def _easy_prime_decomp(p, polynomial):
"""
prime decomposition by factoring polynomial
"""
f = algfield.fppoly(polynomial, p)
d = f.degree()
factor = f.factor()
p_alg = algfield.BasicAlgNumber([[p] + [0] * (d - 1), 1], polynomial)
if len(factor) == 1 and factor[0][1] == 1:
return [(module.Ideal_with_generator([p_alg]), 1,
factor[0][0].degree())]
else:
dlist = []
for i in range(len(factor)):
basis_list = []
for j in range(d):
if factor[i][0][j] == 0:
basis_list.append(0)
else:
basis_list.append(factor[i][0][j].toInteger())
prime_ideal = module.Ideal_with_generator(
[p_alg,
algfield.BasicAlgNumber([basis_list, 1], polynomial)])
dlist.append((prime_ideal, factor[i][1], factor[i][0].degree()))
return dlist
def _check_H(p, H, field, gamma_mul):
"""
If H express the prime ideal, return (H, f),
where f: residual degree.
Else return some column of M_1 (not (1,0,..,0)^t).
"""
F_p = finitefield.FinitePrimeField(p)
if H == None:
f = field.degree
else:
f = H.row - H.column # rank(A), A.column
# CCANT Algo 6.2.9 step 10-11
# step 10
B = matrix.unitMatrix(f, F_p)
M_basis = []
for j in range(1, f + 1):
alpha_pow = _pow_by_base_mul(B[j], p, gamma_mul, f)
alpha_pow -= B[j]
M_basis.append(alpha_pow)
M_1 = matrix.FieldMatrix(f, f, M_basis).kernel()
# step 11
if M_1.column > 1: # dim(M_1) > 1
return M_1[M_1.column]
else:
H_simple = _two_element_repr_prime_ideal(
p, H.map(lambda x: x.getResidue()), field, f)
p_alg = algfield.BasicAlgNumber([[p] + [0] * (field.degree - 1), 1],
field.polynomial)
sol = module.Ideal_with_generator([p_alg, H_simple])
return (sol, f)
def _vector_to_algnumber(vector_repr, field):
"""
vector (w.r.t. theta) -> omega
"""
int_vector, denom = module._toIntegerMatrix(vector_repr)
int_vector = int_vector[1].compo # Submodule -> list
omega = algfield.BasicAlgNumber([int_vector, denom], field.polynomial)
return omega
def _matrix_to_algnumber_list(matrix_repr, field):
"""
matrix (w.r.t. theta) -> [omega_i]
"""
int_matrix, denom = module._toIntegerMatrix(matrix_repr)
omega = []
for i in range(1, int_matrix.column + 1):
omega_i = algfield.BasicAlgNumber(
[int_matrix[i], denom], field.polynomial)
omega.append(omega_i)
return omega
def setUp(self):
self.a = algfield.BasicAlgNumber([[1, 1], 1], [-2, 0, 1])
self.b = algfield.BasicAlgNumber([[-1, 2], 1], [-2, 0, 1])