mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-12-21 18:50:15 +00:00
24d3cf8244
* The black formatter is no longer beta * pre-commit autoupdate * pre-commit autoupdate * Remove project_euler/problem_145 which is killing our CI tests * updating DIRECTORY.md Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
138 lines
4.1 KiB
Python
138 lines
4.1 KiB
Python
"""Created by Nathan Damon, @bizzfitch on github
|
|
>>> test_miller_rabin()
|
|
"""
|
|
|
|
|
|
def miller_rabin(n: int, allow_probable: bool = False) -> bool:
|
|
"""Deterministic Miller-Rabin algorithm for primes ~< 3.32e24.
|
|
|
|
Uses numerical analysis results to return whether or not the passed number
|
|
is prime. If the passed number is above the upper limit, and
|
|
allow_probable is True, then a return value of True indicates that n is
|
|
probably prime. This test does not allow False negatives- a return value
|
|
of False is ALWAYS composite.
|
|
|
|
Parameters
|
|
----------
|
|
n : int
|
|
The integer to be tested. Since we usually care if a number is prime,
|
|
n < 2 returns False instead of raising a ValueError.
|
|
allow_probable: bool, default False
|
|
Whether or not to test n above the upper bound of the deterministic test.
|
|
|
|
Raises
|
|
------
|
|
ValueError
|
|
|
|
Reference
|
|
---------
|
|
https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
|
|
"""
|
|
if n == 2:
|
|
return True
|
|
if not n % 2 or n < 2:
|
|
return False
|
|
if n > 5 and n % 10 not in (1, 3, 7, 9): # can quickly check last digit
|
|
return False
|
|
if n > 3_317_044_064_679_887_385_961_981 and not allow_probable:
|
|
raise ValueError(
|
|
"Warning: upper bound of deterministic test is exceeded. "
|
|
"Pass allow_probable=True to allow probabilistic test. "
|
|
"A return value of True indicates a probable prime."
|
|
)
|
|
# array bounds provided by analysis
|
|
bounds = [
|
|
2_047,
|
|
1_373_653,
|
|
25_326_001,
|
|
3_215_031_751,
|
|
2_152_302_898_747,
|
|
3_474_749_660_383,
|
|
341_550_071_728_321,
|
|
1,
|
|
3_825_123_056_546_413_051,
|
|
1,
|
|
1,
|
|
318_665_857_834_031_151_167_461,
|
|
3_317_044_064_679_887_385_961_981,
|
|
]
|
|
|
|
primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
|
|
for idx, _p in enumerate(bounds, 1):
|
|
if n < _p:
|
|
# then we have our last prime to check
|
|
plist = primes[:idx]
|
|
break
|
|
d, s = n - 1, 0
|
|
# break up n -1 into a power of 2 (s) and
|
|
# remaining odd component
|
|
# essentially, solve for d * 2 ** s == n - 1
|
|
while d % 2 == 0:
|
|
d //= 2
|
|
s += 1
|
|
for prime in plist:
|
|
pr = False
|
|
for r in range(s):
|
|
m = pow(prime, d * 2**r, n)
|
|
# see article for analysis explanation for m
|
|
if (r == 0 and m == 1) or ((m + 1) % n == 0):
|
|
pr = True
|
|
# this loop will not determine compositeness
|
|
break
|
|
if pr:
|
|
continue
|
|
# if pr is False, then the above loop never evaluated to true,
|
|
# and the n MUST be composite
|
|
return False
|
|
return True
|
|
|
|
|
|
def test_miller_rabin() -> None:
|
|
"""Testing a nontrivial (ends in 1, 3, 7, 9) composite
|
|
and a prime in each range.
|
|
"""
|
|
assert not miller_rabin(561)
|
|
assert miller_rabin(563)
|
|
# 2047
|
|
|
|
assert not miller_rabin(838_201)
|
|
assert miller_rabin(838_207)
|
|
# 1_373_653
|
|
|
|
assert not miller_rabin(17_316_001)
|
|
assert miller_rabin(17_316_017)
|
|
# 25_326_001
|
|
|
|
assert not miller_rabin(3_078_386_641)
|
|
assert miller_rabin(3_078_386_653)
|
|
# 3_215_031_751
|
|
|
|
assert not miller_rabin(1_713_045_574_801)
|
|
assert miller_rabin(1_713_045_574_819)
|
|
# 2_152_302_898_747
|
|
|
|
assert not miller_rabin(2_779_799_728_307)
|
|
assert miller_rabin(2_779_799_728_327)
|
|
# 3_474_749_660_383
|
|
|
|
assert not miller_rabin(113_850_023_909_441)
|
|
assert miller_rabin(113_850_023_909_527)
|
|
# 341_550_071_728_321
|
|
|
|
assert not miller_rabin(1_275_041_018_848_804_351)
|
|
assert miller_rabin(1_275_041_018_848_804_391)
|
|
# 3_825_123_056_546_413_051
|
|
|
|
assert not miller_rabin(79_666_464_458_507_787_791_867)
|
|
assert miller_rabin(79_666_464_458_507_787_791_951)
|
|
# 318_665_857_834_031_151_167_461
|
|
|
|
assert not miller_rabin(552_840_677_446_647_897_660_333)
|
|
assert miller_rabin(552_840_677_446_647_897_660_359)
|
|
# 3_317_044_064_679_887_385_961_981
|
|
# upper limit for probabilistic test
|
|
|
|
|
|
if __name__ == "__main__":
|
|
test_miller_rabin()
|