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