/* 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 (C) 1991, 1993, 1994, 1996, 1997, 1998, 1999, 2000 Free Software Foundation, Inc. Miller-Rabin code 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 the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version. 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 Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with the GNU MP Library; see the file COPYING.LIB. If not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. */ #include "gmp.h" #include "gmp-impl.h" #include "longlong.h" static int isprime _PROTO ((unsigned long int t)); static int mpz_millerrabin _PROTO ((mpz_srcptr n, int reps)); int #if __STDC__ mpz_probab_prime_p (mpz_srcptr n, int reps) #else mpz_probab_prime_p (n, reps) mpz_srcptr n; int reps; #endif { mp_limb_t r; /* Handle small and negative n. */ if (mpz_cmp_ui (n, 1000000L) <= 0) { int is_prime; if (mpz_sgn (n) < 0) { /* Negative number. Negate and call ourselves. */ mpz_t n2; mpz_init (n2); mpz_neg (n2, n); is_prime = mpz_probab_prime_p (n2, reps); mpz_clear (n2); return is_prime; } is_prime = isprime (mpz_get_ui (n)); return is_prime ? 2 : 0; } /* If n is now even, it is not a prime. */ if ((mpz_get_ui (n) & 1) == 0) return 0; /* Check if n has small factors. */ if (UDIV_TIME > (2 * UMUL_TIME + 6)) r = mpn_preinv_mod_1 (PTR(n), SIZ(n), (mp_limb_t) PP, (mp_limb_t) PP_INVERTED); else r = mpn_mod_1 (PTR(n), SIZ(n), (mp_limb_t) PP); if (r % 3 == 0 || r % 5 == 0 || r % 7 == 0 || r % 11 == 0 || r % 13 == 0 || r % 17 == 0 || r % 19 == 0 || r % 23 == 0 || r % 29 == 0 #if BITS_PER_MP_LIMB == 64 || r % 31 == 0 || r % 37 == 0 || r % 41 == 0 || r % 43 == 0 || r % 47 == 0 || r % 53 == 0 #endif ) { return 0; } /* 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) / 30; ln2 = ln2 * ln2; for (q = BITS_PER_MP_LIMB == 64 ? 59 : 31; q < ln2; q += 2) { if (isprime (q)) { umul_ppmm (p1, p0, p, q); if (p1 != 0) { r = mpn_mod_1 (PTR(n), SIZ(n), p); while (--nprimes >= 0) if (r % primes[nprimes] == 0) { if (mpn_mod_1 (PTR(n), SIZ(n), (mp_limb_t) primes[nprimes]) != 0) abort (); 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 #if __STDC__ isprime (unsigned long int t) #else isprime (t) unsigned long int t; #endif { unsigned long int q, r, d; if (t < 3 || (t & 1) == 0) return t == 2; for (d = 3, r = 1; r != 0; d += 2) { q = t / d; r = t - q * d; if (q < d) return 1; } return 0; } static int millerrabin _PROTO ((mpz_srcptr n, mpz_srcptr nm1, mpz_ptr x, mpz_ptr y, mpz_srcptr q, unsigned long int k)); static int #if __STDC__ mpz_millerrabin (mpz_srcptr n, int reps) #else mpz_millerrabin (n, reps) mpz_srcptr n; int reps; #endif { int r; mpz_t nm1, x, y, q; unsigned long int k; gmp_randstate_t rstate; int is_prime; TMP_DECL (marker); TMP_MARK (marker); MPZ_TMP_INIT (nm1, SIZ (n) + 1); mpz_sub_ui (nm1, n, 1L); MPZ_TMP_INIT (x, SIZ (n)); MPZ_TMP_INIT (y, 2 * SIZ (n)); /* mpz_powm_ui needs excessive memory!!! */ /* Perform a Fermat test. */ mpz_set_ui (x, 210L); mpz_powm (y, x, nm1, n); if (mpz_cmp_ui (y, 1L) != 0) { return 0; TMP_FREE (marker); } MPZ_TMP_INIT (q, SIZ (n)); /* Find q and k, where q is odd and n = 1 + 2**k * q. */ k = mpz_scan1 (nm1, 0L); mpz_tdiv_q_2exp (q, nm1, k); gmp_randinit (rstate, GMP_RAND_ALG_DEFAULT, 32L); is_prime = 1; for (r = 0; r < reps && is_prime; r++) { do mpz_urandomb (x, rstate, mpz_sizeinbase (n, 2) - 1); while (mpz_cmp_ui (x, 1L) <= 0); is_prime = millerrabin (n, nm1, x, y, q, k); } gmp_randclear (rstate); TMP_FREE (marker); return is_prime; } static int #if __STDC__ millerrabin (mpz_srcptr n, mpz_srcptr nm1, mpz_ptr x, mpz_ptr y, mpz_srcptr q, unsigned long int k) #else millerrabin (n, nm1, x, y, q, k) mpz_srcptr n; mpz_srcptr nm1; mpz_ptr x; mpz_ptr y; mpz_srcptr q; unsigned long int k; #endif { unsigned long int i; mpz_powm (y, x, q, n); if (mpz_cmp_ui (y, 1L) == 0 || mpz_cmp (y, nm1) == 0) return 1; for (i = 1; i < k; i++) { mpz_powm_ui (y, y, 2L, n); if (mpz_cmp (y, nm1) == 0) return 1; if (mpz_cmp_ui (y, 1L) == 0) return 0; } return 0; }