/* 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.

   With the current implementation, the first 24 MR-tests are substituted by a

   Baillie-PSW probable prime test.

   This implementation the Baillie-PSW test was checked up to 31*2^46,

   for smaller values no MR-test is performed, regardless of reps, and

   2 ("surely prime") is returned if the number was not proved composite.

   If GMP_BPSW_NOFALSEPOSITIVES_UPTO_64BITS is defined as non-zero,

   the code assumes that the Baillie-PSW test was checked up to 2^64.

   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, 2019 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"

#ifndef GMP_BPSW_NOFALSEPOSITIVES_UPTO_64BITS

#define GMP_BPSW_NOFALSEPOSITIVES_UPTO_64BITS 0

#endif

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;

  ASSERT (SIZ (n) > 0);

  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)

    {

      if (

#if GMP_BPSW_NOFALSEPOSITIVES_UPTO_64BITS

    /* Consider numbers up to 2^64 that pass the BPSW test as primes. */

#if GMP_NUMB_BITS <= 64

    SIZ (n) <= 64 / GMP_NUMB_BITS

#else

    0

#endif

#if 64 % GMP_NUMB_BITS != 0

    || SIZ (n) - 64 / GMP_NUMB_BITS == (PTR (n) [64 / GMP_NUMB_BITS] < CNST_LIMB(1) << 64 % GMP_NUMB_BITS)

#endif

#else

    /* Consider numbers up to 31*2^46 that pass the BPSW test as primes.

       This implementation was tested up to 31*2^46 */

    /* 2^4 < 31 = 0b11111 < 2^5 */

#define GMP_BPSW_LIMB_CONST CNST_LIMB(31)

#define GMP_BPSW_BITS_CONST (LOG2C(31) - 1)

#define GMP_BPSW_BITS_LIMIT (46 + GMP_BPSW_BITS_CONST)

#define GMP_BPSW_LIMBS_LIMIT (GMP_BPSW_BITS_LIMIT / GMP_NUMB_BITS)

#define GMP_BPSW_BITS_MOD (GMP_BPSW_BITS_LIMIT % GMP_NUMB_BITS)

#if GMP_NUMB_BITS <=  GMP_BPSW_BITS_LIMIT

    SIZ (n) <= GMP_BPSW_LIMBS_LIMIT

#else

    0

#endif

#if GMP_BPSW_BITS_MOD >=  GMP_BPSW_BITS_CONST

    || SIZ (n) - GMP_BPSW_LIMBS_LIMIT == (PTR (n) [GMP_BPSW_LIMBS_LIMIT] < GMP_BPSW_LIMB_CONST << (GMP_BPSW_BITS_MOD - GMP_BPSW_BITS_CONST))

#else

#if GMP_BPSW_BITS_MOD != 0

    || SIZ (n) - GMP_BPSW_LIMBS_LIMIT == (PTR (n) [GMP_BPSW_LIMBS_LIMIT] < GMP_BPSW_LIMB_CONST >> (GMP_BPSW_BITS_CONST -  GMP_BPSW_BITS_MOD))

#else

#if GMP_NUMB_BITS > GMP_BPSW_BITS_CONST

    || SIZ (nm) - GMP_BPSW_LIMBS_LIMIT + 1 == (PTR (nm) [GMP_BPSW_LIMBS_LIMIT - 1] < GMP_BPSW_LIMB_CONST << (GMP_NUMB_BITS - 1 - GMP_BPSW_BITS_CONST))

#endif

#endif

#endif

#undef GMP_BPSW_BITS_LIMIT

#undef GMP_BPSW_LIMB_CONST

#undef GMP_BPSW_BITS_CONST

#undef GMP_BPSW_LIMBS_LIMIT

#undef GMP_BPSW_BITS_MOD

#endif

    )

  is_prime = 2;

      else

  {

    reps -= 24;

    if (reps > 0)

      {

        /* (n-5)/2 */

        mpz_sub_ui (nm, nm, 2L);

        ASSERT (mpz_cmp_ui (nm, 1L) >= 0);

        gmp_randinit_default (rstate);

        do

    {

      /* 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);

    } while (--reps > 0 && is_prime);

        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;

}