"""
This module holds a fast Factorization class.
"""
from random import randint


class Factorization:
    """
    Simple module to factorize large numbers really quickly.
    """
    @classmethod
    def factorize(cls, pq):
        """
        Factorizes the given large integer.

        :param pq: the prime pair pq.
        :return: a tuple containing the two factors p and q.
        """
        if pq % 2 == 0:
            return 2, pq // 2

        y, c, m = randint(1, pq - 1), randint(1, pq - 1), randint(1, pq - 1)
        g = r = q = 1
        x = ys = 0

        while g == 1:
            x = y
            for i in range(r):
                y = (pow(y, 2, pq) + c) % pq

            k = 0
            while k < r and g == 1:
                ys = y
                for i in range(min(m, r - k)):
                    y = (pow(y, 2, pq) + c) % pq
                    q = q * (abs(x - y)) % pq

                g = cls.gcd(q, pq)
                k += m

            r *= 2

        if g == pq:
            while True:
                ys = (pow(ys, 2, pq) + c) % pq
                g = cls.gcd(abs(x - ys), pq)
                if g > 1:
                    break

        return g, pq // g

    @staticmethod
    def gcd(a, b):
        """
        Calculates the Greatest Common Divisor.

        :param a: the first number.
        :param b: the second number.
        :return: GCD(a, b)
        """
        while b:
            a, b = b, a % b

        return a