I'm wrote a method to get the symbolic coefficients of polynomial p \in
GF(2^pp)[X0,...,X_{nvars-1}] translate to GF(2). The method was wrotte
using companion matrix [1] of irreducible polynomial GF(2^pp). For example
for pp = 2. The polynomial p = X0*X1 = (x_00+x_01*t)*(x_10+x_11*t) =
(x_00*x_10+x_01*x_11) + (x_01*x_11+x_01*x_10 + x_00*x_11)*t then the
method get [(x_00*x_10+x_01*x_11),(x_01*x_11+x_01*x_10 + x_00*x_11)]. The
method use the matrix of quadratic form p, n below code that variable is
p_matix. My problem is when I want extend the number pp and nvars to values
for example 8 and 8. Then my question is: How I will be able to accelerate
that code?
p = 2
n = 4 # num vars
K.<t> = GF(2**p, name='t')
VS = K.vector_space()
vars_GF = [var("X%d"%i) for i in range(n)]
#variable for formulas: addition and multiplication
vars_gf_X_ = [var("fX_%d"%i) for i in range(p)]
vars_gf_Y_ = [var("fY_%d"%i) for i in range(p)]
vars_gf_Z_ = [var("fZ_%d"%i) for i in range(p)]
#variables of components of variables vars_GF
vars_gf_X = [[var("x_%d_%d"%(i,k)) for k in range(p)] for i in range(n)]
P=PolynomialRing(K,names=vars_GF)
A = companion_matrix(K.modulus())
sum = 0
sum1 = 0
sum2= 0
for i in range(p):
sum = sum + (A^i)*vars_gf_X_[i]
sum1 = sum1 + (A^i)*vars_gf_Y_[i]
sum2 = sum2 + (A^i)*vars_gf_Z_[i]
#calculate formula to adtion and multiplication with the components of
the vector space GF(2^p)
Msum = sum + sum1
Mmul = sum*sum1*sum2
psum0 = (Msum)[0,0]
psum1 = (Msum)[0,1]
pmul0 = (Mmul)[0,0]
pmul1 = (Mmul)[0,1]
p_matrix = random_matrix(K,n,n)
#saving the element 0,0 of matrix
s0 =
pmul0(VS(p_matrix[0,0])[0],VS(p_matrix[0,0])[1],vars_gf_X[0][0],vars_gf_X[0][1],vars_gf_X[0][0],vars_gf_X[0][1])
s1 =
pmul1(VS(p_matrix[0,0])[0],VS(p_matrix[0,0])[1],vars_gf_X[0][0],vars_gf_X[0][1],vars_gf_X[0][0],vars_gf_X[0][1])
#sum of each monomial
for i in range(n):
for j in range(i+1):
if i != 0 or j !=0:
s0 =
psum0(s0,pmul0(VS(p_matrix[i,j])[0],VS(p_matrix[i,j])[1],vars_gf_X[i][0],vars_gf_X[i][1],vars_gf_X[j][0],vars_gf_X[j][1]))
s1 =
psum1(s1,pmul1(VS(p_matrix[i,j])[0],VS(p_matrix[i,j])[1],vars_gf_X[i][0],vars_gf_X[i][1],vars_gf_X[j][0],vars_gf_X[j][1]))
print s0
print s1
[1]: http://www.gregorybard.com/papers/extension_fields.pdf
--
---------------------------------------------------------------------
MSc. Juan del Carmen Grados Vásquez
Laboratório Nacional de Computação Científica
Tel: +55 21 97633 3228
(http://www.lncc.br/)
http://juaninf.blogspot.com
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