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138 lines
4.1 KiB
Python
138 lines
4.1 KiB
Python
"""Created by Nathan Damon, @bizzfitch on github
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>>> test_miller_rabin()
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"""
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def miller_rabin(n, allow_probable=False):
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"""Deterministic Miller-Rabin algorithm for primes ~< 3.32e24.
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Uses numerical analysis results to return whether or not the passed number
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is prime. If the passed number is above the upper limit, and
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allow_probable is True, then a return value of True indicates that n is
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probably prime. This test does not allow False negatives- a return value
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of False is ALWAYS composite.
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Parameters
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----------
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n : int
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The integer to be tested. Since we usually care if a number is prime,
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n < 2 returns False instead of raising a ValueError.
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allow_probable: bool, default False
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Whether or not to test n above the upper bound of the deterministic test.
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Raises
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------
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ValueError
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Reference
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---------
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https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
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"""
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if n == 2:
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return True
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if not n % 2 or n < 2:
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return False
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if n > 5 and n % 10 not in (1, 3, 7, 9): # can quickly check last digit
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return False
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if n > 3_317_044_064_679_887_385_961_981 and not allow_probable:
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raise ValueError(
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"Warning: upper bound of deterministic test is exceeded. "
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"Pass allow_probable=True to allow probabilistic test. "
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"A return value of True indicates a probable prime."
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)
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# array bounds provided by analysis
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bounds = [
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2_047,
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1_373_653,
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25_326_001,
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3_215_031_751,
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2_152_302_898_747,
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3_474_749_660_383,
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341_550_071_728_321,
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1,
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3_825_123_056_546_413_051,
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1,
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1,
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318_665_857_834_031_151_167_461,
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3_317_044_064_679_887_385_961_981,
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]
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primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
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for idx, _p in enumerate(bounds, 1):
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if n < _p:
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# then we have our last prime to check
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plist = primes[:idx]
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break
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d, s = n - 1, 0
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# break up n -1 into a power of 2 (s) and
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# remaining odd component
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# essentially, solve for d * 2 ** s == n - 1
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while d % 2 == 0:
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d //= 2
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s += 1
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for prime in plist:
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pr = False
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for r in range(s):
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m = pow(prime, d * 2 ** r, n)
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# see article for analysis explanation for m
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if (r == 0 and m == 1) or ((m + 1) % n == 0):
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pr = True
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# this loop will not determine compositeness
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break
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if pr:
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continue
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# if pr is False, then the above loop never evaluated to true,
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# and the n MUST be composite
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return False
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return True
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def test_miller_rabin():
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"""Testing a nontrivial (ends in 1, 3, 7, 9) composite
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and a prime in each range.
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"""
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assert not miller_rabin(561)
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assert miller_rabin(563)
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# 2047
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assert not miller_rabin(838_201)
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assert miller_rabin(838_207)
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# 1_373_653
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assert not miller_rabin(17_316_001)
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assert miller_rabin(17_316_017)
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# 25_326_001
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assert not miller_rabin(3_078_386_641)
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assert miller_rabin(3_078_386_653)
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# 3_215_031_751
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assert not miller_rabin(1_713_045_574_801)
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assert miller_rabin(1_713_045_574_819)
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# 2_152_302_898_747
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assert not miller_rabin(2_779_799_728_307)
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assert miller_rabin(2_779_799_728_327)
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# 3_474_749_660_383
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assert not miller_rabin(113_850_023_909_441)
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assert miller_rabin(113_850_023_909_527)
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# 341_550_071_728_321
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assert not miller_rabin(1_275_041_018_848_804_351)
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assert miller_rabin(1_275_041_018_848_804_391)
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# 3_825_123_056_546_413_051
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assert not miller_rabin(79_666_464_458_507_787_791_867)
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assert miller_rabin(79_666_464_458_507_787_791_951)
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# 318_665_857_834_031_151_167_461
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assert not miller_rabin(552_840_677_446_647_897_660_333)
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assert miller_rabin(552_840_677_446_647_897_660_359)
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# 3_317_044_064_679_887_385_961_981
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# upper limit for probabilistic test
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if __name__ == "__main__":
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test_miller_rabin()
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