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Miller-Rabin Primality Test

06.24.2007
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        The Miller-Rabin primality test.

MillerRabin(n, s = 1000) -> bool Checks whether n is prime or not. This is an extremley fast algorithm designed to test <strong>very large</strong> numbers.

s is the number of tests to perform. The chance that Rabin-Miller is mistaken about a number (i.e. thinks it's prime, but it's not) is 2^(-s). So, a value of 50 for s is more than enough for any imaginable goal (2^(-50) is 8.8817841970012523e-16).

import sys
import random

def toBinary(n):
  r = []
  while (n > 0):
    r.append(n % 2)
    n = n / 2
  return r

def test(a, n):
  """
  test(a, n) -> bool Tests whether n is complex.

  Returns:
    - True, if n is complex.
    - False, if n is probably prime.
  """
  b = toBinary(n - 1)
  d = 1
  for i in xrange(len(b) - 1, -1, -1):
    x = d
    d = (d * d) % n
    if d == 1 and x != 1 and x != n - 1:
      return True # Complex
    if b[i] == 1:
      d = (d * a) % n
  if d != 1:
    return True # Complex
  return False # Prime

def MillerRabin(n, s = 50):
  """
    MillerRabin(n, s = 1000) -> bool Checks whether n is prime or not

    Returns:
      - True, if n is probably prime.
      - False, if n is complex.
  """
  for j in xrange(1, s + 1):
    a = random.randint(1, n - 1)
    if (test(a, n)):
      return False # n is complex
  return True # n is prime

def main(argv):
  print MillerRabin(int(argv[0]), int(argv[1]))

if __name__ == "__main__":
  main(sys.argv[1:])