text: # Primality Testing with the Rabin-Miller Algorithm
  # http://inventwithpython.com/hacking (BSD Licensed)
  
  import random
  
  
  def rabinMiller(num):
      # Returns True if num is a prime number.
  
      s = num - 1
      t = 0
      while s % 2 == 0:
          # keep halving s until 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 num'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 isPrime(num):
      # Return True if num is a prime number. This function does a quicker
      # prime number check before calling rabinMiller().
  
      if (num < 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.
      lowPrimes = [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 num in lowPrimes:
          return True
  
      # See if any of the low prime numbers can divide num
      for prime in lowPrimes:
          if (num % prime == 0):
              return False
  
      # If all else fails, call rabinMiller() to determine if num is a prime.
      return rabinMiller(num)
  
  
  def generateLargePrime(keysize=1024):
      # Return a random prime number of keysize bits in size.
      while True:
          num = random.randrange(2**(keysize-1), 2**(keysize))
          if isPrime(num):
              return num