miller.py

# Python3 program Miller-Rabin primality test
import random


# Utility function to do
# modular exponentiation.
# It returns (x^y) % p
def power(x, y, p):
    # Initialize result
    res = 1;

    # Update x if it is more than or
    # equal to p
    x = x % p;
    while (y > 0):

        # If y is odd, multiply
        # x with result
        if (y & 1):
            res = (res * x) % p;

        # y must be even now
        y = y >> 1;  # y = y/2
        x = (x * x) % p;

    return res;


# This function is called
# for all k trials. It returns
# false if n is composite and
# returns false if n is
# probably prime. d is an odd
# number such that d*2<sup>r</sup> = n-1
# for some r >= 1
def miillerTest(d, n):
    # Pick a random number in [2..n-2]
    # Corner cases make sure that n > 4
    a = 2 + random.randint(1, n - 4);
    print('Choose a random integer a =', a)
    # Compute a^d % n
    x = power(a, d, n);
    print('Compute y = a^r (mod n) = ', x)

    if (x == 1 or x == n - 1):
        return True;

    # Keep squaring x while one
    # of the following doesn't
    # happen
    # (i) d does not reach n-1
    # (ii) (x^2) % n is not 1
    # (iii) (x^2) % n is not n-1
    while (d != n - 1):
        print('d is ', d)
        x = (x * x) % n;
        print('x=', x)
        d *= 2;

        if (x == 1):
            return False;
        if (x == n - 1):
            return True;

    # Return composite
    return False;


# It returns false if n is
# composite and returns true if n
# is probably prime. k is an
# input parameter that determines
# accuracy level. Higher value of
# k indicates more accuracy.
def isPrime(n, k):
    # Corner cases
    if (n <= 1 or n == 4):
        return False;
    if (n <= 3):
        return True;

    # Find r such that n =
    # 2^d * r + 1 for some r >= 1
    d = n - 1;
    while (d % 2 == 0):
        d //= 2;

    # Iterate given number of 'k' times
    for i in range(k):
        if (miillerTest(d, n) == False):
            return False;

    return True;


# Driver Code
# Number of iterations
k = 4;

print("All primes smaller than 100: ");
for n in range(1, 100):
    if (isPrime(n, k)):
        print(n, end=" ");
print()
print()
print(miillerTest(5,33))