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free_module_integer.py
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free_module_integer.py
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# -*- coding: utf-8 -*-
"""
Discrete Subgroups of `\\ZZ^n`.
AUTHORS:
- Martin Albrecht (2014-03): initial version
- Jan Pöschko (2012-08): some code in this module was taken from Jan Pöschko's
2012 GSoC project
TESTS::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: L = IntegerLattice(random_matrix(ZZ, 10, 10))
sage: TestSuite(L).run()
"""
from __future__ import absolute_import
##############################################################################
# Copyright (C) 2012 Jan Poeschko <jan@poeschko.com>
# Copyright (C) 2014 Martin Albrecht <martinralbecht@googlemail.com>
#
# Distributed under the terms of the GNU General Public License (GPL)
#
# This code is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# General Public License for more details.
#
# The full text of the GPL is available at:
#
# http://www.gnu.org/licenses/
##############################################################################
from sage.rings.integer_ring import ZZ
from sage.matrix.constructor import matrix
from sage.misc.cachefunc import cached_method
from sage.modules.free_module import FreeModule_submodule_with_basis_pid, FreeModule_ambient_pid
from sage.modules.free_module_element import vector
from sage.rings.number_field.number_field_element import OrderElement_absolute
def IntegerLattice(basis, lll_reduce=True):
r"""
Construct a new integer lattice from ``basis``.
INPUT:
- ``basis`` -- can be one of the following:
- a list of vectors
- a matrix over the integers
- an element of an absolute order
- ``lll_reduce`` -- (default: ``True``) run LLL reduction on the basis
on construction.
EXAMPLES:
We construct a lattice from a list of rows::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: IntegerLattice([[1,0,3], [0,2,1], [0,2,7]])
Free module of degree 3 and rank 3 over Integer Ring
User basis matrix:
[-2 0 0]
[ 0 2 1]
[ 1 -2 2]
Sage includes a generator for hard lattices from cryptography::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: A = sage.crypto.gen_lattice(type='modular', m=10, seed=1337, dual=True)
sage: IntegerLattice(A)
Free module of degree 10 and rank 10 over Integer Ring
User basis matrix:
[-1 1 2 -2 0 1 0 -1 2 1]
[ 1 0 0 -1 -2 1 -2 3 -1 0]
[ 1 2 0 2 -1 1 -2 2 2 0]
[ 1 0 -1 0 2 3 0 0 -1 -2]
[ 1 -3 0 0 2 1 -2 -1 0 0]
[-3 0 -1 0 -1 2 -2 0 0 2]
[ 0 0 0 1 0 2 -3 -3 -2 -1]
[ 0 -1 -4 -1 -1 1 2 -1 0 1]
[ 1 1 -2 1 1 2 1 1 -2 3]
[ 2 -1 1 2 -3 2 2 1 0 1]
You can also construct the lattice directly::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: sage.crypto.gen_lattice(type='modular', m=10, seed=1337, dual=True, lattice=True)
Free module of degree 10 and rank 10 over Integer Ring
User basis matrix:
[-1 1 2 -2 0 1 0 -1 2 1]
[ 1 0 0 -1 -2 1 -2 3 -1 0]
[ 1 2 0 2 -1 1 -2 2 2 0]
[ 1 0 -1 0 2 3 0 0 -1 -2]
[ 1 -3 0 0 2 1 -2 -1 0 0]
[-3 0 -1 0 -1 2 -2 0 0 2]
[ 0 0 0 1 0 2 -3 -3 -2 -1]
[ 0 -1 -4 -1 -1 1 2 -1 0 1]
[ 1 1 -2 1 1 2 1 1 -2 3]
[ 2 -1 1 2 -3 2 2 1 0 1]
We construct an ideal lattice from an element of an absolute order::
sage: K.<a> = CyclotomicField(17)
sage: O = K.ring_of_integers()
sage: f = O.random_element(); f
-a^15 + a^13 + 4*a^12 - 12*a^11 - 256*a^10 + a^9 - a^7 - 4*a^6 + a^5 + 210*a^4 + 2*a^3 - 2*a^2 + 2*a - 2
sage: from sage.modules.free_module_integer import IntegerLattice
sage: IntegerLattice(f)
Free module of degree 16 and rank 16 over Integer Ring
User basis matrix:
[ -2 2 -2 2 210 1 -4 -1 0 1 -256 -12 4 1 0 -1]
[ 33 48 44 48 256 -209 28 51 45 49 -1 35 44 48 44 48]
[ 1 -1 3 -1 3 211 2 -3 0 1 2 -255 -11 5 2 1]
[-223 34 50 47 258 0 29 45 46 47 2 -11 33 48 44 48]
[ -13 31 46 42 46 -2 -225 32 48 45 256 -2 27 43 44 45]
[ -16 33 42 46 254 1 -19 32 44 45 0 -13 -225 32 48 45]
[ -15 -223 30 50 255 1 -20 32 42 47 -2 -11 -15 33 44 44]
[ -11 -11 33 48 256 3 -17 -222 32 53 1 -9 -14 35 44 48]
[ -12 -13 32 45 257 0 -16 -13 32 48 -1 -10 -14 -222 31 51]
[ -9 -13 -221 32 52 1 -11 -12 33 46 258 1 -15 -12 33 49]
[ -5 -2 -1 0 -257 -13 3 0 -1 -2 -1 -3 1 -3 1 209]
[ -15 -11 -15 33 256 -1 -17 -14 -225 33 4 -12 -13 -14 31 44]
[ 11 11 11 11 -245 -3 17 10 13 220 12 5 12 9 14 -35]
[ -18 -15 -20 29 250 -3 -23 -16 -19 30 -4 -17 -17 -17 -229 28]
[ -15 -11 -15 -223 242 5 -18 -12 -16 34 -2 -11 -15 -11 -15 33]
[ 378 120 92 147 152 462 136 96 99 144 -52 412 133 91 -107 138]
We construct `\ZZ^n`::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: IntegerLattice(ZZ^10)
Free module of degree 10 and rank 10 over Integer Ring
User basis matrix:
[1 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 1]
Sage also interfaces with fpylll's lattice generator::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: from fpylll import IntegerMatrix
sage: A = IntegerMatrix.random(8, "simdioph", bits=20, bits2=10)
sage: A = A.to_matrix(matrix(ZZ, 8, 8))
sage: IntegerLattice(A, lll_reduce=False)
Free module of degree 8 and rank 8 over Integer Ring
User basis matrix:
[ 1024 829556 161099 11567 521155 769480 639201 689979]
[ 0 1048576 0 0 0 0 0 0]
[ 0 0 1048576 0 0 0 0 0]
[ 0 0 0 1048576 0 0 0 0]
[ 0 0 0 0 1048576 0 0 0]
[ 0 0 0 0 0 1048576 0 0]
[ 0 0 0 0 0 0 1048576 0]
[ 0 0 0 0 0 0 0 1048576]
"""
if isinstance(basis, OrderElement_absolute):
basis = basis.matrix()
elif isinstance(basis, FreeModule_ambient_pid):
basis = basis.basis_matrix()
try:
basis = matrix(ZZ, basis)
except TypeError:
raise NotImplementedError("only integer lattices supported")
return FreeModule_submodule_with_basis_integer(ZZ**basis.ncols(),
basis=basis,
lll_reduce=lll_reduce)
class FreeModule_submodule_with_basis_integer(FreeModule_submodule_with_basis_pid):
r"""
This class represents submodules of `\ZZ^n` with a distinguished basis.
However, most functionality in excess of standard submodules over PID
is for these submodules considered as discrete subgroups of `\ZZ^n`, i.e.
as lattices. That is, this class provides functions for computing LLL
and BKZ reduced bases for this free module with respect to the standard
Euclidean norm.
EXAMPLE::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: L = IntegerLattice(sage.crypto.gen_lattice(type='modular', m=10, seed=1337, dual=True)); L
Free module of degree 10 and rank 10 over Integer Ring
User basis matrix:
[-1 1 2 -2 0 1 0 -1 2 1]
[ 1 0 0 -1 -2 1 -2 3 -1 0]
[ 1 2 0 2 -1 1 -2 2 2 0]
[ 1 0 -1 0 2 3 0 0 -1 -2]
[ 1 -3 0 0 2 1 -2 -1 0 0]
[-3 0 -1 0 -1 2 -2 0 0 2]
[ 0 0 0 1 0 2 -3 -3 -2 -1]
[ 0 -1 -4 -1 -1 1 2 -1 0 1]
[ 1 1 -2 1 1 2 1 1 -2 3]
[ 2 -1 1 2 -3 2 2 1 0 1]
sage: L.shortest_vector()
(-1, 1, 2, -2, 0, 1, 0, -1, 2, 1)
"""
def __init__(self, ambient, basis, check=True, echelonize=False,
echelonized_basis=None, already_echelonized=False,
lll_reduce=True):
r"""
Construct a new submodule of `\ZZ^n` with a distinguished basis.
INPUT:
- ``ambient`` -- ambient free module over a principal ideal domain
`\ZZ`, i.e. `\ZZ^n`
- ``basis`` -- either a list of vectors or a matrix over the integers
- ``check`` -- (default: ``True``) if ``False``, correctness of
the input will not be checked and type conversion may be omitted,
use with care
- ``echelonize`` -- (default:``False``) if ``True``, ``basis`` will be
echelonized and the result will be used as the default basis of the
constructed submodule
- `` echelonized_basis`` -- (default: ``None``) if not ``None``, must
be the echelonized basis spanning the same submodule as ``basis``
- ``already_echelonized`` -- (default: ``False``) if ``True``,
``basis`` must be already given in the echelonized form
- ``lll_reduce`` -- (default: ``True``) run LLL reduction on the basis
on construction
EXAMPLES::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: IntegerLattice([[1,0,-2], [0,2,5], [0,0,7]])
Free module of degree 3 and rank 3 over Integer Ring
User basis matrix:
[ 1 0 -2]
[ 1 -2 0]
[ 2 2 1]
sage: IntegerLattice(random_matrix(ZZ, 5, 5, x=-2^20, y=2^20))
Free module of degree 5 and rank 5 over Integer Ring
User basis matrix:
[ -7945 -381123 85872 -225065 12924]
[-158254 120252 189195 -262144 -345323]
[ 232388 -49556 306585 -31340 401528]
[-353460 213748 310673 158140 172810]
[-287787 333937 -145713 -482137 186529]
sage: K.<a> = NumberField(x^8+1)
sage: O = K.ring_of_integers()
sage: f = O.random_element(); f
a^7 - a^6 + 4*a^5 - a^4 + a^3 + 1
sage: IntegerLattice(f)
Free module of degree 8 and rank 8 over Integer Ring
User basis matrix:
[ 0 1 0 1 0 3 3 0]
[ 1 0 0 1 -1 4 -1 1]
[ 0 0 1 0 1 0 3 3]
[-4 1 -1 1 0 0 1 -1]
[ 1 -3 0 0 0 3 0 -2]
[ 0 -1 1 -4 1 -1 1 0]
[ 2 0 -3 -1 0 -3 0 0]
[-1 0 -1 0 -3 -3 0 0]
"""
basis = matrix(ZZ, basis)
self._basis_is_LLL_reduced = False
if lll_reduce:
basis = matrix([v for v in basis.LLL() if v])
self._basis_is_LLL_reduced = True
basis.set_immutable()
FreeModule_submodule_with_basis_pid.__init__(self,
ambient=ambient,
basis=basis,
check=check,
echelonize=echelonize,
echelonized_basis=echelonized_basis,
already_echelonized=already_echelonized)
self._reduced_basis = basis.change_ring(ZZ)
@property
def reduced_basis(self):
"""
This attribute caches the currently best known reduced basis for
``self``, where "best" is defined by the Euclidean norm of the
first row vector.
EXAMPLE::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: L = IntegerLattice(random_matrix(ZZ, 10, 10), lll_reduce=False)
sage: L.reduced_basis
[ -8 2 0 0 1 -1 2 1 -95 -1]
[ -2 -12 0 0 1 -1 1 -1 -2 -1]
[ 4 -4 -6 5 0 0 -2 0 1 -4]
[ -6 1 -1 1 1 -1 1 -1 -3 1]
[ 1 0 0 -3 2 -2 0 -2 1 0]
[ -1 1 0 0 1 -1 4 -1 1 -1]
[ 14 1 -5 4 -1 0 2 4 1 1]
[ -2 -1 0 4 -3 1 -5 0 -2 -1]
[ -9 -1 -1 3 2 1 -1 1 -2 1]
[ -1 2 -7 1 0 2 3 -1955 -22 -1]
sage: _ = L.LLL()
sage: L.reduced_basis
[ 1 0 0 -3 2 -2 0 -2 1 0]
[ -1 1 0 0 1 -1 4 -1 1 -1]
[ -2 0 0 1 0 -2 -1 -3 0 -2]
[ -2 -2 0 -1 3 0 -2 0 2 0]
[ 1 1 1 2 3 -2 -2 0 3 1]
[ -4 1 -1 0 1 1 2 2 -3 3]
[ 1 -3 -7 2 3 -1 0 0 -1 -1]
[ 1 -9 1 3 1 -3 1 -1 -1 0]
[ 8 5 19 3 27 6 -3 8 -25 -22]
[ 172 -25 57 248 261 793 76 -839 -41 376]
"""
return self._reduced_basis
def LLL(self, *args, **kwds):
r"""
Return an LLL reduced basis for ``self``.
A lattice basis `(b_1, b_2, ..., b_d)` is `(\delta, \eta)`-LLL-reduced
if the two following conditions hold:
- For any `i > j`, we have `\lvert \mu_{i, j} \rvert \leq η`.
- For any `i < d`, we have
`\delta \lvert b_i^* \rvert^2 \leq \lvert b_{i+1}^* +
\mu_{i+1, i} b_i^* \rvert^2`,
where `\mu_{i,j} = \langle b_i, b_j^* \rangle / \langle b_j^*,b_j^*
\rangle` and `b_i^*` is the `i`-th vector of the Gram-Schmidt
orthogonalisation of `(b_1, b_2, \ldots, b_d)`.
The default reduction parameters are `\delta = 3/4` and
`\eta = 0.501`.
The parameters `\delta` and `\eta` must satisfy:
`0.25 < \delta \leq 1.0` and `0.5 \leq \eta < \sqrt{\delta}`.
Polynomial time complexity is only guaranteed for `\delta < 1`.
INPUT:
- ``*args`` -- passed through to
:meth:`sage.matrix.matrix_integer_dense.Matrix_integer_dense.LLL`
- ``**kwds`` -- passed through to
:meth:`sage.matrix.matrix_integer_dense.Matrix_integer_dense.LLL`
OUTPUT:
An integer matrix which is an LLL-reduced basis for this lattice.
EXAMPLES::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: A = random_matrix(ZZ, 10, 10, x=-2000, y=2000)
sage: L = IntegerLattice(A, lll_reduce=False); L
Free module of degree 10 and rank 10 over Integer Ring
User basis matrix:
[ -645 -1037 -1775 -1619 1721 -1434 1766 1701 1669 1534]
[ 1303 960 1998 -1838 1683 -1332 149 327 -849 -1562]
[-1113 -1366 1379 669 54 1214 -1750 -605 -1566 1626]
[-1367 1651 926 1731 -913 627 669 -1437 -132 1712]
[ -549 1327 -1353 68 1479 -1803 -456 1090 -606 -317]
[ -221 -1920 -1361 1695 1139 111 -1792 1925 -656 1992]
[-1934 -29 88 890 1859 1820 -1912 -1614 -1724 1606]
[ -590 -1380 1768 774 656 760 -746 -849 1977 -1576]
[ 312 -242 -1732 1594 -439 -1069 458 -1195 1715 35]
[ 391 1229 -1815 607 -413 -860 1408 1656 1651 -628]
sage: min(v.norm().n() for v in L.reduced_basis)
3346.57...
sage: L.LLL()
[ -888 53 -274 243 -19 431 710 -83 928 347]
[ 448 -330 370 -511 242 -584 -8 1220 502 183]
[ -524 -460 402 1338 -247 -279 -1038 -28 -159 -794]
[ 166 -190 -162 1033 -340 -77 -1052 1134 -843 651]
[ -47 -1394 1076 -132 854 -151 297 -396 -580 -220]
[-1064 373 -706 601 -587 -1394 424 796 -22 -133]
[-1126 398 565 -1418 -446 -890 -237 -378 252 247]
[ -339 799 295 800 425 -605 -730 -1160 808 666]
[ 755 -1206 -918 -192 -1063 -37 -525 -75 338 400]
[ 382 -199 -1839 -482 984 -15 -695 136 682 563]
sage: L.reduced_basis[0].norm().n()
1613.74...
"""
basis = self.reduced_basis
basis = [v for v in basis.LLL(*args, **kwds) if v]
basis = matrix(ZZ, len(basis), len(basis[0]), basis)
basis.set_immutable()
if self.reduced_basis[0].norm() > basis[0].norm():
self._reduced_basis = basis
return basis
def BKZ(self, *args, **kwds):
"""
Return a Block Korkine-Zolotareff reduced basis for ``self``.
INPUT:
- ``*args`` -- passed through to
:meth:`sage.matrix.matrix_integer_dense.Matrix_integer_dense.BKZ`
- ``*kwds`` -- passed through to
:meth:`sage.matrix.matrix_integer_dense.Matrix_integer_dense.BKZ`
OUTPUT:
An integer matrix which is a BKZ-reduced basis for this lattice.
EXAMPLES::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: A = sage.crypto.gen_lattice(type='random', n=1, m=60, q=2^60, seed=42)
sage: L = IntegerLattice(A, lll_reduce=False)
sage: min(v.norm().n() for v in L.reduced_basis)
4.17330740711759e15
sage: L.LLL()
60 x 60 dense matrix over Integer Ring (use the '.str()' method to see the entries)
sage: min(v.norm().n() for v in L.reduced_basis)
5.19615242270663
sage: L.BKZ(block_size=10)
60 x 60 dense matrix over Integer Ring (use the '.str()' method to see the entries)
sage: min(v.norm().n() for v in L.reduced_basis)
4.12310562561766
.. NOTE::
If ``block_size == L.rank()`` where ``L`` is this lattice, then
this function performs Hermite-Korkine-Zolotareff (HKZ) reduction.
"""
basis = self.reduced_basis
basis = [v for v in basis.BKZ(*args, **kwds) if v]
basis = matrix(ZZ, len(basis), len(basis[0]), basis)
basis.set_immutable()
if self.reduced_basis[0].norm() > basis[0].norm():
self._reduced_basis = basis
return basis
def HKZ(self, *args, **kwds):
r"""
Hermite-Korkine-Zolotarev (HKZ) reduce the basis.
A basis `B` of a lattice `L`, with orthogonalized basis `B^*` such
that `B = M \cdot B^*` is HKZ reduced, if and only if, the following
properties are satisfied:
#. The basis `B` is size-reduced, i.e., all off-diagonal
coefficients of `M` satisfy `|\mu_{i,j}| \leq 1/2`
#. The vector `b_1` realizes the first minimum `\lambda_1(L)`.
#. The projection of the vectors `b_2, \ldots,b_r` orthogonally to
`b_1` form an HKZ reduced basis.
.. NOTE::
This is realized by calling
:func:`sage.modules.free_module_integer.FreeModule_submodule_with_basis_integer.BKZ` with
``block_size == self.rank()``.
INPUT:
- ``*args`` -- passed through to :meth:`BKZ`
- ``*kwds`` -- passed through to :meth:`BKZ`
OUTPUT:
An integer matrix which is a HKZ-reduced basis for this lattice.
EXAMPLE::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: L = sage.crypto.gen_lattice(type='random', n=1, m=40, q=2^60, seed=1337, lattice=True)
sage: L.HKZ()
40 x 40 dense matrix over Integer Ring (use the '.str()' method to see the entries)
sage: L.reduced_basis[0]
(0, 0, -1, -1, 0, 0, -1, 1, 0, 0, -1, 1, 1, 0, 0, 1, 1, 1, -1, 0, 0, 1, -1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 0, 1, 1, 0, 0, -2)
"""
return self.BKZ(block_size=self.rank())
@cached_method
def volume(self):
r"""
Return `vol(L)` which is `\sqrt{\det(B \cdot B^T)}` for any basis `B`.
OUTPUT:
An integer.
EXAMPLE::
sage: L = sage.crypto.gen_lattice(m=10, seed=1337, lattice=True)
sage: L.volume()
14641
"""
if self.rank() == self.degree():
return abs(self.reduced_basis.determinant())
else:
return self.gram_matrix().determinant().sqrt()
@cached_method
def discriminant(self):
r"""
Return `|\det(G)|`, i.e. the absolute value of the determinant of the
Gram matrix `B \cdot B^T` for any basis `B`.
OUTPUT:
An integer.
EXAMPLE::
sage: L = sage.crypto.gen_lattice(m=10, seed=1337, lattice=True)
sage: L.discriminant()
214358881
"""
return abs(self.gram_matrix().determinant())
@cached_method
def is_unimodular(self):
"""
Return ``True`` if this lattice is unimodular.
OUTPUT:
A boolean.
EXAMPLES::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: L = IntegerLattice([[1, 0], [0, 1]])
sage: L.is_unimodular()
True
sage: IntegerLattice([[2, 0], [0, 3]]).is_unimodular()
False
"""
return self.volume() == 1
@cached_method
def shortest_vector(self, update_reduced_basis=True, algorithm="fplll", *args, **kwds):
r"""
Return a shortest vector.
INPUT:
- ``update_reduced_basis`` -- (default: ``True``) set this flag if
the found vector should be used to improve the basis
- ``algorithm`` -- (default: ``"fplll"``) either ``"fplll"`` or
``"pari"``
- ``*args`` -- passed through to underlying implementation
- ``**kwds`` -- passed through to underlying implementation
OUTPUT:
A shortest non-zero vector for this lattice.
EXAMPLES::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: A = sage.crypto.gen_lattice(type='random', n=1, m=30, q=2^40, seed=42)
sage: L = IntegerLattice(A, lll_reduce=False)
sage: min(v.norm().n() for v in L.reduced_basis)
6.03890756700000e10
sage: L.shortest_vector().norm().n()
3.74165738677394
sage: L = IntegerLattice(A, lll_reduce=False)
sage: min(v.norm().n() for v in L.reduced_basis)
6.03890756700000e10
sage: L.shortest_vector(algorithm="pari").norm().n()
3.74165738677394
sage: L = IntegerLattice(A, lll_reduce=True)
sage: L.shortest_vector(algorithm="pari").norm().n()
3.74165738677394
"""
if algorithm == "pari":
if self._basis_is_LLL_reduced:
B = self.basis_matrix().change_ring(ZZ)
qf = self.gram_matrix()
else:
B = self.reduced_basis.LLL()
qf = B*B.transpose()
count, length, vectors = qf._pari_().qfminim()
v = vectors.sage().columns()[0]
w = v*B
elif algorithm == "fplll":
from fpylll import IntegerMatrix, SVP
L = IntegerMatrix.from_matrix(self.reduced_basis)
w = vector(ZZ, SVP.shortest_vector(L, *args, **kwds))
else:
raise ValueError("algorithm '{}' unknown".format(algorithm))
if update_reduced_basis:
self.update_reduced_basis(w)
return w
def update_reduced_basis(self, w):
"""
Inject the vector ``w`` and run LLL to update the basis.
INPUT:
- ``w`` -- a vector
OUTPUT:
Nothing is returned but the internal state is modified.
EXAMPLE::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: A = sage.crypto.gen_lattice(type='random', n=1, m=30, q=2^40, seed=42)
sage: L = IntegerLattice(A)
sage: B = L.reduced_basis
sage: v = L.shortest_vector(update_reduced_basis=False)
sage: L.update_reduced_basis(v)
sage: bool(L.reduced_basis[0].norm() < B[0].norm())
True
"""
w = matrix(ZZ, w)
L = w.stack(self.reduced_basis).LLL()
assert(L[0] == 0)
self._reduced_basis = L.matrix_from_rows(range(1, L.nrows()))
@cached_method
def voronoi_cell(self, radius=None):
"""
Compute the Voronoi cell of a lattice, returning a Polyhedron.
INPUT:
- ``radius`` -- (default: automatic determination) radius of ball
containing considered vertices
OUTPUT:
The Voronoi cell as a Polyhedron instance.
The result is cached so that subsequent calls to this function
return instantly.
EXAMPLES::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: L = IntegerLattice([[1, 0], [0, 1]])
sage: V = L.voronoi_cell()
sage: V.Vrepresentation()
(A vertex at (1/2, -1/2), A vertex at (1/2, 1/2), A vertex at (-1/2, 1/2), A vertex at (-1/2, -1/2))
The volume of the Voronoi cell is the square root of the
discriminant of the lattice::
sage: L = IntegerLattice(Matrix(ZZ, 4, 4, [[0,0,1,-1],[1,-1,2,1],[-6,0,3,3,],[-6,-24,-6,-5]])); L
Free module of degree 4 and rank 4 over Integer Ring
User basis matrix:
[ 0 0 1 -1]
[ 1 -1 2 1]
[ -6 0 3 3]
[ -6 -24 -6 -5]
sage: V = L.voronoi_cell() # long time
sage: V.volume() # long time
678
sage: sqrt(L.discriminant())
678
Lattices not having full dimension are handled as well::
sage: L = IntegerLattice([[2, 0, 0], [0, 2, 0]])
sage: V = L.voronoi_cell()
sage: V.Hrepresentation()
(An inequality (-1, 0, 0) x + 1 >= 0, An inequality (0, -1, 0) x + 1 >= 0, An inequality (1, 0, 0) x + 1 >= 0, An inequality (0, 1, 0) x + 1 >= 0)
ALGORITHM:
Uses parts of the algorithm from [VB1996]_.
"""
if not self._basis_is_LLL_reduced:
self.LLL()
B = self.reduced_basis
from .diamond_cutting import calculate_voronoi_cell
return calculate_voronoi_cell(B, radius=radius)
def voronoi_relevant_vectors(self):
"""
Compute the embedded vectors inducing the Voronoi cell.
OUTPUT:
The list of Voronoi relevant vectors.
EXAMPLES::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: L = IntegerLattice([[3, 0], [4, 0]])
sage: L.voronoi_relevant_vectors()
[(-1, 0), (1, 0)]
"""
V = self.voronoi_cell()
def defining_point(ieq):
"""
Compute the point defining an inequality.
INPUT:
- ``ieq`` -- an inequality in the form [c, a1, a2, ...]
meaning a1 * x1 + a2 * x2 + ... ≦ c
OUTPUT:
The point orthogonal to the hyperplane defined by ``ieq``
in twice the distance from the origin.
"""
c = ieq[0]
a = ieq[1:]
n = sum(y ** 2 for y in a)
return vector([2 * y * c / n for y in a])
return [defining_point(ieq) for ieq in V.inequality_generator()]
def closest_vector(self, t):
"""
Compute the closest vector in the embedded lattice to a given vector.
INPUT:
- ``t`` -- the target vector to compute the closest vector to
OUTPUT:
The vector in the lattice closest to ``t``.
EXAMPLES::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: L = IntegerLattice([[1, 0], [0, 1]])
sage: L.closest_vector((-6, 5/3))
(-6, 2)
ALGORITHM:
Uses the algorithm from [MV2010]_.
"""
voronoi_cell = self.voronoi_cell()
def projection(M, v):
Mt = M.transpose()
P = Mt * (M * Mt) ** (-1) * M
return P * v
t = projection(matrix(self.reduced_basis), vector(t))
def CVPP_2V(t, V, voronoi_cell):
t_new = t
while not voronoi_cell.contains(t_new.list()):
v = max(V, key=lambda v: t_new * v / v.norm() ** 2)
t_new = t_new - v
return t - t_new
V = self.voronoi_relevant_vectors()
t = vector(t)
p = 0
while not (ZZ(2 ** p) * voronoi_cell).contains(t):
p += 1
t_new = t
i = p
while i >= 1:
V_scaled = [v * (2 ** (i - 1)) for v in V]
t_new = t_new - CVPP_2V(t_new, V_scaled, ZZ(2 ** (i - 1)) * voronoi_cell)
i -= 1
return t - t_new