def baillie_psw(candidate):
"""Perform the Baillie-PSW probabilistic primality test on candidate"""
# Check divisibility by a short list of primes less than 50
for known_prime in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]:
if candidate == known_prime:
return True
elif candidate % known_prime == 0:
return False
# Now perform the Miller-Rabin primality test base 2
if not miller_rabin_base_2(candidate):
return False
# Check that the number isn't a square number, as this will throw out
# calculating the correct value of D later on (and means we have a
# composite number)
# the slight ugliness is from having to deal with floating point numbers
if int(sqrt(candidate) + 0.5) ** 2 == candidate:
return False
# Finally perform the Lucas primality test
D = D_chooser(candidate)
if not lucas_pp(candidate, D, 1, (1-D)/4):
return False
# You've probably got a prime!
return True
def baillie_psw(candidate):
"""Perform the Baillie-PSW probabilistic primality test on candidate"""
# Check divisibility by a short list of primes less than 50
for known_prime in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]:
if candidate == known_prime:
return True
elif candidate % known_prime == 0:
return False
# Now perform the Miller-Rabin primality test base 2
if not miller_rabin_base_2(candidate):
return False
# These two composite numbers are Miller-Rabin pseudoprimes and cause
# an infinite loop in the below implementation of the Lucas Test
if candidate in set([1194649, 12327121]):
return False
# Finally perform the Lucas primality test
D = D_chooser(candidate)
if not lucas_pp(candidate, D, 1, (1-D)/4):
return False
# You've probably got a prime!
return True
def baillie_psw(candidate):
"""Perform the Baillie-PSW probabilistic primality test on candidate"""
# Check divisibility by a short list of primes less than 50
for known_prime in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 33, 37, 41, 43, 47]:
if candidate == known_prime:
return True
elif candidate % known_prime == 0:
return False
# Now perform the Miller-Rabin primality test base 2
if not miller_rabin_base_2(candidate):
return False
# Finally perform the Lucas primality test
D = D_chooser(candidate)
if not lucas_pp(candidate, D, 1, (1-D)/4):
return False
# You've probably got a prime!
return True
def baillie_psw(candidate):
"""Perform the Baillie-PSW probabilistic primality test on candidate"""
for known_prime in [
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
]:
if candidate == known_prime:
return True
elif candidate % known_prime == 0:
return False
if not miller_rabin_base_2(candidate):
return False
if int(candidate**0.5 + 0.5)**2 == candidate:
return False
D = D_chooser(candidate)
if not lucas_pp(candidate, D, 1, (1 - D) // 4):
return False
return True
def testMillerRabinOnEvens():
for x in range(30, 10000000, 2):
assert not miller_rabin_base_2(x)
def testMillerRabinFailsForStrongPseudoprimes():
for x in [2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141]:
assert miller_rabin_base_2(x)
def testMillerRabinFailsForNonPrimes():
for x in [
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28,
30
]:
assert not miller_rabin_base_2(x)
def testMillerRabinPassesForLargerPrimes():
"""Largest primes for which the base-2 MR test definitely pass"""
for x in [2003, 2011, 2017, 2029, 2039]:
assert miller_rabin_base_2(x)
def testMillerRabinPassesForSmallPrimes():
for x in [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]:
assert miller_rabin_base_2(x)
def testMillerRabinOnEvens():
for x in range(30, 10000000, 2):
assert not miller_rabin_base_2(x)
def testMillerRabinFailsForStrongPseudoprimes():
for x in [2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141]:
assert miller_rabin_base_2(x)
def testMillerRabinFailsForNonPrimes():
for x in [4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30]:
assert not miller_rabin_base_2(x)
def testMillerRabinPassesForLargerPrimes():
"""These are the largest primes for which the base-2 MR test will definitely pass"""
for x in [2003, 2011, 2017, 2029, 2039]:
assert miller_rabin_base_2(x)
def testMillerRabinPassesForSmallPrimes():
for x in [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]:
assert miller_rabin_base_2(x)
