def gen_lattice(type='modular', n=4, m=8, q=11, seed=None, quotient=None, dual=False, ntl=False, lattice=False): """ This function generates different types of integral lattice bases of row vectors relevant in cryptography. Randomness can be set either with ``seed``, or by using :func:`sage.misc.randstate.set_random_seed`. INPUT: - ``type`` -- one of the following strings - ``'modular'`` (default) -- A class of lattices for which asymptotic worst-case to average-case connections hold. For more refer to [A96]_. - ``'random'`` -- Special case of modular (n=1). A dense class of lattice used for testing basis reduction algorithms proposed by Goldstein and Mayer [GM02]_. - ``'ideal'`` -- Special case of modular. Allows for a more compact representation proposed by [LM06]_. - ``'cyclotomic'`` -- Special case of ideal. Allows for efficient processing proposed by [LM06]_. - ``n`` -- Determinant size, primal:`det(L) = q^n`, dual:`det(L) = q^{m-n}`. For ideal lattices this is also the degree of the quotient polynomial. - ``m`` -- Lattice dimension, `L \subseteq Z^m`. - ``q`` -- Coefficient size, `q-Z^m \subseteq L`. - ``seed`` -- Randomness seed. - ``quotient`` -- For the type ideal, this determines the quotient polynomial. Ignored for all other types. - ``dual`` -- Set this flag if you want a basis for `q-dual(L)`, for example for Regev's LWE bases [R05]_. - ``ntl`` -- Set this flag if you want the lattice basis in NTL readable format. - ``lattice`` -- Set this flag if you want a :class:`FreeModule_submodule_with_basis_integer` object instead of an integer matrix representing the basis. OUTPUT: ``B`` a unique size-reduced triangular (primal: lower_left, dual: lower_right) basis of row vectors for the lattice in question. EXAMPLES: Modular basis:: sage: sage.crypto.gen_lattice(m=10, seed=42) [11 0 0 0 0 0 0 0 0 0] [ 0 11 0 0 0 0 0 0 0 0] [ 0 0 11 0 0 0 0 0 0 0] [ 0 0 0 11 0 0 0 0 0 0] [ 2 4 3 5 1 0 0 0 0 0] [ 1 -5 -4 2 0 1 0 0 0 0] [-4 3 -1 1 0 0 1 0 0 0] [-2 -3 -4 -1 0 0 0 1 0 0] [-5 -5 3 3 0 0 0 0 1 0] [-4 -3 2 -5 0 0 0 0 0 1] Random basis:: sage: sage.crypto.gen_lattice(type='random', n=1, m=10, q=11^4, seed=42) [14641 0 0 0 0 0 0 0 0 0] [ 431 1 0 0 0 0 0 0 0 0] [-4792 0 1 0 0 0 0 0 0 0] [ 1015 0 0 1 0 0 0 0 0 0] [-3086 0 0 0 1 0 0 0 0 0] [-5378 0 0 0 0 1 0 0 0 0] [ 4769 0 0 0 0 0 1 0 0 0] [-1159 0 0 0 0 0 0 1 0 0] [ 3082 0 0 0 0 0 0 0 1 0] [-4580 0 0 0 0 0 0 0 0 1] Ideal bases with quotient x^n-1, m=2*n are NTRU bases:: sage: sage.crypto.gen_lattice(type='ideal', seed=42, quotient=x^4-1) [11 0 0 0 0 0 0 0] [ 0 11 0 0 0 0 0 0] [ 0 0 11 0 0 0 0 0] [ 0 0 0 11 0 0 0 0] [ 4 -2 -3 -3 1 0 0 0] [-3 4 -2 -3 0 1 0 0] [-3 -3 4 -2 0 0 1 0] [-2 -3 -3 4 0 0 0 1] Ideal bases also work with polynomials:: sage: R.<t> = PolynomialRing(ZZ) sage: sage.crypto.gen_lattice(type='ideal', seed=1234, quotient=t^4-1) [11 0 0 0 0 0 0 0] [ 0 11 0 0 0 0 0 0] [ 0 0 11 0 0 0 0 0] [ 0 0 0 11 0 0 0 0] [ 4 1 4 -3 1 0 0 0] [-3 4 1 4 0 1 0 0] [ 4 -3 4 1 0 0 1 0] [ 1 4 -3 4 0 0 0 1] Cyclotomic bases with n=2^k are SWIFFT bases:: sage: sage.crypto.gen_lattice(type='cyclotomic', seed=42) [11 0 0 0 0 0 0 0] [ 0 11 0 0 0 0 0 0] [ 0 0 11 0 0 0 0 0] [ 0 0 0 11 0 0 0 0] [ 4 -2 -3 -3 1 0 0 0] [ 3 4 -2 -3 0 1 0 0] [ 3 3 4 -2 0 0 1 0] [ 2 3 3 4 0 0 0 1] Dual modular bases are related to Regev's famous public-key encryption [R05]_:: sage: sage.crypto.gen_lattice(type='modular', m=10, seed=42, dual=True) [ 0 0 0 0 0 0 0 0 0 11] [ 0 0 0 0 0 0 0 0 11 0] [ 0 0 0 0 0 0 0 11 0 0] [ 0 0 0 0 0 0 11 0 0 0] [ 0 0 0 0 0 11 0 0 0 0] [ 0 0 0 0 11 0 0 0 0 0] [ 0 0 0 1 -5 -2 -1 1 -3 5] [ 0 0 1 0 -3 4 1 4 -3 -2] [ 0 1 0 0 -4 5 -3 3 5 3] [ 1 0 0 0 -2 -1 4 2 5 4] Relation of primal and dual bases:: sage: B_primal=sage.crypto.gen_lattice(m=10, q=11, seed=42) sage: B_dual=sage.crypto.gen_lattice(m=10, q=11, seed=42, dual=True) sage: B_dual_alt=transpose(11*B_primal.inverse()).change_ring(ZZ) sage: B_dual_alt.hermite_form() == B_dual.hermite_form() True TESTS: Test some bad quotient polynomials:: sage: sage.crypto.gen_lattice(type='ideal', seed=1234, quotient=cos(x)) Traceback (most recent call last): ... TypeError: unable to convert cos(x) to an integer sage: sage.crypto.gen_lattice(type='ideal', seed=1234, quotient=x^23-1) Traceback (most recent call last): ... ValueError: ideal basis requires n = quotient.degree() sage: R.<u,v> = ZZ[] sage: sage.crypto.gen_lattice(type='ideal', seed=1234, quotient=u+v) Traceback (most recent call last): ... TypeError: quotient should be a univariate polynomial We are testing output format choices:: sage: sage.crypto.gen_lattice(m=10, q=11, seed=42) [11 0 0 0 0 0 0 0 0 0] [ 0 11 0 0 0 0 0 0 0 0] [ 0 0 11 0 0 0 0 0 0 0] [ 0 0 0 11 0 0 0 0 0 0] [ 2 4 3 5 1 0 0 0 0 0] [ 1 -5 -4 2 0 1 0 0 0 0] [-4 3 -1 1 0 0 1 0 0 0] [-2 -3 -4 -1 0 0 0 1 0 0] [-5 -5 3 3 0 0 0 0 1 0] [-4 -3 2 -5 0 0 0 0 0 1] sage: sage.crypto.gen_lattice(m=10, q=11, seed=42, ntl=True) [ [11 0 0 0 0 0 0 0 0 0] [0 11 0 0 0 0 0 0 0 0] [0 0 11 0 0 0 0 0 0 0] [0 0 0 11 0 0 0 0 0 0] [2 4 3 5 1 0 0 0 0 0] [1 -5 -4 2 0 1 0 0 0 0] [-4 3 -1 1 0 0 1 0 0 0] [-2 -3 -4 -1 0 0 0 1 0 0] [-5 -5 3 3 0 0 0 0 1 0] [-4 -3 2 -5 0 0 0 0 0 1] ] sage: sage.crypto.gen_lattice(m=10, q=11, seed=42, lattice=True) Free module of degree 10 and rank 10 over Integer Ring User basis matrix: [ 0 0 1 1 0 -1 -1 -1 1 0] [-1 1 0 1 0 1 1 0 1 1] [-1 0 0 0 -1 1 1 -2 0 0] [-1 -1 0 1 1 0 0 1 1 -1] [ 1 0 -1 0 0 0 -2 -2 0 0] [ 2 -1 0 0 1 0 1 0 0 -1] [-1 1 -1 0 1 -1 1 0 -1 -2] [ 0 0 -1 3 0 0 0 -1 -1 -1] [ 0 -1 0 -1 2 0 -1 0 0 2] [ 0 1 1 0 1 1 -2 1 -1 -2] REFERENCES: .. [A96] Miklos Ajtai. Generating hard instances of lattice problems (extended abstract). STOC, pp. 99--108, ACM, 1996. .. [GM02] Daniel Goldstein and Andrew Mayer. On the equidistribution of Hecke points. Forum Mathematicum, 15:2, pp. 165--189, De Gruyter, 2003. .. [LM06] Vadim Lyubashevsky and Daniele Micciancio. Generalized compact knapsacks are collision resistant. ICALP, pp. 144--155, Springer, 2006. .. [R05] Oded Regev. On lattices, learning with errors, random linear codes, and cryptography. STOC, pp. 84--93, ACM, 2005. """ from sage.rings.finite_rings.integer_mod_ring import IntegerModRing from sage.matrix.constructor import identity_matrix, block_matrix from sage.matrix.matrix_space import MatrixSpace from sage.rings.integer_ring import IntegerRing if seed is not None: from sage.misc.randstate import set_random_seed set_random_seed(seed) if type == 'random': if n != 1: raise ValueError('random bases require n = 1') ZZ = IntegerRing() ZZ_q = IntegerModRing(q) A = identity_matrix(ZZ_q, n) if type == 'random' or type == 'modular': R = MatrixSpace(ZZ_q, m-n, n) A = A.stack(R.random_element()) elif type == 'ideal': if quotient is None: raise ValueError('ideal bases require a quotient polynomial') try: quotient = quotient.change_ring(ZZ_q) except (AttributeError, TypeError): quotient = quotient.polynomial(base_ring=ZZ_q) P = quotient.parent() # P should be a univariate polynomial ring over ZZ_q if not is_PolynomialRing(P): raise TypeError("quotient should be a univariate polynomial") assert P.base_ring() is ZZ_q if quotient.degree() != n: raise ValueError('ideal basis requires n = quotient.degree()') R = P.quotient(quotient) for i in range(m//n): A = A.stack(R.random_element().matrix()) elif type == 'cyclotomic': from sage.arith.all import euler_phi from sage.misc.functional import cyclotomic_polynomial # we assume that n+1 <= min( euler_phi^{-1}(n) ) <= 2*n found = False for k in range(2*n,n,-1): if euler_phi(k) == n: found = True break if not found: raise ValueError("cyclotomic bases require that n " "is an image of Euler's totient function") R = ZZ_q['x'].quotient(cyclotomic_polynomial(k, 'x'), 'x') for i in range(m//n): A = A.stack(R.random_element().matrix()) # switch from representatives 0,...,(q-1) to (1-q)/2,....,(q-1)/2 def minrep(a): if abs(a-q) < abs(a): return a-q else: return a A_prime = A[n:m].lift().apply_map(minrep) if not dual: B = block_matrix([[ZZ(q), ZZ.zero()], [A_prime, ZZ.one()] ], subdivide=False) else: B = block_matrix([[ZZ.one(), -A_prime.transpose()], [ZZ.zero(), ZZ(q)]], subdivide=False) for i in range(m//2): B.swap_rows(i,m-i-1) if ntl and lattice: raise ValueError("Cannot specify ntl=True and lattice=True " "at the same time") if ntl: return B._ntl_() elif lattice: from sage.modules.free_module_integer import IntegerLattice return IntegerLattice(B) else: return B
def gen_lattice(type='modular', n=4, m=8, q=11, seed=None, quotient=None, dual=False, ntl=False, lattice=False): r""" This function generates different types of integral lattice bases of row vectors relevant in cryptography. Randomness can be set either with ``seed``, or by using :func:`sage.misc.randstate.set_random_seed`. INPUT: - ``type`` -- one of the following strings - ``'modular'`` (default) -- A class of lattices for which asymptotic worst-case to average-case connections hold. For more refer to [Aj1996]_. - ``'random'`` -- Special case of modular (n=1). A dense class of lattice used for testing basis reduction algorithms proposed by Goldstein and Mayer [GM2002]_. - ``'ideal'`` -- Special case of modular. Allows for a more compact representation proposed by [LM2006]_. - ``'cyclotomic'`` -- Special case of ideal. Allows for efficient processing proposed by [LM2006]_. - ``n`` -- Determinant size, primal:`det(L) = q^n`, dual:`det(L) = q^{m-n}`. For ideal lattices this is also the degree of the quotient polynomial. - ``m`` -- Lattice dimension, `L \subseteq Z^m`. - ``q`` -- Coefficient size, `q-Z^m \subseteq L`. - ``seed`` -- Randomness seed. - ``quotient`` -- For the type ideal, this determines the quotient polynomial. Ignored for all other types. - ``dual`` -- Set this flag if you want a basis for `q-dual(L)`, for example for Regev's LWE bases [Reg2005]_. - ``ntl`` -- Set this flag if you want the lattice basis in NTL readable format. - ``lattice`` -- Set this flag if you want a :class:`FreeModule_submodule_with_basis_integer` object instead of an integer matrix representing the basis. OUTPUT: ``B`` a unique size-reduced triangular (primal: lower_left, dual: lower_right) basis of row vectors for the lattice in question. EXAMPLES: Modular basis:: sage: sage.crypto.gen_lattice(m=10, seed=42) [11 0 0 0 0 0 0 0 0 0] [ 0 11 0 0 0 0 0 0 0 0] [ 0 0 11 0 0 0 0 0 0 0] [ 0 0 0 11 0 0 0 0 0 0] [ 2 4 3 5 1 0 0 0 0 0] [ 1 -5 -4 2 0 1 0 0 0 0] [-4 3 -1 1 0 0 1 0 0 0] [-2 -3 -4 -1 0 0 0 1 0 0] [-5 -5 3 3 0 0 0 0 1 0] [-4 -3 2 -5 0 0 0 0 0 1] Random basis:: sage: sage.crypto.gen_lattice(type='random', n=1, m=10, q=11^4, seed=42) [14641 0 0 0 0 0 0 0 0 0] [ 431 1 0 0 0 0 0 0 0 0] [-4792 0 1 0 0 0 0 0 0 0] [ 1015 0 0 1 0 0 0 0 0 0] [-3086 0 0 0 1 0 0 0 0 0] [-5378 0 0 0 0 1 0 0 0 0] [ 4769 0 0 0 0 0 1 0 0 0] [-1159 0 0 0 0 0 0 1 0 0] [ 3082 0 0 0 0 0 0 0 1 0] [-4580 0 0 0 0 0 0 0 0 1] Ideal bases with quotient x^n-1, m=2*n are NTRU bases:: sage: sage.crypto.gen_lattice(type='ideal', seed=42, quotient=x^4-1) [11 0 0 0 0 0 0 0] [ 0 11 0 0 0 0 0 0] [ 0 0 11 0 0 0 0 0] [ 0 0 0 11 0 0 0 0] [-2 -3 -3 4 1 0 0 0] [ 4 -2 -3 -3 0 1 0 0] [-3 4 -2 -3 0 0 1 0] [-3 -3 4 -2 0 0 0 1] Ideal bases also work with polynomials:: sage: R.<t> = PolynomialRing(ZZ) sage: sage.crypto.gen_lattice(type='ideal', seed=1234, quotient=t^4-1) [11 0 0 0 0 0 0 0] [ 0 11 0 0 0 0 0 0] [ 0 0 11 0 0 0 0 0] [ 0 0 0 11 0 0 0 0] [ 1 4 -3 3 1 0 0 0] [ 3 1 4 -3 0 1 0 0] [-3 3 1 4 0 0 1 0] [ 4 -3 3 1 0 0 0 1] Cyclotomic bases with n=2^k are SWIFFT bases:: sage: sage.crypto.gen_lattice(type='cyclotomic', seed=42) [11 0 0 0 0 0 0 0] [ 0 11 0 0 0 0 0 0] [ 0 0 11 0 0 0 0 0] [ 0 0 0 11 0 0 0 0] [-2 -3 -3 4 1 0 0 0] [-4 -2 -3 -3 0 1 0 0] [ 3 -4 -2 -3 0 0 1 0] [ 3 3 -4 -2 0 0 0 1] Dual modular bases are related to Regev's famous public-key encryption [Reg2005]_:: sage: sage.crypto.gen_lattice(type='modular', m=10, seed=42, dual=True) [ 0 0 0 0 0 0 0 0 0 11] [ 0 0 0 0 0 0 0 0 11 0] [ 0 0 0 0 0 0 0 11 0 0] [ 0 0 0 0 0 0 11 0 0 0] [ 0 0 0 0 0 11 0 0 0 0] [ 0 0 0 0 11 0 0 0 0 0] [ 0 0 0 1 -5 -2 -1 1 -3 5] [ 0 0 1 0 -3 4 1 4 -3 -2] [ 0 1 0 0 -4 5 -3 3 5 3] [ 1 0 0 0 -2 -1 4 2 5 4] Relation of primal and dual bases:: sage: B_primal=sage.crypto.gen_lattice(m=10, q=11, seed=42) sage: B_dual=sage.crypto.gen_lattice(m=10, q=11, seed=42, dual=True) sage: B_dual_alt=transpose(11*B_primal.inverse()).change_ring(ZZ) sage: B_dual_alt.hermite_form() == B_dual.hermite_form() True TESTS: Test some bad quotient polynomials:: sage: sage.crypto.gen_lattice(type='ideal', seed=1234, quotient=cos(x)) Traceback (most recent call last): ... TypeError: unable to convert cos(x) to an integer sage: sage.crypto.gen_lattice(type='ideal', seed=1234, quotient=x^23-1) Traceback (most recent call last): ... ValueError: ideal basis requires n = quotient.degree() sage: R.<u,v> = ZZ[] sage: sage.crypto.gen_lattice(type='ideal', seed=1234, quotient=u+v) Traceback (most recent call last): ... TypeError: quotient should be a univariate polynomial We are testing output format choices:: sage: sage.crypto.gen_lattice(m=10, q=11, seed=42) [11 0 0 0 0 0 0 0 0 0] [ 0 11 0 0 0 0 0 0 0 0] [ 0 0 11 0 0 0 0 0 0 0] [ 0 0 0 11 0 0 0 0 0 0] [ 2 4 3 5 1 0 0 0 0 0] [ 1 -5 -4 2 0 1 0 0 0 0] [-4 3 -1 1 0 0 1 0 0 0] [-2 -3 -4 -1 0 0 0 1 0 0] [-5 -5 3 3 0 0 0 0 1 0] [-4 -3 2 -5 0 0 0 0 0 1] sage: sage.crypto.gen_lattice(m=10, q=11, seed=42, ntl=True) [ [11 0 0 0 0 0 0 0 0 0] [0 11 0 0 0 0 0 0 0 0] [0 0 11 0 0 0 0 0 0 0] [0 0 0 11 0 0 0 0 0 0] [2 4 3 5 1 0 0 0 0 0] [1 -5 -4 2 0 1 0 0 0 0] [-4 3 -1 1 0 0 1 0 0 0] [-2 -3 -4 -1 0 0 0 1 0 0] [-5 -5 3 3 0 0 0 0 1 0] [-4 -3 2 -5 0 0 0 0 0 1] ] sage: sage.crypto.gen_lattice(m=10, q=11, seed=42, lattice=True) Free module of degree 10 and rank 10 over Integer Ring User basis matrix: [ 0 0 1 1 0 -1 -1 -1 1 0] [-1 1 0 1 0 1 1 0 1 1] [-1 0 0 0 -1 1 1 -2 0 0] [-1 -1 0 1 1 0 0 1 1 -1] [ 1 0 -1 0 0 0 -2 -2 0 0] [ 2 -1 0 0 1 0 1 0 0 -1] [-1 1 -1 0 1 -1 1 0 -1 -2] [ 0 0 -1 3 0 0 0 -1 -1 -1] [ 0 -1 0 -1 2 0 -1 0 0 2] [ 0 1 1 0 1 1 -2 1 -1 -2] """ from sage.rings.finite_rings.integer_mod_ring import IntegerModRing from sage.matrix.constructor import identity_matrix, block_matrix from sage.matrix.matrix_space import MatrixSpace from sage.rings.integer_ring import IntegerRing if seed is not None: from sage.misc.randstate import set_random_seed set_random_seed(seed) if type == 'random': if n != 1: raise ValueError('random bases require n = 1') ZZ = IntegerRing() ZZ_q = IntegerModRing(q) A = identity_matrix(ZZ_q, n) if type == 'random' or type == 'modular': R = MatrixSpace(ZZ_q, m-n, n) A = A.stack(R.random_element()) elif type == 'ideal': if quotient is None: raise ValueError('ideal bases require a quotient polynomial') try: quotient = quotient.change_ring(ZZ_q) except (AttributeError, TypeError): quotient = quotient.polynomial(base_ring=ZZ_q) P = quotient.parent() # P should be a univariate polynomial ring over ZZ_q if not is_PolynomialRing(P): raise TypeError("quotient should be a univariate polynomial") assert P.base_ring() is ZZ_q if quotient.degree() != n: raise ValueError('ideal basis requires n = quotient.degree()') R = P.quotient(quotient) for i in range(m//n): A = A.stack(R.random_element().matrix()) elif type == 'cyclotomic': from sage.arith.all import euler_phi from sage.misc.functional import cyclotomic_polynomial # we assume that n+1 <= min( euler_phi^{-1}(n) ) <= 2*n found = False for k in range(2*n,n,-1): if euler_phi(k) == n: found = True break if not found: raise ValueError("cyclotomic bases require that n " "is an image of Euler's totient function") R = ZZ_q['x'].quotient(cyclotomic_polynomial(k, 'x'), 'x') for i in range(m//n): A = A.stack(R.random_element().matrix()) # switch from representatives 0,...,(q-1) to (1-q)/2,....,(q-1)/2 def minrep(a): if abs(a-q) < abs(a): return a-q else: return a A_prime = A[n:m].lift().apply_map(minrep) if not dual: B = block_matrix([[ZZ(q), ZZ.zero()], [A_prime, ZZ.one()] ], subdivide=False) else: B = block_matrix([[ZZ.one(), -A_prime.transpose()], [ZZ.zero(), ZZ(q)]], subdivide=False) for i in range(m//2): B.swap_rows(i,m-i-1) if ntl and lattice: raise ValueError("Cannot specify ntl=True and lattice=True " "at the same time") if ntl: return B._ntl_() elif lattice: from sage.modules.free_module_integer import IntegerLattice return IntegerLattice(B) else: return B
g=x**(n/2)+1 T=ZZ_q['x'].quotient(x**(n/2)+1) a_pol=R.random_element() s_pol=sample_noise(R) e_pol=sample_noise(R) s_pol2=T((s_pol)) e_pol2=T((e_pol)) print("s={0},e={1}".format(T(s_pol),T(e_pol))) Z_mat=e_pol2.matrix().augment(s_pol2.matrix()) Z_mattop=Z_mat[0:1].augment(matrix(1,1,[ZZ.one()*-1])) b_pol=(a_pol*s_pol+e_pol) print("s_pol={0}\ne_pol={1}".format((s_pol2).list(),(e_pol2).list())) # Does a linear mapping change the shortest vector size for the rest?/ a_pol=a_pol#*x_pol b_pol=b_pol#*x_pol a_pol2 = T(a_pol.list())# % S(g.list()) b_pol2 = T((b_pol).list())# % S(g.list()) # print("a={0}\nb={1}".format(a_pol2,b_pol2)) A=identity_matrix(ZZ_q,n/2) A=A.stack(a_pol2.matrix())
def gen_lattice(type='modular', n=4, m=8, q=11, seed=None, \ quotient=None, dual=False, ntl=False): """ This function generates different types of integral lattice bases of row vectors relevant in cryptography. Randomness can be set either with ``seed``, or by using :func:`sage.misc.randstate.set_random_seed`. INPUT: * ``type`` - one of the following strings * ``'modular'`` (default). A class of lattices for which asymptotic worst-case to average-case connections hold. For more refer to [A96]_. * ``'random'`` - Special case of modular (n=1). A dense class of lattice used for testing basis reduction algorithms proposed by Goldstein and Mayer [GM02]_. * ``'ideal'`` - Special case of modular. Allows for a more compact representation proposed by [LM06]_. * ``'cyclotomic'`` - Special case of ideal. Allows for efficient processing proposed by [LM06]_. * ``n`` - Determinant size, primal:`det(L) = q^n`, dual:`det(L) = q^{m-n}`. For ideal lattices this is also the degree of the quotient polynomial. * ``m`` - Lattice dimension, `L \subseteq Z^m`. * ``q`` - Coefficent size, `q*Z^m \subseteq L`. * ``seed`` - Randomness seed. * ``quotient`` - For the type ideal, this determines the quotient polynomial. Ignored for all other types. * ``dual`` - Set this flag if you want a basis for `q*dual(L)`, for example for Regev's LWE bases [R05]_. * ``ntl`` - Set this flag if you want the lattice basis in NTL readable format. OUTPUT: ``B`` a unique size-reduced triangular (primal: lower_left, dual: lower_right) basis of row vectors for the lattice in question. EXAMPLES: * Modular basis :: sage: sage.crypto.gen_lattice(m=10, seed=42) [11 0 0 0 0 0 0 0 0 0] [ 0 11 0 0 0 0 0 0 0 0] [ 0 0 11 0 0 0 0 0 0 0] [ 0 0 0 11 0 0 0 0 0 0] [ 2 4 3 5 1 0 0 0 0 0] [ 1 -5 -4 2 0 1 0 0 0 0] [-4 3 -1 1 0 0 1 0 0 0] [-2 -3 -4 -1 0 0 0 1 0 0] [-5 -5 3 3 0 0 0 0 1 0] [-4 -3 2 -5 0 0 0 0 0 1] * Random basis :: sage: sage.crypto.gen_lattice(type='random', n=1, m=10, q=11^4, seed=42) [14641 0 0 0 0 0 0 0 0 0] [ 431 1 0 0 0 0 0 0 0 0] [-4792 0 1 0 0 0 0 0 0 0] [ 1015 0 0 1 0 0 0 0 0 0] [-3086 0 0 0 1 0 0 0 0 0] [-5378 0 0 0 0 1 0 0 0 0] [ 4769 0 0 0 0 0 1 0 0 0] [-1159 0 0 0 0 0 0 1 0 0] [ 3082 0 0 0 0 0 0 0 1 0] [-4580 0 0 0 0 0 0 0 0 1] * Ideal bases with quotient x^n-1, m=2*n are NTRU bases :: sage: sage.crypto.gen_lattice(type='ideal', seed=42, quotient=x^4-1) [11 0 0 0 0 0 0 0] [ 0 11 0 0 0 0 0 0] [ 0 0 11 0 0 0 0 0] [ 0 0 0 11 0 0 0 0] [ 4 -2 -3 -3 1 0 0 0] [-3 4 -2 -3 0 1 0 0] [-3 -3 4 -2 0 0 1 0] [-2 -3 -3 4 0 0 0 1] * Cyclotomic bases with n=2^k are SWIFFT bases :: sage: sage.crypto.gen_lattice(type='cyclotomic', seed=42) [11 0 0 0 0 0 0 0] [ 0 11 0 0 0 0 0 0] [ 0 0 11 0 0 0 0 0] [ 0 0 0 11 0 0 0 0] [ 4 -2 -3 -3 1 0 0 0] [ 3 4 -2 -3 0 1 0 0] [ 3 3 4 -2 0 0 1 0] [ 2 3 3 4 0 0 0 1] * Dual modular bases are related to Regev's famous public-key encryption [R05]_ :: sage: sage.crypto.gen_lattice(type='modular', m=10, seed=42, dual=True) [ 0 0 0 0 0 0 0 0 0 11] [ 0 0 0 0 0 0 0 0 11 0] [ 0 0 0 0 0 0 0 11 0 0] [ 0 0 0 0 0 0 11 0 0 0] [ 0 0 0 0 0 11 0 0 0 0] [ 0 0 0 0 11 0 0 0 0 0] [ 0 0 0 1 -5 -2 -1 1 -3 5] [ 0 0 1 0 -3 4 1 4 -3 -2] [ 0 1 0 0 -4 5 -3 3 5 3] [ 1 0 0 0 -2 -1 4 2 5 4] * Relation of primal and dual bases :: sage: B_primal=sage.crypto.gen_lattice(m=10, q=11, seed=42) sage: B_dual=sage.crypto.gen_lattice(m=10, q=11, seed=42, dual=True) sage: B_dual_alt=transpose(11*B_primal.inverse()).change_ring(ZZ) sage: B_dual_alt.hermite_form() == B_dual.hermite_form() True REFERENCES: .. [A96] Miklos Ajtai. Generating hard instances of lattice problems (extended abstract). STOC, pp. 99--108, ACM, 1996. .. [GM02] Daniel Goldstein and Andrew Mayer. On the equidistribution of Hecke points. Forum Mathematicum, 15:2, pp. 165--189, De Gruyter, 2003. .. [LM06] Vadim Lyubashevsky and Daniele Micciancio. Generalized compact knapsacks are collision resistant. ICALP, pp. 144--155, Springer, 2006. .. [R05] Oded Regev. On lattices, learning with errors, random linear codes, and cryptography. STOC, pp. 84--93, ACM, 2005. """ from sage.rings.finite_rings.integer_mod_ring \ import IntegerModRing from sage.matrix.constructor import matrix, \ identity_matrix, block_matrix from sage.matrix.matrix_space import MatrixSpace from sage.rings.integer_ring import IntegerRing if seed != None: from sage.misc.randstate import set_random_seed set_random_seed(seed) if type == 'random': if n != 1: raise ValueError('random bases require n = 1') ZZ = IntegerRing() ZZ_q = IntegerModRing(q) A = identity_matrix(ZZ_q, n) if type == 'random' or type == 'modular': R = MatrixSpace(ZZ_q, m-n, n) A = A.stack(R.random_element()) elif type == 'ideal': if quotient == None: raise \ ValueError('ideal bases require a quotient polynomial') x = quotient.default_variable() if n != quotient.degree(x): raise \ ValueError('ideal bases require n = quotient.degree()') R = ZZ_q[x].quotient(quotient, x) for i in range(m//n): A = A.stack(R.random_element().matrix()) elif type == 'cyclotomic': from sage.rings.arith import euler_phi from sage.misc.functional import cyclotomic_polynomial # we assume that n+1 <= min( euler_phi^{-1}(n) ) <= 2*n found = False for k in range(2*n,n,-1): if euler_phi(k) == n: found = True break if not found: raise \ ValueError('cyclotomic bases require that n is an image of' + \ 'Euler\'s totient function') R = ZZ_q['x'].quotient(cyclotomic_polynomial(k, 'x'), 'x') for i in range(m//n): A = A.stack(R.random_element().matrix()) # switch from representatives 0,...,(q-1) to (1-q)/2,....,(q-1)/2 def minrep(a): if abs(a-q) < abs(a): return a-q else: return a A_prime = A[n:m].lift().apply_map(minrep) if not dual: B = block_matrix([[ZZ(q), ZZ.zero()], [A_prime, ZZ.one()] ], \ subdivide=False) else: B = block_matrix([[ZZ.one(), -A_prime.transpose()], [ZZ.zero(), \ ZZ(q)]], subdivide=False) for i in range(m//2): B.swap_rows(i,m-i-1) if not ntl: return B else: return B._ntl_()
def my_gen_lattice2(n=4, q=11, seed=None, quotient=None, dual=False, ntl=False, lattice=False, GuessStuff=True): """ This is a modification of the code for the gen_lattice function from Sage Randomness can be set either with ``seed``, or by using :func:`sage.misc.randstate.set_random_seed`. INPUT: - ``type`` -- one of the following strings - ``'cyclotomic'`` -- Special case of ideal. Allows for efficient processing proposed by [LM2006]_. - ``n`` -- Determinant size, primal:`det(L) = q^n`, dual:`det(L) = q^{m-n}`. For ideal lattices this is also the degree of the quotient polynomial. - ``m`` -- Lattice dimension, `L \subseteq Z^m`. - ``q`` -- Coefficient size, `q-Z^m \subseteq L`. - ``t`` -- BKZ Block Size - ``seed`` -- Randomness seed. - ``quotient`` -- For the type ideal, this determines the quotient polynomial. Ignored for all other types. - ``dual`` -- Set this flag if you want a basis for `q-dual(L)`, for example for Regev's LWE bases [Reg2005]_. - ``ntl`` -- Set this flag if you want the lattice basis in NTL readable format. - ``lattice`` -- Set this flag if you want a :class:`FreeModule_submodule_with_basis_integer` object instead of an integer matrix representing the basis. OUTPUT: ``B`` a unique size-reduced triangular (primal: lower_left, dual: lower_right) basis of row vectors for the lattice in question. EXAMPLES: Cyclotomic bases with n=2^k are SWIFFT bases:: sage: sage.crypto.gen_lattice(type='cyclotomic', seed=42) [11 0 0 0 0 0 0 0] [ 0 11 0 0 0 0 0 0] [ 0 0 11 0 0 0 0 0] [ 0 0 0 11 0 0 0 0] [ 4 -2 -3 -3 1 0 0 0] [ 3 4 -2 -3 0 1 0 0] [ 3 3 4 -2 0 0 1 0] [ 2 3 3 4 0 0 0 1] Dual modular bases are related to Regev's famous public-key encryption [Reg2005]_:: sage: sage.crypto.gen_lattice(type='modular', m=10, seed=42, dual=True) [ 0 0 0 0 0 0 0 0 0 11] [ 0 0 0 0 0 0 0 0 11 0] [ 0 0 0 0 0 0 0 11 0 0] [ 0 0 0 0 0 0 11 0 0 0] [ 0 0 0 0 0 11 0 0 0 0] [ 0 0 0 0 11 0 0 0 0 0] [ 0 0 0 1 -5 -2 -1 1 -3 5] [ 0 0 1 0 -3 4 1 4 -3 -2] [ 0 1 0 0 -4 5 -3 3 5 3] [ 1 0 0 0 -2 -1 4 2 5 4] """ from sage.rings.finite_rings.integer_mod_ring import IntegerModRing from sage.matrix.constructor import identity_matrix, block_matrix from sage.matrix.matrix_space import MatrixSpace from sage.rings.integer_ring import IntegerRing from sage.modules.free_module_integer import IntegerLattice if seed is not None: from sage.misc.randstate import set_random_seed set_random_seed(seed) m=n+1 ZZ = IntegerRing() ZZ_q = IntegerModRing(q) from sage.arith.all import euler_phi from sage.misc.functional import cyclotomic_polynomial # we assume that n+1 <= min( euler_phi^{-1}(n) ) <= 2*n found = False for k in range(2*n,n,-1): if euler_phi(k) == n: found = True break if not found: raise ValueError("cyclotomic bases require that n " "is an image of Euler's totient function") R = ZZ_q['x'].quotient(cyclotomic_polynomial(2*n, 'x'), 'x') g=x**(n/2)+1 T=ZZ_q['x'].quotient(x**(n/2)+1) a_pol=R.random_element() s_pol=sample_noise(R) e_pol=sample_noise(R) s_pol2=T((s_pol)) e_pol2=T((e_pol)) print("s={0},e={1}".format(T(s_pol),T(e_pol))) Z_mat=e_pol2.matrix().augment(s_pol2.matrix()) Z_mattop=Z_mat[0:1].augment(matrix(1,1,[ZZ.one()*-1])) b_pol=(a_pol*s_pol+e_pol) print("s_pol={0}\ne_pol={1}".format((s_pol2).list(),(e_pol2).list())) # Does a linear mapping change the shortest vector size for the rest?/ a_pol=a_pol#*x_pol b_pol=b_pol#*x_pol a_pol2 = T(a_pol.list())# % S(g.list()) b_pol2 = T((b_pol).list())# % S(g.list()) # print("a={0}\nb={1}".format(a_pol2,b_pol2)) A=identity_matrix(ZZ_q,n/2) A=A.stack(a_pol2.matrix()) b_prime=b_pol2.matrix()[0:1] b_prime=b_prime - 11*A[8:9] A=A.stack(b_pol2.matrix()[0:1]) # print("X=\n{0}".format(X)) # A = A.stack(identity_matrix(ZZ_q, n/2)) print("A=\n{0}\n".format(A)) # switch from representatives 0,...,(q-1) to (1-q)/2,....,(q-1)/2 def minrepnegative(a): if abs(a-q) < abs(a): return (a-q)*-1 else: return a*-1 def minrep(a): if abs(a-q) < abs(a): return (a-q) else: return a A_prime = A[(n/2):(n+1)].lift().apply_map(minrep) # b_neg= A[(n):(n+1)].lift().apply_map(minrepnegative) Z_fixed=Z_mattop.lift().apply_map(minrep) print("Z_fixed={0}\n||Z_fixed||={1}".format(Z_fixed,float(Z_fixed[0].norm()))) print('Z_fixed*A={0}\n\n'.format(Z_fixed*A)) print("z_fixed[0].norm()={0}".format(float(Z_fixed[0].norm()))) # B=block_matrix([[ZZ(q),ZZ.zero()],[A_neg,ZZ.one()]], subdivide=False) # B = block_matrix([[ZZ.one(), -A_prime.transpose()], # [ZZ.zero(), ZZ(q)]], subdivide=False) B = block_matrix([[ZZ(q), ZZ.zero()], [-A_prime, ZZ.one()]], subdivide=False) # for i in range(m//2): # B.swap_rows(i,m-i-1) # print("{0}\n".format(A_neg)) # B=block_matrix([[ZZ(q), ZZ.zero(),ZZ.zero()],[ZZ.one(),A_neg,ZZ.zero() ],[ZZ.zero(),b_neg,ZZ.one()]], # subdivide=False) #print("B=\n{0}".format(B)) print("B*A=\n{0}\n\n".format(B*A)) #print("A=\n{0}\n".format(A)) def remap(x): return minrep((x*251)%251) BL=B.BKZ(block_size=n/2.) y=(BL.solve_left(Z_fixed))#.apply_map(remap)) # print("y*B={0}".format(y*B)) print("y:=B.solve_left(Z_fixed)={0}".format(y)) # BL=B.BKZ(block_size=n/2.) print(BL[0]) print("shortest norm={0}".format(float(BL[0].norm()))) # L = IntegerLattice(B) # p # v=L.shortest_vector() # print("L.shortest_vector={0}, norm={1}".format(v,float(v.norm()))) if ntl and lattice: raise ValueError("Cannot specify ntl=True and lattice=True ") if ntl: return B._ntl_() elif lattice: from sage.modules.free_module_integer import IntegerLattice return IntegerLattice(B) else: return B