primality_testing_with_rabin_miller.py

# Primality Testing with the Rabin-Miller Algorithm


import random


def rabin_miller(num):
    # Returns True if num is a prime number.
    s = num - 1
    t = 0
    while s % 2 == 0:
        # keep halving s while it is even (and use t
        # to count how many times we halve s)
        s = s // 2
        t += 1
        for trials in range(5):
            # try to falsify number's primality 5 times
            a = random.randrange(2, num - 1)

            v = pow(a, s, num)
            if v != 1:  # this test does not apply if v is 1.
                i = 0
                while v != (num - 1):
                    if i == t - 1:
                        return False
                    else:
                        i = i + 1
                        v = (v ** 2) % num
                        return True


def is_prime(number):
    # Return True if num is a prime number. This function does a quicker
    # prime number check before calling rabinMiller().
    if number < 2:
        return False  # 0, 1, and negative numbers are not prime

    # About 1/3 of the time we can quickly determine if num is not prime
    # by dividing by the first few dozen prime numbers. This is quicker
    # than rabinMiller(), but unlike rabinMiller() is not guaranteed to
    # prove that a number is prime.
    low_primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101,
                  103,
                  107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211,
                  223,
                  227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337,
                  347,
                  349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461,
                  463,
                  467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601,
                  607,
                  613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739,
                  743,
                  751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881,
                  883,
                  887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997]
    if number in low_primes:
        return True

    # See if any of the low prime numbers can divide num
    for prime in low_primes:
        if number % prime == 0:
            return False

    # If all else fails, call rabinMiller() to determine if num is a prime.
    return rabin_miller(number)


def generate_large_prime(key_size=1024):
    # Return a random prime number of key_size bits in size.
    while True:
        num = random.randrange(2 ** (key_size - 1), 2 ** (key_size))
        if is_prime(num):
            return num