97 lines
2.2 KiB
Python
97 lines
2.2 KiB
Python
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k = 31
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F = GF(2)
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E = GF(2^k)
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PR = PolynomialRing(F, 'x')
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p = PR.irreducible_element(k)
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while(True):
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gamma = E.random_element()
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if gamma not in F:
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break
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# gamma = PR('x^3+x')
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# print('0b'+ ''.join(map(str, gamma.coefficients(sparse=False)[::-1])))
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eqs = [PR(gamma)^i % p for i in range(k+1)]
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for eq in eqs:
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print(eq)
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print()
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eqs = list(map(lambda eq: eq.coefficients(sparse=False) + [0]*(k-eq.degree()-1), eqs))
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for eq in eqs:
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print(eq)
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print()
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eqs = matrix(eqs[::-1])
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print(eqs)
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print()
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eqs = eqs.transpose()
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print(eqs)
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print()
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rref = eqs.rref()
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print(rref)
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irrpol = PR(rref[:,-1].list() + [1])
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print(irrpol)
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print(irrpol.is_irreducible())
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print()
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# manual rref
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def calc_rref_Z2(A, m, n):
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"""Calculate the reduced row echelon form of a (mxn)-matrix"""
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A = copy(A)
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# calculate row echelon form
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# https://en.wikipedia.org/wiki/Gaussian_elimination#Pseudocode
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h = 0 # Initialization of the pivot row
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k = 0 # Initialization of the pivot column
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while h < m and k < n:
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# Find the k-th pivot:
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i_max = -1
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for i in range(h, m):
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if A[i,k] == 1:
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i_max = i
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break
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if i_max == -1:
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# No pivot in this column, pass to next column
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k += 1
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else:
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tmp = A[i_max, :]
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A[i_max, :] = A[h, :]
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A[h, :] = tmp
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# Do for all rows below pivot:
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for i in range(h+1, m):
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f = A[i][k];
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if f == 0:
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continue;
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A[i, k] = 0;
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for j in range(k+1, n):
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A[i, j] = A[i, j] - A[h, j]
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# Increase pivot row and column
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h += 1
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k += 1
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# perform back substitution
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for i in range(1, m):
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for j in range(i):
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if A[j, i] == 1:
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A[j, :] -= A[i, :]
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return A
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print('Manual RREF:\n')
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rref = calc_rref_Z2(eqs, k, k+1) # cols = rows+1, since we transposed the matrix
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print(rref)
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print()
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irrpol = PR(rref[:,-1].list() + [1])
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print(irrpol)
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print(irrpol.is_irreducible())
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print()
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print('0b'+ ''.join(map(str, irrpol.coefficients(sparse=False)[::-1])))
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