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/* mpz_millerrabin(n,reps) -- An implementation of the probabilistic primality
test found in Knuth's Seminumerical Algorithms book. If the function
mpz_millerrabin() returns 0 then n is not prime. If it returns 1, then n is
'probably' 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.
THE FUNCTIONS IN THIS FILE ARE FOR INTERNAL USE ONLY. THEY'RE ALMOST
CERTAIN TO BE SUBJECT TO INCOMPATIBLE CHANGES OR DISAPPEAR COMPLETELY IN
FUTURE GNU MP RELEASES.
Copyright 1991, 1993, 1994, 1996-2002, 2005, 2014, 2018 Free Software
Foundation, Inc.
Contributed by John Amanatides.
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"
static int millerrabin (mpz_srcptr,
mpz_ptr, mpz_ptr,
mpz_srcptr, unsigned long int);
int
mpz_millerrabin (mpz_srcptr n, int reps)
{
mpz_t nm, x, y, q;
unsigned long int k;
gmp_randstate_t rstate;
int is_prime;
TMP_DECL;
TMP_MARK;
MPZ_TMP_INIT (nm, SIZ (n) + 1);
mpz_tdiv_q_2exp (nm, n, 1);
MPZ_TMP_INIT (x, SIZ (n) + 1);
MPZ_TMP_INIT (y, 2 * SIZ (n)); /* mpz_powm_ui needs excessive memory!!! */
MPZ_TMP_INIT (q, SIZ (n));
/* Find q and k, where q is odd and n = 1 + 2**k * q. */
k = mpz_scan1 (nm, 0L);
mpz_tdiv_q_2exp (q, nm, k);
++k;
/* BPSW test */
mpz_set_ui (x, 2);
is_prime = millerrabin (n, x, y, q, k) && mpz_stronglucas (n, x, y);
if (is_prime)
{
/* (n-5)/2 */
mpz_sub_ui (nm, nm, 2L);
ASSERT (mpz_cmp_ui (nm, 1L) >= 0);
gmp_randinit_default (rstate);
for (reps -= 24; reps > 0 && is_prime; --reps)
{
/* 3 to (n-1)/2 inclusive, don't want 1, 0 or 2 */
mpz_urandomm (x, rstate, nm);
mpz_add_ui (x, x, 3L);
is_prime = millerrabin (n, x, y, q, k);
}
gmp_randclear (rstate);
}
TMP_FREE;
return is_prime;
}
static int
mod_eq_m1 (mpz_srcptr x, mpz_srcptr m)
{
mp_size_t ms;
mp_srcptr mp, xp;
ms = SIZ (m);
if (SIZ (x) != ms)
return 0;
ASSERT (ms > 0);
mp = PTR (m);
xp = PTR (x);
ASSERT ((mp[0] - 1) == (mp[0] ^ 1)); /* n is odd */
if ((*xp ^ CNST_LIMB(1) ^ *mp) != CNST_LIMB(0)) /* xp[0] != mp[0] - 1 */
return 0;
else
{
int cmp;
--ms;
++xp;
++mp;
MPN_CMP (cmp, xp, mp, ms);
return cmp == 0;
}
}
static int
millerrabin (mpz_srcptr n, mpz_ptr x, mpz_ptr y,
mpz_srcptr q, unsigned long int k)
{
unsigned long int i;
mpz_powm (y, x, q, n);
if (mpz_cmp_ui (y, 1L) == 0 || mod_eq_m1 (y, n))
return 1;
for (i = 1; i < k; i++)
{
mpz_powm_ui (y, y, 2L, n);
if (mod_eq_m1 (y, n))
return 1;
/* y == 1 means that the previous y was a non-trivial square root
of 1 (mod n). y == 0 means that n is a power of the base.
In either case, n is not prime. */
if (mpz_cmp_ui (y, 1L) <= 0)
return 0;
}
return 0;
}
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