Hi,
I want to make a python function to use in sagemath that converts from polynomial to matrix using a normal basis generator in a GF. I am now running
F.<a> = GF(2^4)
a = F.gen()
R.<x> = PolynomialRing(F)
poly = x^3
b = a^3
M = polynomial_to_matrix(F, poly, b)
poly_rec = matrix_to_polynomial(F, M, basis)
Then expecting a matrix like [ 0, a^6, 1, a^3 ] [ a^6, 0, a^12, 1 ] [ 1, a^12, 0, a^9 ] [ a^3, 1, a^9, 0 ]
Rather I get [ 0 a^3 + a^2 + a a^2 + a + 1 a^3 + a^2 + 1] [a^3 + a^2 + a 0 a^3 + a + 1 a^2 + a] [ a^2 + a + 1 a^3 + a + 1 0 a^3 + 1] [a^3 + a^2 + 1 a^2 + a a^3 + 1 0
What concept do I not get or have misinterpreted? All help is greatly appreciated
The code
def polynomial_to_matrix(F, poly, basis_gen):
n = F.degree()
basis = [basis_gen**(2**i) for i in range(n)]
M = [[poly(bi) + poly(bj) + poly(bi + bj) + poly(0) for bj in basis] for bi in basis]
return Matrix(F, M)
def matrix_to_polynomial(F, M, basis):
n = F.degree()
R = PolynomialRing(F, 'x')
x = R.gen()
M_alpha = Matrix(F, [[basis[i]**(2**j) for j in range(n)] for i in range(n)])
M_alpha_inv = M_alpha.inverse()
CF = M_alpha_inv.transpose() * M * M_alpha_inv
poly = 0
for i in range(n):
for j in range(i):
c = CF[i,j]
if c != 0:
poly += c * x**(2**i + 2**j)
return poly
