Adding deterministic miller rabin primality test (#1453)

* Adding deterministic miller rabin primality test

* Moved to ciphers directory and renamed for clarity. Changed docstring to add test

Author: bizzfitch

ciphers/deterministic_miller_rabin.py

@@ -0,0 +1,135 @@
+"""Created by Nathan Damon, @bizzfitch on github
+>>> test_miller_rabin()
+"""
+
+
+def miller_rabin(n, allow_probable=False):
+    """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():
+    """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()