -B_jj/2 and B_jj/2 for all j<i.

The generated matrix is printed in stdout.

Note that by default, the random bits always use the same seed, to ensure reproducibility. The seed may be changed with the option -randseed <integer> or by using the current time (in seconds) -randseed time. If you use this option, it must be the first one on the command line.

fplll

fplll does LLL, BKZ, HKZ or SVP on a matrix (considered as a set of row vectors) given in stdin or in a file as parameter.

The options are:

  • -a lll : LLL-reduction (default).

  • -a bkz : BKZ-reduction.

  • -a hkz : HKZ-reduction.

  • -a svp : prints a shortest non-zero vector of the lattice.

  • -a sdb : self dual variant of BKZ-reduction.

  • -a sld : slide reduction.

  • -a cvp : prints the vector in the lattice closest to the input vector.

  • -v : verbose mode.

  • -nolll : does not apply to LLL-reduction. In the case of bkz, hkz and svp, by default, the input basis is LLL-reduced before anything else. This option allows to remove that initial LLL-reduction (note that other calls to LLL-reduction may occur during the execution). In the cas of hlll, verify if the input basis is HLLL-reduced.

  • -a hlll : HLLL-reduction.

Options for LLL-reduction:

  • -d delta : δ (default=0.99)

  • -e eta : η (default=0.51). See [NS09] for the definition of (δ,η)-LLL-reduced bases.

  • -l lovasz : if !=0 Lovasz's condition. Otherwise, Siegel's condition (default: Lovasz). See [A02] for the definition of Siegel condition.

  • -f mpfr : sets the floating-point type to MPFR (default if m=proved).

  • -p precision : precision of the floating-point arithmetic, works only with -f mpfr.

  • -f dd : sets the floating-point type to double-double.

  • -f qd : sets the floating-point type to quad-double.

  • -f dpe : sets the floating-point type to DPE (default if m=heuristic).

  • -f double : sets the floating-point type to double (default if m=fast).

  • -f longdouble : sets the floating-point type to long double.

  • -z mpz : sets the integer type to mpz, the integer type of GMP (default).

  • -z int : sets the integer type to int.

  • -z long : as -z int.

  • -z double : sets the integer type to double.

  • -m wrapper : uses the wrapper. (default if z=mpz).

  • -m fast : uses the fast method, works only with -f double.

  • -m heuristic : uses the heuristic method.

  • -m proved : uses the proved version of the algorithm.

  • -y : early reduction.

With the wrapper or the proved version, it is guaranteed that the basis is LLL-reduced with δ'=2×δ-1 and η'=2×η-1/2. For instance, with the default options, it is guaranteed that the basis is (0.98,0.52)-LLL-reduced.

Options for BKZ-reduction:

  • -b block_size : block size, mandatory, between 2 and the number of vectors.

  • -f float_type : same as LLL (-p is required if float_type=mpfr).

  • -p precision : precision of the floating-point arithmetic with -f mpfr.

  • -bkzmaxloops loops : maximum number of full loop iterations.

  • -bkzmaxtime time : stops after time seconds (up to completion of the current loop iteration).

  • -bkzautoabort : stops when the average slope of the log ||b_i*||'s does not decrease fast enough.

Without any of the last three options, BKZ runs until no block has been updated for a full loop iteration.

  • -s filename.json : use strategies for preprocessing and pruning paramater (/strategies/default.json provided). Experimental.

  • -bkzghbound factor : multiplies the Gaussian heuristic by factor (of float type) to set the enumeration radius of the SVP calls.

  • -bkzboundedlll : restricts the LLL call before considering a block to vector indices within that block.

  • -bkzdumgso file_name : dumps the log ||b_i*|| 's in specified file.

Output formats:

  • -of : prints new line (if -a [lll|bkz])
  • -of b : prints the basis (if -a [lll|bkz], this value by default)
  • -of bk : prints the basis (if -a [lll|bkz], format compatible with sage)
  • -of c : prints the closest vector (if -a cvp, this value by default)
  • -of s : prints the closest vector (if -a svp, this value by default)
  • -of t : prints status (if -a [lll|bkz|cvp|svp])
  • -of u : prints unimodular matrix (if -a [lll|bkz])
  • -of uk : prints unimodular matrix (if -a [lll|bkz], format compatible with sage)
  • -of v : prints inverse of u (if -a lll)
  • -of vk : prints inverse of u (if -a lll, format compatible with sage)

A combination of these option is allowed (e.g., -of bkut).

Only for -a hlll:

  • -t theta : θ (default=0.001). See [MSV09] for the definition of (δ,η,θ)-HLLL-reduced bases.
  • -c c : constant for HLLL during the size-reduction (only used if fplll is compiled with -DHOUSEHOLDER_USE_SIZE_REDUCTION_TEST)

llldiff

llldiff compares two bases (b1,...,bd) and (c1,...c_d'): they are considered equal iff d=d' and for any i, bi = +- ci. Concretely, if basis B is in file 'B.txt' and if basis C is in file 'C.txt' (in the fplll format), then one may run cat B.txt C.txt | ./llldiff.

latsieve

latsieve does (tuple) lattice sieve on a matrix (considered as a set of row vectors) given in a file as parameter. You may compile it by using make latsieve. It will generate the executable file latsieve in fplll/ which is a wrapper of fplll/.libs/latsieve.

The options are:

  • -a nnn : nnn is the tuple algorithm to use (default 2 corresponding to GaussSieve)
  • -f filename : follows input matrix
  • -b nnn : BKZ preprocessing of blocksize nnn (optional)
  • -t nnn : targeted square norm for stoping sieving (optional)
  • -s nnn : using seed=nnn (optional)
  • -v : verbose toggle

How to use as a library

See API documentation.

Multicore support

This library does not currently use multiple cores and running multiple threads working on the same object IntegerMatrix, LLLReduction, MatGSO etc. is not supported. Running multiple threads working on different objects, however, is supported. That is, there are no global variables and it is safe to e.g. reduce several lattices in parallel in the same process.

Examples

  1. LLL reduction

    ./latticegen r 10 1000 | ./fplll

  2. Fileinput for reduction. If the file matrix contains

    [[ 10 11]
[11 12]]

    then

    ./fplll matrix

    produces

    [[0 1 ]
 [1 0 ]
]

  3. Random generator

    ./latticegen -randseed 1234 r 10 1000 | ./fplll
./latticegen -randseed time u 10 16 | ./fplll

  4. Solving SVP

    ./latticegen r 30 3000 | ./fplll -a svp

  5. Solving CVP

    echo "[[17 42 4][50 75 108][11 47 33]][100 101 102]" | ./fplll -a cvp

Alternative interfaces

Credit

Maintainers

fplll is currently maintained by:

Contributors

The following people have contributed to fplll:

  • Martin Albrecht
  • Shi Bai
  • Guillaume Bonnoron
  • David Cade
  • Léo Ducas
  • Joop van de Pol
  • Xavier Pujol
  • Damien Stehlé
  • Marc Stevens
  • Gilles Villard
  • Michael Walter

Please add yourself here if you make a contribution.

Acknowledgments

  • Patrick Pelissier and Paul Zimmermann for dpe.

  • David H. Bailey for QD.

  • Sylvain Chevillard, Christoph Lauter and Gilles Villard for the configure/make/make install packaging.

  • Timothy Abbott, Michael Abshoff, Bill Allombert, John Cannon, Sylvain Chevillard, Julien Clement, Andreas Enge, Jean-Pierre Flori, Laurent Fousse, Guillaume Hanrot, Jens Hermans, Jerry James, Christoph Lauter, Tancrède Lepoint, Andrew Novocin, Willem Jan Palenstijn, Patrick Pelissier, Julien Puydt, Michael Schneider, Thiemo Seufer, Allan Steel, Gilles Villard and Paul Zimmermann for their support and for many suggestions that helped debugging and improving this code.

  • CONTRIBUTING.md is taken, almost verbatim, from https://github.com/pydanny/djangopackages/blob/master/docs/contributing.rst

  • json.hpp is taken from https://github.com/nlohmann/json

  • This project has been supported by ERC Starting Grant ERC-2013-StG-335086-LATTAC.

Contributing

fplll welcomes contributions. See CONTRIBUTING.md for details.

New releases and bug reports

New releases will be announced on https://groups.google.com/forum/#!forum/fplll-devel.

Bug reports may be sent to https://groups.google.com/forum/#!forum/fplll-devel or via https://github.com/fplll/fplll/issues.

Bibliography

[A02] A. Akhavi. Random lattices, threshold phenomena and efficient reduction algorithms. Theor. Comput. Sci. 287(2): 359-385 (2002)

[Chen13] Y. Chen, Lattice reduction and concrete security of fully homomorphic encryption.

[CN11] Y. Chen and P. Q. Nguyen. BKZ 2.0: Better Lattice Security Estimates. ASIACRYPT 2011: 1-20

[GM03] D. Goldstein and A. Mayer. On the equidistribution of Hecke points. Forum Mathematicum, 15:165–189 (2003)

[GN08] N. Gama and P. Q. Nguyen. Finding Short Lattice Vectors within Mordell's Inequality. STOC 2008: 207-216

[GNR13] N. Gama, P. Q. Nguyen and Oded Regev. Lattice Enumeration Using Extreme Pruning.

[HPS98] J. Hoffstein, J. Pipher, J. H. Silverman. NTRU: A Ring-Based Public Key Cryptosystem. ANTS 1998: 267-288

[K83] R. Kannan. Improved algorithms for integer programming and related lattice problems. STOC 1983, 99-108

[FP85] U. Fincke and M. Pohst. Improved methods for calculating vectors of short length in a lattice, including a complexity analysis. Math. Comp., 44(170):463–471 (1985)

[LLL82] A. K. Lenstra, H. W. Lenstra, Jr. and L. Lovasz. Factoring polynomials with rational coefficients. Math. Ann., 261: 515–534 (1982)

[MSV09] I. Morel, D. Stehle and G. Villard. H-LLL: using Householder inside LLL. ISSAC 2009: 271-278

[MV10] D. Micciancio and P. Voulgaris. Faster Exponential Time Algorithms for the Shortest Vector Problem. SODA 2010: 1468-1480

[MW16] D. Micciancio and M. Walter. Practical, Predictable Lattice Basis Reduction. EUROCRYPT 2016: 820-849

[MR09] D. Micciancio and O. Regev. Post-Quantum Cryptography. Chapter of Lattice-based Cryptography, 147-191 (2009)

[NS09] P. Q. Nguyen and D. Stehle. An LLL Algorithm with Quadratic Complexity. SIAM J. Comput. 39(3): 874-903 (2009)

[S10] D. Stehle. Floating-Point LLL: Theoretical and Practical Aspects. The LLL Algorithm 2010: 179-213

[SE94]: C.-P. Schnorr and M. Euchner. Lattice basis reduction: Improved practical algorithms and solving subset sum problems. Math. Program. 66: 181-199 (1994)

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Lattice algorithms using floating-point arithmetic

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