def _separable_algebra(p, H, base_multiply):
"""
return A=O/H (as matrix representation)
"""
# CCANT Algo 6.2.9 step 8-9
F_p = finitefield.FinitePrimeField(p)
n = base_multiply.row
if H == None:
H_new = matrix.zeroMatrix(n, 1, F_p)
r = 0
else:
H_new = H
r = H.column
# step 8
H_add_one = H_new.copy()
if H_add_one.column == n:
raise ValueError
H_add_one.extendColumn(vector.Vector([F_p.one] + [F_p.zero] * (n - 1)))
H_add_one.toSubspace(True)
full_base = H_add_one.supplementBasis()
full_base.toFieldMatrix()
# step 9
A = full_base.subMatrix(
range(1, n + 1), range(r + 1, n + 1))
gamma_mul = full_base.inverseImage(
_matrix_mul_by_base_mul(A, A, base_multiply))
gamma_mul = gamma_mul.subMatrix(
range(r + 1, n + 1), range(1, gamma_mul.column + 1))
return A, gamma_mul
def _div_mod_pO(self, other, base_multiply, p):
"""
ideal division modulo pO by using base_multiply
"""
n = other.row
m = other.column
F_p = finitefield.FinitePrimeField(p)
#Algo 2.3.7
if self == None:
self_new = matrix.zeroMatrix(n, F_p)
r = 0
else:
self_new = self
r = self.column
if r == m:
return matrix.unitMatrix(n, F_p)
# if r > m, raise NoInverseImage error
X = matrix.Subspace.fromMatrix(other.inverseImage(self_new))
B = X.supplementBasis()
other_self = other * B # first r columns are self, others are supplement
other_self.toFieldMatrix()
gamma_part = other_self.subMatrix(
range(1, n + 1), range(r + 1, m + 1))
omega_gamma = other_self.inverseImage(_matrix_mul_by_base_mul(
matrix.unitMatrix(n, F_p), gamma_part, base_multiply))
vect_list = []
for k in range(1, n + 1):
vect = vector.Vector([F_p.zero] * ((m - r) ** 2))
for i in range(1, m - r + 1):
for j in range(1, m - r + 1):
vect[(m - r) * (i - 1) + j] = _mul_place(
k, i, omega_gamma, m - r)[j + r]
vect_list.append(vect)
return matrix.FieldMatrix( (m - r) ** 2, n, vect_list).kernel()
def _div_mod_pO(self, other, base_multiply, p):
"""
ideal division modulo pO by using base_multiply
"""
n = other.row
m = other.column
F_p = finitefield.FinitePrimeField(p)
#Algo 2.3.7
if self == None:
self_new = matrix.zeroMatrix(n, F_p)
r = 0
else:
self_new = self
r = self.column
if r == m:
return matrix.unitMatrix(n, F_p)
# if r > m, raise NoInverseImage error
X = matrix.Subspace.fromMatrix(other.inverseImage(self_new))
B = X.supplementBasis()
other_self = other * B # first r columns are self, others are supplement
other_self.toFieldMatrix()
gamma_part = other_self.subMatrix(
range(1, n + 1), range(r + 1, m + 1))
omega_gamma = other_self.inverseImage(_matrix_mul_by_base_mul(
matrix.unitMatrix(n, F_p), gamma_part, base_multiply))
vect_list = []
for k in range(1, n + 1):
vect = vector.Vector([F_p.zero] * ((m - r) ** 2))
for i in range(1, m - r + 1):
for j in range(1, m - r + 1):
vect[(m - r) * (i - 1) + j] = _mul_place(
k, i, omega_gamma, m - r)[j + r]
vect_list.append(vect)
return matrix.FieldMatrix( (m - r) ** 2, n, vect_list).kernel()
def _separable_algebra(p, H, base_multiply):
"""
return A=O/H (as matrix representation)
"""
# CCANT Algo 6.2.9 step 8-9
F_p = finitefield.FinitePrimeField(p)
n = base_multiply.row
if H == None:
H_new = matrix.zeroMatrix(n, 1, F_p)
r = 0
else:
H_new = H
r = H.column
# step 8
H_add_one = H_new.copy()
if H_add_one.column == n:
raise ValueError
H_add_one.extendColumn(vector.Vector([F_p.one] + [F_p.zero] * (n - 1)))
H_add_one.toSubspace(True)
full_base = H_add_one.supplementBasis()
full_base.toFieldMatrix()
# step 9
A = full_base.subMatrix(
range(1, n + 1), range(r + 1, n + 1))
gamma_mul = full_base.inverseImage(
_matrix_mul_by_base_mul(A, A, base_multiply))
gamma_mul = gamma_mul.subMatrix(
range(r + 1, n + 1), range(1, gamma_mul.column + 1))
return A, gamma_mul
def intersect(self, S1, S2):
cvectors = [ flatten_matrix(S) for S in S1 ] + \
[ flatten_matrix(-S) for S in S2 ]
coeffs = matrix.Matrix(len(cvectors[0]), len(cvectors),
cvectors, self.field)
v = flatten_matrix(matrix.zeroMatrix(S1[0].row, S1[0].column))
self.addEquations(coeffs, v)
def intersect_solutions(S1, S2):
if not S1 or not S2: return []
field = S1[0].coeff_ring
lineq = LinearEquations(field, len(S1) + len(S2))
lineq.intersect(S1, S2)
K = lineq.kernel()
if K is None: return []
Kp = [ vector.Vector(v.compo[0:len(S1)]) for v in K ]
intersection = [ vector_to_matrix(k, S1) for k in Kp ]
return [ X for X in intersection if X != matrix.zeroMatrix(X.row) ]
def toHNF(self):
"""
Change matrix to HNF(hermite normal form)
Note that HNF do not always give basis of self
(i.e. HNF may be redundant)
"""
if not self.ishnf:
hnf = self.hermiteNormalForm(True)
if hnf == None: #zero module
hnf = matrix.zeroMatrix(self.row, 1, self.coeff_ring)
self.compo = hnf.compo
self.column = hnf.column
self.ishnf = True
def toHNF(self):
"""
Change matrix to HNF(hermite normal form)
Note that HNF do not always give basis of self
(i.e. HNF may be redundant)
"""
if not self.ishnf:
hnf = self.hermiteNormalForm(True)
if hnf == None: #zero module
hnf = matrix.zeroMatrix(self.row, 1, self.coeff_ring)
self.compo = hnf.compo
self.column = hnf.column
self.ishnf = True
def testDigitalMethod(self):
zero1 = matrix.zeroMatrix(3,0)
one1 = matrix.identityMatrix(3,1)
d_func1 = digital_method_func(
lambda a,b:a+b, lambda a,b:a*b, lambda i,a:i*a, lambda a,i:a**i,
zero1, one1)
coefficients11 = []
coefficients12 = [(2,1), (1,2), (0,1)]
coefficients13 = [(3,1), (2,2), (1,3)]
A = matrix.SquareMatrix(3, [1,2,3]+[4,5,6]+[7,8,9])
self.assertEqual(d_func1(coefficients11, A), zero1)
self.assertEqual(d_func1(coefficients12, A), A**2+2*A+one1)
self.assertEqual(d_func1(coefficients13, A), (A**3+2*A**2+3*A))
def _kernel_of_qpow(omega, q, p, minpoly, n):
"""
Return the kernel of q-th powering, which is a linear map over Fp.
q is a power of p which exceeds n.
(omega_j^q (mod theminpoly) = \sum a_i_j omega_i a_i_j in Fp)
"""
omega_poly = omega.get_polynomials()
denom = omega.denominator
theminpoly = minpoly.to_field_polynomial()
field_p = finitefield.FinitePrimeField.getInstance(p)
zero = field_p.zero
qpow = matrix.zeroMatrix(n, n, field_p) # Fp matrix
for j in range(n):
a_j = [zero] * n
omega_poly_j = uniutil.polynomial(enumerate(omega.basis[j]), Z)
omega_j_qpow = pow(omega_poly_j, q, minpoly)
redundancy = gcd.gcd(omega_j_qpow.content(), denom ** q)
omega_j_qpow = omega_j_qpow.coefficients_map(lambda c: c // redundancy)
essential = denom ** q // redundancy
while omega_j_qpow:
i = omega_j_qpow.degree()
a_ji = int(omega_j_qpow[i] / (omega_poly[i][i] * essential))
omega_j_qpow -= a_ji * (omega_poly[i] * essential)
if omega_j_qpow.degree() < i:
a_j[i] = field_p.createElement(a_ji)
else:
_log.debug("%s / %d" % (str(omega_j_qpow), essential))
_log.debug("j = %d, a_ji = %s" % (j, a_ji))
raise ValueError("omega is not a basis")
qpow.setColumn(j + 1, a_j)
result = qpow.kernel()
if result is None:
_log.debug("(_k_q): trivial kernel")
return matrix.zeroMatrix(n, 1, field_p)
else:
return result
def _kernel_of_qpow(omega, q, p, minpoly, n):
"""
Return the kernel of q-th powering, which is a linear map over Fp.
q is a power of p which exceeds n.
(omega_j^q (mod theminpoly) = \sum a_i_j omega_i a_i_j in Fp)
"""
omega_poly = omega.get_polynomials()
denom = omega.denominator
theminpoly = minpoly.to_field_polynomial()
field_p = finitefield.FinitePrimeField.getInstance(p)
zero = field_p.zero
qpow = matrix.zeroMatrix(n, n, field_p) # Fp matrix
for j in range(n):
a_j = [zero] * n
omega_poly_j = uniutil.polynomial(enumerate(omega.basis[j]), Z)
omega_j_qpow = pow(omega_poly_j, q, minpoly)
redundancy = gcd.gcd(omega_j_qpow.content(), denom ** q)
omega_j_qpow = omega_j_qpow.coefficients_map(lambda c: c // redundancy)
essential = denom ** q // redundancy
while omega_j_qpow:
i = omega_j_qpow.degree()
a_ji = int(omega_j_qpow[i] / (omega_poly[i][i] * essential))
omega_j_qpow -= a_ji * (omega_poly[i] * essential)
if omega_j_qpow.degree() < i:
a_j[i] = field_p.createElement(a_ji)
else:
_log.debug("%s / %d" % (str(omega_j_qpow), essential))
_log.debug("j = %d, a_ji = %s" % (j, a_ji))
raise ValueError("omega is not a basis")
qpow.setColumn(j + 1, a_j)
result = qpow.kernel()
if result is None:
_log.debug("(_k_q): trivial kernel")
return matrix.zeroMatrix(n, 1, field_p)
else:
return result
def intersectionOfSubmodules(self, other):
"""
Return module which is intersection of self and other.
"""
if self.row != other.row:
raise matrix.MatrixSizeError()
M1 = self.copy()
M1.extendColumn(other)
N = M1.kernelAsModule()
if N == None: # zero kernel
N = matrix.zeroMatrix(self.row, 1, self.coeff_ring)
N1 = N.getBlock(1, 1, self.column, N.column) # N.column is dim(ker(M1))
module = Submodule.fromMatrix(self * N1)
module.toHNF()
return module
def intersectionOfSubmodules(self, other):
"""
Return module which is intersection of self and other.
"""
if self.row != other.row:
raise matrix.MatrixSizeError()
M1 = self.copy()
M1.extendColumn(other)
N = M1.kernelAsModule()
if N == None: # zero kernel
N = matrix.zeroMatrix(self.row, 1, self.coeff_ring)
N1 = N.getBlock(1, 1, self.column, N.column) # N.column is dim(ker(M1))
module = Submodule.fromMatrix(self * N1)
module.toHNF()
return module
def _splitting_squarefree(p, H, base_multiply, A, gamma_mul, alpha):
"""
split squarefree part
"""
F_p = finitefield.FinitePrimeField(p)
if H == None:
n = base_multiply.row
f = n
H_new = matrix.zeroMatrix(n, 1, F_p)
else:
n = H.row
f = n - H.column
H_new = H.copy()
# step 12
one_gamma = vector.Vector([F_p.one] + [F_p.zero] *
(f - 1)) # w.r.t. gamma_1
minpoly_matrix_alpha = matrix.FieldMatrix(f, 2, [one_gamma, alpha])
ker = minpoly_matrix_alpha.kernel()
alpha_pow = alpha
while ker == None:
alpha_pow = _vect_mul_by_base_mul(alpha_pow, alpha, gamma_mul, f)
minpoly_matrix_alpha.extendColumn(alpha_pow)
ker = minpoly_matrix_alpha.kernel()
minpoly_alpha = uniutil.FinitePrimeFieldPolynomial(enumerate(ker[1].compo),
F_p)
# step 13
m_list = minpoly_alpha.factor()
# step 14
L = []
for (m_s, i) in m_list:
M_s = H_new.copy()
beta_s_gamma = _substitution_by_base_mul(m_s, alpha, gamma_mul,
one_gamma, f)
# beta_s_gamma (w.r.t. gamma) -> beta_s (w.r.t. omega)
beta_s = A * beta_s_gamma
omega_beta_s = _matrix_mul_by_base_mul(matrix.unitMatrix(n, F_p),
beta_s.toMatrix(True),
base_multiply)
M_s.extendColumn(omega_beta_s)
L.append(M_s.image())
return L
def _splitting_squarefree(p, H, base_multiply, A, gamma_mul, alpha):
"""
split squarefree part
"""
F_p = finitefield.FinitePrimeField(p)
if H == None:
n = base_multiply.row
f = n
H_new = matrix.zeroMatrix(n, 1, F_p)
else:
n = H.row
f = n - H.column
H_new = H.copy()
# step 12
one_gamma = vector.Vector([F_p.one] + [F_p.zero] * (f - 1)) # w.r.t. gamma_1
minpoly_matrix_alpha = matrix.FieldMatrix(f, 2, [one_gamma, alpha])
ker = minpoly_matrix_alpha.kernel()
alpha_pow = alpha
while ker == None:
alpha_pow = _vect_mul_by_base_mul(alpha_pow, alpha, gamma_mul, f)
minpoly_matrix_alpha.extendColumn(alpha_pow)
ker = minpoly_matrix_alpha.kernel()
minpoly_alpha = uniutil.FinitePrimeFieldPolynomial(
enumerate(ker[1].compo), F_p)
# step 13
m_list = minpoly_alpha.factor()
# step 14
L = []
for (m_s, i) in m_list:
M_s = H_new.copy()
beta_s_gamma = _substitution_by_base_mul(
m_s, alpha, gamma_mul, one_gamma, f)
# beta_s_gamma (w.r.t. gamma) -> beta_s (w.r.t. omega)
beta_s = A * beta_s_gamma
omega_beta_s = _matrix_mul_by_base_mul(
matrix.unitMatrix(n, F_p), beta_s.toMatrix(True), base_multiply)
M_s.extendColumn(omega_beta_s)
L.append(M_s.image())
return L
