RabinMiller算法
# -*- coding:utf-8 -*-
"""
@Author: [email protected]
@Date: 2017-10-30
https://zh.wikipedia.org/zh-cn/%E7%B1%B3%E5%8B%92-%E6%8B%89%E5%AE%BE%E6%A3%80%E9%AA%8C
http://blog.miskcoo.com/2014/07/miller-rabin-primality-test
https://rosettacode.org/wiki/Miller%E2%80%93Rabin_primality_test
https://inventwithpython.com/rabinMiller.py
# 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 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 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
"""
# Primality Testing with the Rabin-Miller Algorithm
# http://inventwithpython.com/hacking (BSD Licensed)
import sys
import time
import random
import logging
import datetime
LOGGING_LEVEL = logging.DEBUG
def logging_config():
# log_format = "%(asctime)s [line: %(lineno)d] - %(levelname)s - %(message)s"
log_format = "[line: %(lineno)d] - %(levelname)s - %(message)s"
logging.basicConfig(level=LOGGING_LEVEL, format=log_format)
# https://en.wikibooks.org/wiki/Algorithm_Implementation/Mathematics/Primality_Testing
def is_probable_prime(n, k=7):
"""use Rabin-Miller algorithm to return True (n is probably prime)
or False (n is definitely composite)"""
if n < 6: # assuming n >= 0 in all cases... shortcut small cases here
return [False, False, True, True, False, True][n]
elif n & 1 == 0: # should be faster than n % 2
return False
else:
s, d = 0, n - 1
while d & 1 == 0:
s, d = s + 1, d >> 1
# Use random.randint(2, n-2) for very large numbers
for a in random.sample(range(2, min(n - 2, sys.maxsize)), min(n - 4, k)):
x = pow(a, d, n)
if x != 1 and x + 1 != n:
for r in range(1, s):
x = pow(x, 2, n)
if x == 1:
return False # composite for sure
elif x == n - 1:
a = 0 # so we know loop didn't continue to end
break # could be strong liar, try another a
if a:
return False # composite if we reached end of this loop
return True # probably prime if reached end of outer loop
def rabinMiller(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 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
def main():
prime_list = [955963, 955967, 955987, 955991, 955993, 956003, 956051, 956057, 956083, 956107, 956113, 956119,
956143, 956147, 956177, 956231, 956237]
print(generateLargePrime())
print(isPrime(8148143905337944345073782753637512644205873574663745002544561797417525199053346824733589523))
print(is_probable_prime(8148143905337944345073782753637512644205873574663745002544561797417525199053346824733589523))
if __name__ == "__main__":
print("Script start execution at %s\n\n" % str(datetime.datetime.now()))
time_start = time.time()
logging_config()
main()
print("\n\nScript end execution at %s" % str(datetime.datetime.now()))
print("Total Elapsed Time: %s seconds\n" % (time.time() - time_start))