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)