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/* mpz_probab_prime_p --
An implementation of the probabilistic primality test found in Knuth's
Seminumerical Algorithms book. If the function mpz_probab_prime_p()
returns 0 then n is not prime. If it returns 1, then n is 'probably'
prime. If it returns 2, n is surely prime. The probability of a false
positive is (1/4)**reps, where reps is the number of internal passes of the
probabilistic algorithm. Knuth indicates that 25 passes are reasonable.
Copyright 1991, 1993, 1994, 1996-2002, 2005, 2015, 2016 Free Software
Foundation, Inc.
This file is part of the GNU MP Library.
The GNU MP Library is free software; you can redistribute it and/or modify
it under the terms of either:
* the GNU Lesser General Public License as published by the Free
Software Foundation; either version 3 of the License, or (at your
option) any later version.
or
* the GNU General Public License as published by the Free Software
Foundation; either version 2 of the License, or (at your option) any
later version.
or both in parallel, as here.
The GNU MP Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
for more details.
You should have received copies of the GNU General Public License and the
GNU Lesser General Public License along with the GNU MP Library. If not,
see https://www.gnu.org/licenses/. */
#include "gmp-impl.h"
#include "longlong.h"
static int isprime (unsigned long int);
/* MPN_MOD_OR_MODEXACT_1_ODD can be used instead of mpn_mod_1 for the trial
division. It gives a result which is not the actual remainder r but a
value congruent to r*2^n mod d. Since all the primes being tested are
odd, r*2^n mod p will be 0 if and only if r mod p is 0. */
int
mpz_probab_prime_p (mpz_srcptr n, int reps)
{
mp_limb_t r;
mpz_t n2;
/* Handle small and negative n. */
if (mpz_cmp_ui (n, 1000000L) <= 0)
{
if (mpz_cmpabs_ui (n, 1000000L) <= 0)
{
int is_prime;
unsigned long n0;
n0 = mpz_get_ui (n);
is_prime = n0 & (n0 > 1) ? isprime (n0) : n0 == 2;
return is_prime ? 2 : 0;
}
/* Negative number. Negate and fall out. */
PTR(n2) = PTR(n);
SIZ(n2) = -SIZ(n);
n = n2;
}
/* If n is now even, it is not a prime. */
if (mpz_even_p (n))
return 0;
#if defined (PP)
/* Check if n has small factors. */
#if defined (PP_INVERTED)
r = MPN_MOD_OR_PREINV_MOD_1 (PTR(n), (mp_size_t) SIZ(n), (mp_limb_t) PP,
(mp_limb_t) PP_INVERTED);
#else
r = mpn_mod_1 (PTR(n), (mp_size_t) SIZ(n), (mp_limb_t) PP);
#endif
if (r % 3 == 0
#if GMP_LIMB_BITS >= 4
|| r % 5 == 0
#endif
#if GMP_LIMB_BITS >= 8
|| r % 7 == 0
#endif
#if GMP_LIMB_BITS >= 16
|| r % 11 == 0 || r % 13 == 0
#endif
#if GMP_LIMB_BITS >= 32
|| r % 17 == 0 || r % 19 == 0 || r % 23 == 0 || r % 29 == 0
#endif
#if GMP_LIMB_BITS >= 64
|| r % 31 == 0 || r % 37 == 0 || r % 41 == 0 || r % 43 == 0
|| r % 47 == 0 || r % 53 == 0
#endif
)
{
return 0;
}
#endif /* PP */
/* Do more dividing. We collect small primes, using umul_ppmm, until we
overflow a single limb. We divide our number by the small primes product,
and look for factors in the remainder. */
{
unsigned long int ln2;
unsigned long int q;
mp_limb_t p1, p0, p;
unsigned int primes[15];
int nprimes;
nprimes = 0;
p = 1;
ln2 = mpz_sizeinbase (n, 2); /* FIXME: tune this limit */
for (q = PP_FIRST_OMITTED; q < ln2; q += 2)
{
if (isprime (q))
{
umul_ppmm (p1, p0, p, q);
if (p1 != 0)
{
r = MPN_MOD_OR_MODEXACT_1_ODD (PTR(n), (mp_size_t) SIZ(n), p);
while (--nprimes >= 0)
if (r % primes[nprimes] == 0)
{
ASSERT_ALWAYS (mpn_mod_1 (PTR(n), (mp_size_t) SIZ(n), (mp_limb_t) primes[nprimes]) == 0);
return 0;
}
p = q;
nprimes = 0;
}
else
{
p = p0;
}
primes[nprimes++] = q;
}
}
}
/* Perform a number of Miller-Rabin tests. */
return mpz_millerrabin (n, reps);
}
static int
isprime (unsigned long int t)
{
unsigned long int q, r, d;
ASSERT (t >= 3 && (t & 1) != 0);
d = 3;
do {
q = t / d;
r = t - q * d;
if (q < d)
return 1;
d += 2;
} while (r != 0);
return 0;
}
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