A bit of cleanup

Move some helper files to other folders.

Actually remove the .sage files from the main directory, as I wrote I
would in the readme.

generate_initial_irreducible_polynomials.sage

@@ -1,12 +0,0 @@
-F = GF(2)
-PR = PolynomialRing(F, 'x')
-
-max_k = 31
-whitespace = max_k+1
-
-for i in range(1, max_k+1):
-    poly_int = '0b'+ ''.join(map(str, PR.irreducible_element(i).coefficients(sparse=False)[::-1]))
-    poly_int += ','
-    poly_int += ' '*(max_k-i)
-    print(f'    {poly_int}// irreducible polynomial of degree {i}')
-

generate_random_irreducible_polynomial.sage

@@ -1,96 +0,0 @@
-k = 31
-F = GF(2)
-E = GF(2^k)
-PR = PolynomialRing(F, 'x')
-p = PR.irreducible_element(k)
-
-while(True):
-    gamma = E.random_element()
-    if gamma not in F:
-        break
-
-# gamma = PR('x^3+x')
-# print('0b'+ ''.join(map(str, gamma.coefficients(sparse=False)[::-1])))
-
-eqs = [PR(gamma)^i % p for i in range(k+1)]
-for eq in eqs:
-    print(eq)
-print()
-
-eqs = list(map(lambda eq: eq.coefficients(sparse=False) + [0]*(k-eq.degree()-1), eqs))
-for eq in eqs:
-    print(eq)
-print()
-eqs = matrix(eqs[::-1])
-print(eqs)
-print()
-eqs = eqs.transpose()
-print(eqs)
-print()
-
-rref = eqs.rref()
-print(rref)
-irrpol = PR(rref[:,-1].list() + [1])
-print(irrpol)
-print(irrpol.is_irreducible())
-print()
-
-# manual rref
-def calc_rref_Z2(A, m, n):
-    """Calculate the reduced row echelon form of a (mxn)-matrix"""
-    A = copy(A)
-
-    # calculate row echelon form
-    # https://en.wikipedia.org/wiki/Gaussian_elimination#Pseudocode
-    h = 0 # Initialization of the pivot row
-    k = 0 # Initialization of the pivot column
-
-    while h < m and k < n:
-        # Find the k-th pivot:
-        i_max = -1
-        for i in range(h, m):
-            if A[i,k] == 1:
-                i_max = i
-                break
-
-        if i_max == -1:
-            # No pivot in this column, pass to next column
-            k += 1
-        else:
-            tmp = A[i_max, :]
-            A[i_max, :] = A[h, :]
-            A[h, :] = tmp
-
-
-            # Do for all rows below pivot:
-            for i in range(h+1, m):
-                f = A[i][k];
-                if f == 0:
-                    continue;
-                A[i, k] = 0;
-                for j in range(k+1, n):
-                    A[i, j] = A[i, j] - A[h, j]
-
-            # Increase pivot row and column
-            h += 1
-            k += 1
-
-    # perform back substitution
-    for i in range(1, m):
-        for j in range(i):
-            if A[j, i] == 1:
-                A[j, :] -= A[i, :]
-
-    return A
-
-print('Manual RREF:\n')
-
-rref = calc_rref_Z2(eqs, k, k+1) # cols = rows+1, since we transposed the matrix
-print(rref)
-print()
-irrpol = PR(rref[:,-1].list() + [1])
-print(irrpol)
-print(irrpol.is_irreducible())
-print()
-
-print('0b'+ ''.join(map(str, irrpol.coefficients(sparse=False)[::-1])))

multiply_polynomials_modulo_polynomial.sage

@@ -1,68 +0,0 @@
-import math
-
-k = 31
-F = GF(2)
-var = 'x'
-PR = PolynomialRing(F, var)
-p = PR.irreducible_element(k)
-gamma = PR.random_element(k)
-
-exp = 10
-target = gamma^exp % p
-
-def polynomial_power_in_Z2(q, exp, p, var):
-    if exp == 0:
-        return 1
-
-    if q.degree() == p.degree():
-        q += p
-
-    res = q
-
-    for i in range(2, exp+1):
-        res = multiply_polynomials_in_Z2(res, q, p, var)
-
-    return res
-
-def polynomial_power_in_Z2_V2(q, exp, p, var):
-    if exp == 0:
-        return 1
-
-    if q.degree() == p.degree():
-        q += p
-
-    log2deg = math.floor(math.log2(exp)) + 1
-    arr = [0]*log2deg
-    arr[0] = q
-    for i in range(1, log2deg):
-        arr[i] = multiply_polynomials_in_Z2(arr[i-1], arr[i-1], p, var)
-
-    res = PR(1)
-    for i, b in enumerate(bin(exp)[:1:-1]):
-        if b == '1':
-            res = multiply_polynomials_in_Z2(res, arr[i], p, var)
-
-    return res
-
-
-def multiply_polynomials_in_Z2(q1, q2, p, var):
-    if q1.degree() > p.degree():
-        raise ValueError('Unsupported!')
-
-    if q1.degree() == p.degree():
-        q1 += p
-
-    arr = [0]*(q2.degree()+1)
-    arr[0] = q1
-    for i in range(1, q2.degree()+1):
-        arr[i] = arr[i-1]*PR(var)
-        if arr[i].degree() == p.degree():
-            arr[i] += p
-
-    return sum(arr[i] for i in q2.exponents())
-
-print(target)
-res = polynomial_power_in_Z2(gamma, exp, p, var)
-print(res)
-res = polynomial_power_in_Z2_V2(gamma, exp, p, var)
-print(res)