def mr(n, bases):
"""Perform a Miller-Rabin strong pseudoprime test on n using a
given list of bases/witnesses.
References
==========
- Richard Crandall & Carl Pomerance (2005), "Prime Numbers:
A Computational Perspective", Springer, 2nd edition, 135-138
A list of thresholds and the bases they require are here:
http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Deterministic_variants_of_the_test
Examples
========
>>> from sympy.ntheory.primetest import mr
>>> mr(1373651, [2, 3])
False
>>> mr(479001599, [31, 73])
True
"""
from sympy.ntheory.factor_ import trailing
from sympy.polys.domains import ZZ
n = as_int(n)
if n < 2:
return False
# remove powers of 2 from n-1 (= t * 2**s)
s = trailing(n - 1)
t = n >> s
for base in bases:
# Bases >= n are wrapped, bases < 2 are invalid
if base >= n:
base %= n
if base >= 2:
base = ZZ(base)
if not _test(n, base, s, t):
return False
return True
def _info(bases):
"""
Analyze the list of bases, reporting the number of 'j-loops' that
will be required if this list is passed to _test (stot) and the primes
that must be cleared by a previous test.
This info tag should then be appended to any new mr_safe line
that is added so someone can easily see whether that line satisfies
the requirements of mr_safe (see docstring there for details).
"""
from sympy.ntheory.factor_ import factorint, trailing
factors = []
tot = 0
for b in bases:
tot += trailing(b - 1)
f = factorint(b)
factors.extend(f)
factors = sorted(set(factors))
bases = sorted(set(bases))
if bases == factors:
factors = '== bases'
else:
factors = str(factors)
return ' # %s stot = %s clear %s' % tuple(
[str(x).replace('L', '') for x in (list(bases), tot, factors)])
def _info(bases):
"""
Analyze the list of bases, reporting the number of 'j-loops' that
will be required if this list is passed to _test (stot) and the primes
that must be cleared by a previous test.
This info tag should then be appended to any new mr_safe line
that is added so someone can easily see whether that line satisfies
the requirements of mr_safe (see docstring there for details).
"""
from sympy.ntheory.factor_ import factorint, trailing
factors = []
tot = 0
for b in bases:
tot += trailing(b - 1)
f = factorint(b)
factors.extend(f)
factors = sorted(set(factors))
bases = sorted(set(bases))
if bases == factors:
factors = '== bases'
else:
factors = str(factors)
return ' # %s stot = %s clear %s' % tuple(
[str(x).replace('L', '') for x in (list(bases), tot, factors)])
def is_strong_lucas_prp(n):
"""Strong Lucas compositeness test with Selfridge parameters. Returns
False if n is definitely composite, and True if n is a strong Lucas
probable prime.
This is often used in combination with the Miller-Rabin test, and
in particular, when combined with M-R base 2 creates the strong BPSW test.
References
==========
- "Lucas Pseudoprimes", Baillie and Wagstaff, 1980.
http://mpqs.free.fr/LucasPseudoprimes.pdf
- OEIS A217255: Strong Lucas Pseudoprimes
https://oeis.org/A217255
- https://en.wikipedia.org/wiki/Lucas_pseudoprime
- https://en.wikipedia.org/wiki/Baillie-PSW_primality_test
Examples
========
>>> from sympy.ntheory.primetest import isprime, is_strong_lucas_prp
>>> for i in range(20000):
... if is_strong_lucas_prp(i) and not isprime(i):
... print(i)
5459
5777
10877
16109
18971
"""
from sympy.ntheory.factor_ import trailing
n = as_int(n)
if n == 2:
return True
if n < 2 or (n % 2) == 0:
return False
if is_square(n, False):
return False
D, P, Q = _lucas_selfridge_params(n)
if D == 0:
return False
# remove powers of 2 from n+1 (= k * 2**s)
s = trailing(n + 1)
k = (n + 1) >> s
U, V, Qk = _lucas_sequence(n, P, Q, k)
if U == 0 or V == 0:
return True
for r in range(1, s):
V = (V * V - 2 * Qk) % n
if V == 0:
return True
Qk = pow(Qk, 2, n)
return False
def is_strong_lucas_prp(n):
"""Strong Lucas compositeness test with Selfridge parameters. Returns
False if n is definitely composite, and True if n is a strong Lucas
probable prime.
This is often used in combination with the Miller-Rabin test, and
in particular, when combined with M-R base 2 creates the strong BPSW test.
References
==========
- "Lucas Pseudoprimes", Baillie and Wagstaff, 1980.
http://mpqs.free.fr/LucasPseudoprimes.pdf
- OEIS A217255: Strong Lucas Pseudoprimes
https://oeis.org/A217255
- https://en.wikipedia.org/wiki/Lucas_pseudoprime
- https://en.wikipedia.org/wiki/Baillie-PSW_primality_test
Examples
========
>>> from sympy.ntheory.primetest import isprime, is_strong_lucas_prp
>>> for i in range(20000):
... if is_strong_lucas_prp(i) and not isprime(i):
... print(i)
5459
5777
10877
16109
18971
"""
from sympy.ntheory.factor_ import trailing
n = as_int(n)
if n == 2:
return True
if n < 2 or (n % 2) == 0:
return False
if is_square(n, False):
return False
D, P, Q = _lucas_selfridge_params(n)
if D == 0:
return False
# remove powers of 2 from n+1 (= k * 2**s)
s = trailing(n + 1)
k = (n + 1) >> s
U, V, Qk = _lucas_sequence(n, P, Q, k)
if U == 0 or V == 0:
return True
for r in range(1, s):
V = (V * V - 2 * Qk) % n
if V == 0:
return True
Qk = pow(Qk, 2, n)
return False
def _test(n, base):
from sympy.ntheory.factor_ import trailing
n = int(n)
if n < 2:
return False
s = trailing(n - 1)
t = n >> s
#Ферма
b = pow(base, t, n)
if b == 1 or b == n - 1:
return True
else:
for j in range(1, s):
b = (b**2) % n
if b == n - 1:
return True
return False
def is_euler_pseudoprime(n, b):
"""Return True if n is prime or an Euler pseudoprime to base b, else False."""
from sympy.ntheory.factor_ import trailing
if not mr(n, [b]):
return False
n = as_int(n)
r = n - 1
c = pow(b, r >> trailing(r), n)
if c == 1:
return True
while True:
if c == n - 1:
return True
c = pow(c, 2, n)
if c == 1:
return False
def is_euler_pseudoprime(n, b):
"""Returns True if n is prime or an Euler pseudoprime to base b, else False.
Euler Pseudoprime : In arithmetic, an odd composite integer n is called an
euler pseudoprime to base a, if a and n are coprime and satisfy the modular
arithmetic congruence relation :
a ^ (n-1)/2 = + 1(mod n) or
a ^ (n-1)/2 = - 1(mod n)
(where mod refers to the modulo operation).
Examples
========
>>> from sympy.ntheory.primetest import is_euler_pseudoprime
>>> is_euler_pseudoprime(2, 5)
True
References
==========
.. [1] https://en.wikipedia.org/wiki/Euler_pseudoprime
"""
from sympy.ntheory.factor_ import trailing
if not mr(n, [b]):
return False
n = as_int(n)
r = n - 1
c = pow(b, r >> trailing(r), n)
if c == 1:
return True
while True:
if c == n - 1:
return True
c = pow(c, 2, n)
if c == 1:
return False
def is_euler_pseudoprime(n, b):
"""Returns True if n is prime or an Euler pseudoprime to base b, else False.
Euler Pseudoprime : In arithmetic, an odd composite integer n is called an
euler pseudoprime to base a, if a and n are coprime and satisfy the modular
arithmetic congruence relation :
a ^ (n-1)/2 = + 1(mod n) or
a ^ (n-1)/2 = - 1(mod n)
(where mod refers to the modulo operation).
Examples
========
>>> from sympy.ntheory.primetest import is_euler_pseudoprime
>>> is_euler_pseudoprime(2, 5)
True
References
==========
.. [1] https://en.wikipedia.org/wiki/Euler_pseudoprime
"""
from sympy.ntheory.factor_ import trailing
if not mr(n, [b]):
return False
n = as_int(n)
r = n - 1
c = pow(b, r >> trailing(r), n)
if c == 1:
return True
while True:
if c == n - 1:
return True
c = pow(c, 2, n)
if c == 1:
return False
def mr(n, bases):
"""Perform a Miller-Rabin strong pseudoprime test on n using a
given list of bases/witnesses.
References
==========
- Richard Crandall & Carl Pomerance (2005), "Prime Numbers:
A Computational Perspective", Springer, 2nd edition, 135-138
A list of thresholds and the bases they require are here:
https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Deterministic_variants_of_the_test
Examples
========
>>> from sympy.ntheory.primetest import mr
>>> mr(1373651, [2, 3])
False
>>> mr(479001599, [31, 73])
True
"""
from sympy.ntheory.factor_ import trailing
from sympy.polys.domains import ZZ
n = as_int(n)
if n < 2:
return False
# remove powers of 2 from n-1 (= t * 2**s)
s = trailing(n - 1)
t = n >> s
for base in bases:
# Bases >= n are wrapped, bases < 2 are invalid
if base >= n:
base %= n
if base >= 2:
base = ZZ(base)
if not _test(n, base, s, t):
return False
return True
def _test(n, base):
"""Miller-Rabin strong pseudoprime test for one base.
Return False if n is definitely composite, True if n is
probably prime, with a probability greater than 3/4.
"""
from sympy.ntheory.factor_ import trailing
n = int(n)
if n < 2:
return False
# remove powers of 2 from n (= t * 2**s)
s = trailing(n - 1)
t = n >> s
# do the Fermat test
b = pow(base, t, n)
if b == 1 or b == n - 1:
return True
else:
for j in xrange(1, s):
b = (b**2) % n
if b == n - 1:
return True
return False
def _test(n, base):
"""Miller-Rabin strong pseudoprime test for one base.
Return False if n is definitely composite, True if n is
probably prime, with a probability greater than 3/4.
"""
from sympy.ntheory.factor_ import trailing
n = int(n)
if n < 2:
return False
# remove powers of 2 from n (= t * 2**s)
s = trailing(n - 1)
t = n >> s
# do the Fermat test
b = pow(base, t, n)
if b == 1 or b == n - 1:
return True
else:
for j in xrange(1, s):
b = (b**2) % n
if b == n - 1:
return True
return False
def is_extra_strong_lucas_prp(n):
"""Extra Strong Lucas compositeness test. Returns False if n is
definitely composite, and True if n is a "extra strong" Lucas probable
prime.
The parameters are selected using P = 3, Q = 1, then incrementing P until
(D|n) == -1. The test itself is as defined in Grantham 2000, from the
Mo and Jones preprint. The parameter selection and test are the same as
used in OEIS A217719, Perl's Math::Prime::Util, and the Lucas pseudoprime
page on Wikipedia.
With these parameters, there are no counterexamples below 2^64 nor any
known above that range. It is 20-50% faster than the strong test.
Because of the different parameters selected, there is no relationship
between the strong Lucas pseudoprimes and extra strong Lucas pseudoprimes.
In particular, one is not a subset of the other.
References
==========
- "Frobenius Pseudoprimes", Jon Grantham, 2000.
http://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01197-2/
- OEIS A217719: Extra Strong Lucas Pseudoprimes
https://oeis.org/A217719
- https://en.wikipedia.org/wiki/Lucas_pseudoprime
Examples
========
>>> from sympy.ntheory.primetest import isprime, is_extra_strong_lucas_prp
>>> for i in range(20000):
... if is_extra_strong_lucas_prp(i) and not isprime(i):
... print(i)
989
3239
5777
10877
"""
# Implementation notes:
# 1) the parameters differ from Thomas R. Nicely's. His parameter
# selection leads to pseudoprimes that overlap M-R tests, and
# contradict Baillie and Wagstaff's suggestion of (D|n) = -1.
# 2) The MathWorld page as of June 2013 specifies Q=-1. The Lucas
# sequence must have Q=1. See Grantham theorem 2.3, any of the
# references on the MathWorld page, or run it and see Q=-1 is wrong.
from sympy.ntheory.factor_ import trailing
n = as_int(n)
if n == 2:
return True
if n < 2 or (n % 2) == 0:
return False
if is_square(n, False):
return False
D, P, Q = _lucas_extrastrong_params(n)
if D == 0:
return False
# remove powers of 2 from n+1 (= k * 2**s)
s = trailing(n + 1)
k = (n + 1) >> s
U, V, Qk = _lucas_sequence(n, P, Q, k)
if U == 0 and (V == 2 or V == n - 2):
return True
if V == 0:
return True
for r in range(1, s):
V = (V * V - 2) % n
if V == 0:
return True
return False
def is_extra_strong_lucas_prp(n):
"""Extra Strong Lucas compositeness test. Returns False if n is
definitely composite, and True if n is a "extra strong" Lucas probable
prime.
The parameters are selected using P = 3, Q = 1, then incrementing P until
(D|n) == -1. The test itself is as defined in Grantham 2000, from the
Mo and Jones preprint. The parameter selection and test are the same as
used in OEIS A217719, Perl's Math::Prime::Util, and the Lucas pseudoprime
page on Wikipedia.
With these parameters, there are no counterexamples below 2^64 nor any
known above that range. It is 20-50% faster than the strong test.
Because of the different parameters selected, there is no relationship
between the strong Lucas pseudoprimes and extra strong Lucas pseudoprimes.
In particular, one is not a subset of the other.
References
==========
- "Frobenius Pseudoprimes", Jon Grantham, 2000.
http://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01197-2/
- OEIS A217719: Extra Strong Lucas Pseudoprimes
https://oeis.org/A217719
- https://en.wikipedia.org/wiki/Lucas_pseudoprime
Examples
========
>>> from sympy.ntheory.primetest import isprime, is_extra_strong_lucas_prp
>>> for i in range(20000):
... if is_extra_strong_lucas_prp(i) and not isprime(i):
... print(i)
989
3239
5777
10877
"""
# Implementation notes:
# 1) the parameters differ from Thomas R. Nicely's. His parameter
# selection leads to pseudoprimes that overlap M-R tests, and
# contradict Baillie and Wagstaff's suggestion of (D|n) = -1.
# 2) The MathWorld page as of June 2013 specifies Q=-1. The Lucas
# sequence must have Q=1. See Grantham theorem 2.3, any of the
# references on the MathWorld page, or run it and see Q=-1 is wrong.
from sympy.ntheory.factor_ import trailing
n = as_int(n)
if n == 2:
return True
if n < 2 or (n % 2) == 0:
return False
if is_square(n, False):
return False
D, P, Q = _lucas_extrastrong_params(n)
if D == 0:
return False
# remove powers of 2 from n+1 (= k * 2**s)
s = trailing(n + 1)
k = (n+1) >> s
U, V, Qk = _lucas_sequence(n, P, Q, k)
if U == 0 and (V == 2 or V == n - 2):
return True
if V == 0:
return True
for r in range(1, s):
V = (V*V - 2) % n
if V == 0:
return True
return False
