def __init__(self, params, asym=False): (self.n, self.q, sigma, self.sigma_prime, self.k) = params S, x = PolynomialRing(ZZ, 'x').objgen() self.R = S.quotient_ring(S.ideal(x**self.n + 1)) Sq = PolynomialRing(Zmod(self.q), 'x') self.Rq = Sq.quotient_ring(Sq.ideal(x**self.n + 1)) # draw z_is uniformly from Rq and compute its inverse in Rq if asym: z = [self.Rq.random_element() for i in range(self.k)] self.zinv = [z_i**(-1) for z_i in z] else: # or do symmetric version z = self.Rq.random_element() zinv = z**(-1) z, self.zinv = zip(*[(z, zinv) for i in range(self.k)]) # set up some discrete Gaussians DGSL_sigma = DGSL(ZZ**self.n, sigma) self.D_sigma = lambda: self.Rq(list(DGSL_sigma())) # discrete Gaussian in ZZ^n with stddev sigma_prime, yields random level-0 encodings DGSL_sigmap_ZZ = DGSL(ZZ**self.n, self.sigma_prime) self.D_sigmap_ZZ = lambda: self.Rq(list(DGSL_sigmap_ZZ())) # draw g repeatedly from a Gaussian distribution of Z^n (with param sigma) # until g^(-1) in QQ[x]/<x^n + 1> is small (< n^2) Sk = PolynomialRing(QQ, 'x') K = Sk.quotient_ring(Sk.ideal(x**self.n + 1)) while True: l = self.D_sigma() ginv_K = K(mod_near_poly(l, self.q))**(-1) ginv_size = vector(ginv_K).norm() if ginv_size < self.n**2: g = self.Rq(l) self.ginv = g**(-1) break # discrete Gaussian in I = <g>, yields random encodings of 0 short_g = vector(ZZ, mod_near_poly(g, self.q)) DGSL_sigmap_I = DGSL(short_g, self.sigma_prime) self.D_sigmap_I = lambda: self.Rq(list(DGSL_sigmap_I())) # compute zero-testing parameter p_zt # randomly draw h (in Rq) from a discrete Gaussian with param q^(1/2) self.h = self.Rq(list(DGSL(ZZ**self.n, round(sqrt(self.q)))())) # create p_zt self.p_zt = self.ginv * self.h * prod(z)
def __init__(self, params, asym=False): (self.n, self.q, sigma, self.sigma_prime, self.k) = params S, x = PolynomialRing(ZZ, 'x').objgen() self.R = S.quotient_ring(S.ideal(x**self.n + 1)) Sq = PolynomialRing(Zmod(self.q), 'x') self.Rq = Sq.quotient_ring(Sq.ideal(x**self.n + 1)) # draw z_is uniformly from Rq and compute its inverse in Rq if asym: z = [self.Rq.random_element() for i in range(self.k)] self.zinv = [z_i**(-1) for z_i in z] else: # or do symmetric version z = self.Rq.random_element() zinv = z**(-1) z, self.zinv = zip(*[(z,zinv) for i in range(self.k)]) # set up some discrete Gaussians DGSL_sigma = DGSL(ZZ**self.n, sigma) self.D_sigma = lambda: self.Rq(list(DGSL_sigma())) # discrete Gaussian in ZZ^n with stddev sigma_prime, yields random level-0 encodings DGSL_sigmap_ZZ = DGSL(ZZ**self.n, self.sigma_prime) self.D_sigmap_ZZ = lambda: self.Rq(list(DGSL_sigmap_ZZ())) # draw g repeatedly from a Gaussian distribution of Z^n (with param sigma) # until g^(-1) in QQ[x]/<x^n + 1> is small (< n^2) Sk = PolynomialRing(QQ, 'x') K = Sk.quotient_ring(Sk.ideal(x**self.n + 1)) while True: l = self.D_sigma() ginv_K = K(mod_near_poly(l, self.q))**(-1) ginv_size = vector(ginv_K).norm() if ginv_size < self.n**2: g = self.Rq(l) self.ginv = g**(-1) break # discrete Gaussian in I = <g>, yields random encodings of 0 short_g = vector(ZZ, mod_near_poly(g,self.q)) DGSL_sigmap_I = DGSL(short_g, self.sigma_prime) self.D_sigmap_I = lambda: self.Rq(list(DGSL_sigmap_I())) # compute zero-testing parameter p_zt # randomly draw h (in Rq) from a discrete Gaussian with param q^(1/2) self.h = self.Rq(list(DGSL(ZZ**self.n, round(sqrt(self.q)))())) # create p_zt self.p_zt = self.ginv * self.h * prod(z)
def generator_relations(self, K): """ An ideal `I` in a polynomial ring `R` over `K`, such that the associated ring is `R / I` surjects onto the ring of modular forms with coefficients in `K`. INPUT: - `K` -- A ring. OUTPUT: An ideal in a polynomial ring. TESTS:: sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import * sage: t = ModularFormTestType_vectorvalued() sage: t.generator_relations(QQ) Ideal (g1^2 - g2, g1^3 - g3, g1^4 - g4, g1^5 - g5) of Multivariate Polynomial Ring in g1, g2, g3, g4, g5, v1, v2, v3 over Rational Field """ if K.has_coerce_map_from(ZZ): R = PolynomialRing( K, self.non_vector_valued()._generator_names(K) + self._generator_names(K)) return R.ideal().parent()( self.non_vector_valued().generator_relations(K)) raise NotImplementedError
def generator_relations(self, K): """ An ideal `I` in a polynomial ring `R` over `K`, such that the associated ring is `R / I` surjects onto the ring of modular forms with coefficients in `K`. INPUT: - `K` -- A ring. OUTPUT: An ideal in a polynomial ring. TESTS:: sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import * sage: t = ModularFormTestType_scalar() sage: t.generator_relations(QQ) Ideal (g1^2 - g2, g1^3 - g3, g1^4 - g4, g1^5 - g5) of Multivariate Polynomial Ring in g1, g2, g3, g4, g5 over Rational Field """ if K.has_coerce_map_from(ZZ): R = PolynomialRing(K, self._generator_names(K)) g1 = R.gen(0) return R.ideal([ g1**i - g for (i, g) in list(enumerate([None] + list(R.gens())))[2:] ]) raise NotImplementedError
def generator_relations(self, K) : """ An ideal `I` in a polynomial ring `R` over `K`, such that the associated ring is `R / I` surjects onto the ring of modular forms with coefficients in `K`. INPUT: - `K` -- A ring. OUTPUT: An ideal in a polynomial ring. TESTS:: sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import * sage: t = ModularFormTestType_vectorvalued() sage: t.generator_relations(QQ) Ideal (g1^2 - g2, g1^3 - g3, g1^4 - g4, g1^5 - g5) of Multivariate Polynomial Ring in g1, g2, g3, g4, g5, v1, v2, v3 over Rational Field """ if K.has_coerce_map_from(ZZ) : R = PolynomialRing(K, self.non_vector_valued()._generator_names(K) + self._generator_names(K)) return R.ideal().parent()(self.non_vector_valued().generator_relations(K)) raise NotImplementedError
def generator_relations(self, K) : """ An ideal `I` in a polynomial ring `R` over `K`, such that the associated ring is `R / I` surjects onto the ring of modular forms with coefficients in `K`. INPUT: - `K` -- A ring. OUTPUT: An ideal in a polynomial ring. TESTS:: sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import * sage: t = ModularFormTestType_scalar() sage: t.generator_relations(QQ) Ideal (g1^2 - g2, g1^3 - g3, g1^4 - g4, g1^5 - g5) of Multivariate Polynomial Ring in g1, g2, g3, g4, g5 over Rational Field """ if K.has_coerce_map_from(ZZ) : R = PolynomialRing(K, self._generator_names(K)) g1 = R.gen(0) return R.ideal([g1**i - g for (i,g) in list(enumerate([None] + list(R.gens())))[2:]]) raise NotImplementedError
def segre_embedding(self, PP=None, var='u'): r""" Return the Segre embedding of this space into the appropriate projective space. INPUT: - ``PP`` -- (default: ``None``) ambient image projective space; this is constructed if it is not given. - ``var`` -- string, variable name of the image projective space, default `u` (optional). OUTPUT: Hom -- from this space to the appropriate subscheme of projective space. .. TODO:: Cartesian products with more than two components. EXAMPLES:: sage: X.<y0,y1,y2,y3,y4,y5> = ProductProjectiveSpaces(ZZ, [2, 2]) sage: phi = X.segre_embedding(); phi Scheme morphism: From: Product of projective spaces P^2 x P^2 over Integer Ring To: Closed subscheme of Projective Space of dimension 8 over Integer Ring defined by: -u5*u7 + u4*u8, -u5*u6 + u3*u8, -u4*u6 + u3*u7, -u2*u7 + u1*u8, -u2*u4 + u1*u5, -u2*u6 + u0*u8, -u1*u6 + u0*u7, -u2*u3 + u0*u5, -u1*u3 + u0*u4 Defn: Defined by sending (y0 : y1 : y2 , y3 : y4 : y5) to (y0*y3 : y0*y4 : y0*y5 : y1*y3 : y1*y4 : y1*y5 : y2*y3 : y2*y4 : y2*y5). :: sage: T = ProductProjectiveSpaces([1, 2], CC, 'z') sage: T.segre_embedding() Scheme morphism: From: Product of projective spaces P^1 x P^2 over Complex Field with 53 bits of precision To: Closed subscheme of Projective Space of dimension 5 over Complex Field with 53 bits of precision defined by: -u2*u4 + u1*u5, -u2*u3 + u0*u5, -u1*u3 + u0*u4 Defn: Defined by sending (z0 : z1 , z2 : z3 : z4) to (z0*z2 : z0*z3 : z0*z4 : z1*z2 : z1*z3 : z1*z4). :: sage: T = ProductProjectiveSpaces([1, 2, 1], QQ, 'z') sage: T.segre_embedding() Scheme morphism: From: Product of projective spaces P^1 x P^2 x P^1 over Rational Field To: Closed subscheme of Projective Space of dimension 11 over Rational Field defined by: -u9*u10 + u8*u11, -u7*u10 + u6*u11, -u7*u8 + u6*u9, -u5*u10 + u4*u11, -u5*u8 + u4*u9, -u5*u6 + u4*u7, -u5*u9 + u3*u11, -u5*u8 + u3*u10, -u5*u8 + u2*u11, -u4*u8 + u2*u10, -u3*u8 + u2*u9, -u3*u6 + u2*u7, -u3*u4 + u2*u5, -u5*u7 + u1*u11, -u5*u6 + u1*u10, -u3*u7 + u1*u9, -u3*u6 + u1*u8, -u5*u6 + u0*u11, -u4*u6 + u0*u10, -u3*u6 + u0*u9, -u2*u6 + u0*u8, -u1*u6 + u0*u7, -u1*u4 + u0*u5, -u1*u2 + u0*u3 Defn: Defined by sending (z0 : z1 , z2 : z3 : z4 , z5 : z6) to (z0*z2*z5 : z0*z2*z6 : z0*z3*z5 : z0*z3*z6 : z0*z4*z5 : z0*z4*z6 : z1*z2*z5 : z1*z2*z6 : z1*z3*z5 : z1*z3*z6 : z1*z4*z5 : z1*z4*z6). """ N = self._dims M = prod([n + 1 for n in N]) - 1 CR = self.coordinate_ring() vars = list(self.coordinate_ring().variable_names()) + [ var + str(i) for i in range(M + 1) ] R = PolynomialRing(self.base_ring(), self.ngens() + M + 1, vars, order='lex') #set-up the elimination for the segre embedding mapping = [] k = self.ngens() index = self.num_components() * [0] for count in range(M + 1): mapping.append( R.gen(k + count) - prod([CR(self[i].gen(index[i])) for i in range(len(index))])) for i in range(len(index) - 1, -1, -1): if index[i] == N[i]: index[i] = 0 else: index[i] += 1 break #only increment once #change the defining ideal of the subscheme into the variables I = R.ideal(list(self.defining_polynomials()) + mapping) J = I.groebner_basis() s = set(R.gens()[:self.ngens()]) n = len(J) - 1 L = [] while s.isdisjoint(J[n].variables()): L.append(J[n]) n = n - 1 #create new subscheme if PP is None: PS = ProjectiveSpace(self.base_ring(), M, R.variable_names()[self.ngens():]) Y = PS.subscheme(L) else: if PP.dimension_relative() != M: raise ValueError( "projective Space %s must be dimension %s") % (PP, M) S = PP.coordinate_ring() psi = R.hom([0] * k + list(S.gens()), S) L = [psi(l) for l in L] Y = PP.subscheme(L) #create embedding for points mapping = [] index = self.num_components() * [0] for count in range(M + 1): mapping.append( prod([CR(self[i].gen(index[i])) for i in range(len(index))])) for i in range(len(index) - 1, -1, -1): if index[i] == N[i]: index[i] = 0 else: index[i] += 1 break #only increment once phi = self.hom(mapping, Y) return phi
def segre_embedding(self, PP=None, var='u'): r""" Return the Segre embedding of ``self`` into the appropriate projective space. INPUT: - ``PP`` -- (default: ``None``) ambient image projective space; this is constructed if it is not given. - ``var`` -- string, variable name of the image projective space, default `u` (optional) OUTPUT: Hom -- from ``self`` to the appropriate subscheme of projective space .. TODO:: Cartesian products with more than two components EXAMPLES:: sage: X.<y0,y1,y2,y3,y4,y5> = ProductProjectiveSpaces(ZZ,[2,2]) sage: phi = X.segre_embedding(); phi Scheme morphism: From: Product of projective spaces P^2 x P^2 over Integer Ring To: Closed subscheme of Projective Space of dimension 8 over Integer Ring defined by: -u5*u7 + u4*u8, -u5*u6 + u3*u8, -u4*u6 + u3*u7, -u2*u7 + u1*u8, -u2*u4 + u1*u5, -u2*u6 + u0*u8, -u1*u6 + u0*u7, -u2*u3 + u0*u5, -u1*u3 + u0*u4 Defn: Defined by sending (y0 : y1 : y2 , y3 : y4 : y5) to (y0*y3 : y0*y4 : y0*y5 : y1*y3 : y1*y4 : y1*y5 : y2*y3 : y2*y4 : y2*y5). :: sage: T = ProductProjectiveSpaces([1,2],CC,'z') sage: T.segre_embedding() Scheme morphism: From: Product of projective spaces P^1 x P^2 over Complex Field with 53 bits of precision To: Closed subscheme of Projective Space of dimension 5 over Complex Field with 53 bits of precision defined by: -u2*u4 + u1*u5, -u2*u3 + u0*u5, -u1*u3 + u0*u4 Defn: Defined by sending (z0 : z1 , z2 : z3 : z4) to (z0*z2 : z0*z3 : z0*z4 : z1*z2 : z1*z3 : z1*z4). """ N = self._dims if len(N) > 2: raise NotImplementedError("Cannot have more than two components.") M = (N[0]+1)*(N[1]+1)-1 vars = list(self.coordinate_ring().variable_names()) + [var + str(i) for i in range(M+1)] R = PolynomialRing(self.base_ring(),self.ngens()+M+1, vars, order='lex') #set-up the elimination for the segre embedding mapping = [] k = self.ngens() for i in range(N[0]+1): for j in range(N[0]+1,N[0]+N[1]+2): mapping.append(R.gen(k)-R(self.gen(i)*self.gen(j))) k+=1 #change the defining ideal of the subscheme into the variables I = R.ideal(list(self.defining_polynomials()) + mapping) J = I.groebner_basis() s = set(R.gens()[:self.ngens()]) n = len(J)-1 L = [] while s.isdisjoint(J[n].variables()): L.append(J[n]) n = n-1 #create new subscheme if PP is None: PS = ProjectiveSpace(self.base_ring(),M,R.gens()[self.ngens():]) Y = PS.subscheme(L) else: if PP.dimension_relative()!= M: raise ValueError("Projective Space %s must be dimension %s")%(PP, M) S = PP.coordinate_ring() psi = R.hom([0]*(N[0]+N[1]+2) + list(S.gens()),S) L = [psi(l) for l in L] Y = PP.subscheme(L) #create embedding for points mapping = [] for i in range(N[0]+1): for j in range(N[0]+1,N[0]+N[1]+2): mapping.append(self.gen(i)*self.gen(j)) phi = self.hom(mapping,Y) return phi
def segre_embedding(self, PP=None, var='u'): r""" Return the Segre embedding of this space into the appropriate projective space. INPUT: - ``PP`` -- (default: ``None``) ambient image projective space; this is constructed if it is not given. - ``var`` -- string, variable name of the image projective space, default `u` (optional). OUTPUT: Hom -- from this space to the appropriate subscheme of projective space. .. TODO:: Cartesian products with more than two components. EXAMPLES:: sage: X.<y0,y1,y2,y3,y4,y5> = ProductProjectiveSpaces(ZZ, [2, 2]) sage: phi = X.segre_embedding(); phi Scheme morphism: From: Product of projective spaces P^2 x P^2 over Integer Ring To: Closed subscheme of Projective Space of dimension 8 over Integer Ring defined by: -u5*u7 + u4*u8, -u5*u6 + u3*u8, -u4*u6 + u3*u7, -u2*u7 + u1*u8, -u2*u4 + u1*u5, -u2*u6 + u0*u8, -u1*u6 + u0*u7, -u2*u3 + u0*u5, -u1*u3 + u0*u4 Defn: Defined by sending (y0 : y1 : y2 , y3 : y4 : y5) to (y0*y3 : y0*y4 : y0*y5 : y1*y3 : y1*y4 : y1*y5 : y2*y3 : y2*y4 : y2*y5). :: sage: T = ProductProjectiveSpaces([1, 2], CC, 'z') sage: T.segre_embedding() Scheme morphism: From: Product of projective spaces P^1 x P^2 over Complex Field with 53 bits of precision To: Closed subscheme of Projective Space of dimension 5 over Complex Field with 53 bits of precision defined by: -u2*u4 + u1*u5, -u2*u3 + u0*u5, -u1*u3 + u0*u4 Defn: Defined by sending (z0 : z1 , z2 : z3 : z4) to (z0*z2 : z0*z3 : z0*z4 : z1*z2 : z1*z3 : z1*z4). :: sage: T = ProductProjectiveSpaces([1, 2, 1], QQ, 'z') sage: T.segre_embedding() Scheme morphism: From: Product of projective spaces P^1 x P^2 x P^1 over Rational Field To: Closed subscheme of Projective Space of dimension 11 over Rational Field defined by: -u9*u10 + u8*u11, -u7*u10 + u6*u11, -u7*u8 + u6*u9, -u5*u10 + u4*u11, -u5*u8 + u4*u9, -u5*u6 + u4*u7, -u5*u9 + u3*u11, -u5*u8 + u3*u10, -u5*u8 + u2*u11, -u4*u8 + u2*u10, -u3*u8 + u2*u9, -u3*u6 + u2*u7, -u3*u4 + u2*u5, -u5*u7 + u1*u11, -u5*u6 + u1*u10, -u3*u7 + u1*u9, -u3*u6 + u1*u8, -u5*u6 + u0*u11, -u4*u6 + u0*u10, -u3*u6 + u0*u9, -u2*u6 + u0*u8, -u1*u6 + u0*u7, -u1*u4 + u0*u5, -u1*u2 + u0*u3 Defn: Defined by sending (z0 : z1 , z2 : z3 : z4 , z5 : z6) to (z0*z2*z5 : z0*z2*z6 : z0*z3*z5 : z0*z3*z6 : z0*z4*z5 : z0*z4*z6 : z1*z2*z5 : z1*z2*z6 : z1*z3*z5 : z1*z3*z6 : z1*z4*z5 : z1*z4*z6). """ N = self._dims M = prod([n+1 for n in N]) - 1 CR = self.coordinate_ring() vars = list(self.coordinate_ring().variable_names()) + [var + str(i) for i in range(M+1)] R = PolynomialRing(self.base_ring(), self.ngens()+M+1, vars, order='lex') #set-up the elimination for the segre embedding mapping = [] k = self.ngens() index = self.num_components()*[0] for count in range(M + 1): mapping.append(R.gen(k+count)-prod([CR(self[i].gen(index[i])) for i in range(len(index))])) for i in range(len(index)-1, -1, -1): if index[i] == N[i]: index[i] = 0 else: index[i] += 1 break #only increment once #change the defining ideal of the subscheme into the variables I = R.ideal(list(self.defining_polynomials()) + mapping) J = I.groebner_basis() s = set(R.gens()[:self.ngens()]) n = len(J)-1 L = [] while s.isdisjoint(J[n].variables()): L.append(J[n]) n = n-1 #create new subscheme if PP is None: PS = ProjectiveSpace(self.base_ring(), M, R.variable_names()[self.ngens():]) Y = PS.subscheme(L) else: if PP.dimension_relative() != M: raise ValueError("projective Space %s must be dimension %s")%(PP, M) S = PP.coordinate_ring() psi = R.hom([0]*k + list(S.gens()), S) L = [psi(l) for l in L] Y = PP.subscheme(L) #create embedding for points mapping = [] index = self.num_components()*[0] for count in range(M + 1): mapping.append(prod([CR(self[i].gen(index[i])) for i in range(len(index))])) for i in range(len(index)-1, -1, -1): if index[i] == N[i]: index[i] = 0 else: index[i] += 1 break #only increment once phi = self.hom(mapping, Y) return phi