1 /* mpz_probab_prime_p --
2 An implementation of the probabilistic primality test found in Knuth's
3 Seminumerical Algorithms book. If the function mpz_probab_prime_p()
4 returns 0 then n is not prime. If it returns 1, then n is 'probably'
5 prime. If it returns 2, n is surely prime. The probability of a false
6 positive is (1/4)**reps, where reps is the number of internal passes of the
7 probabilistic algorithm. Knuth indicates that 25 passes are reasonable.
8
9 Copyright 1991, 1993, 1994, 1996-2002, 2005, 2015, 2016 Free Software
10 Foundation, Inc.
11
12 This file is part of the GNU MP Library.
13
14 The GNU MP Library is free software; you can redistribute it and/or modify
15 it under the terms of either:
16
17 * the GNU Lesser General Public License as published by the Free
18 Software Foundation; either version 3 of the License, or (at your
19 option) any later version.
20
21 or
22
23 * the GNU General Public License as published by the Free Software
24 Foundation; either version 2 of the License, or (at your option) any
25 later version.
26
27 or both in parallel, as here.
28
29 The GNU MP Library is distributed in the hope that it will be useful, but
30 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
31 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
32 for more details.
33
34 You should have received copies of the GNU General Public License and the
35 GNU Lesser General Public License along with the GNU MP Library. If not,
36 see https://www.gnu.org/licenses/. */
37
38 #include "gmp-impl.h"
39 #include "longlong.h"
40
41 static int isprime (unsigned long int);
42
43
44 /* MPN_MOD_OR_MODEXACT_1_ODD can be used instead of mpn_mod_1 for the trial
45 division. It gives a result which is not the actual remainder r but a
46 value congruent to r*2^n mod d. Since all the primes being tested are
47 odd, r*2^n mod p will be 0 if and only if r mod p is 0. */
48
49 int
50 mpz_probab_prime_p (mpz_srcptr n, int reps)
51 {
52 mp_limb_t r;
53 mpz_t n2;
54
55 /* Handle small and negative n. */
56 if (mpz_cmp_ui (n, 1000000L) <= 0)
57 {
58 if (mpz_cmpabs_ui (n, 1000000L) <= 0)
59 {
60 int is_prime;
61 unsigned long n0;
62 n0 = mpz_get_ui (n);
63 is_prime = n0 & (n0 > 1) ? isprime (n0) : n0 == 2;
64 return is_prime ? 2 : 0;
65 }
66 /* Negative number. Negate and fall out. */
67 PTR(n2) = PTR(n);
68 SIZ(n2) = -SIZ(n);
69 n = n2;
70 }
71
72 /* If n is now even, it is not a prime. */
73 if (mpz_even_p (n))
74 return 0;
75
76 #if defined (PP)
77 /* Check if n has small factors. */
78 #if defined (PP_INVERTED)
79 r = MPN_MOD_OR_PREINV_MOD_1 (PTR(n), (mp_size_t) SIZ(n), (mp_limb_t) PP,
80 (mp_limb_t) PP_INVERTED);
81 #else
82 r = mpn_mod_1 (PTR(n), (mp_size_t) SIZ(n), (mp_limb_t) PP);
83 #endif
84 if (r % 3 == 0
85 #if GMP_LIMB_BITS >= 4
86 || r % 5 == 0
87 #endif
88 #if GMP_LIMB_BITS >= 8
89 || r % 7 == 0
90 #endif
91 #if GMP_LIMB_BITS >= 16
92 || r % 11 == 0 || r % 13 == 0
93 #endif
94 #if GMP_LIMB_BITS >= 32
95 || r % 17 == 0 || r % 19 == 0 || r % 23 == 0 || r % 29 == 0
96 #endif
97 #if GMP_LIMB_BITS >= 64
98 || r % 31 == 0 || r % 37 == 0 || r % 41 == 0 || r % 43 == 0
99 || r % 47 == 0 || r % 53 == 0
100 #endif
101 )
102 {
103 return 0;
104 }
105 #endif /* PP */
106
107 /* Do more dividing. We collect small primes, using umul_ppmm, until we
108 overflow a single limb. We divide our number by the small primes product,
109 and look for factors in the remainder. */
110 {
111 unsigned long int ln2;
112 unsigned long int q;
113 mp_limb_t p1, p0, p;
114 unsigned int primes[15];
115 int nprimes;
116
117 nprimes = 0;
118 p = 1;
119 ln2 = mpz_sizeinbase (n, 2); /* FIXME: tune this limit */
120 for (q = PP_FIRST_OMITTED; q < ln2; q += 2)
121 {
122 if (isprime (q))
123 {
124 umul_ppmm (p1, p0, p, q);
125 if (p1 != 0)
126 {
127 r = MPN_MOD_OR_MODEXACT_1_ODD (PTR(n), (mp_size_t) SIZ(n), p);
128 while (--nprimes >= 0)
129 if (r % primes[nprimes] == 0)
130 {
131 ASSERT_ALWAYS (mpn_mod_1 (PTR(n), (mp_size_t) SIZ(n), (mp_limb_t) primes[nprimes]) == 0);
132 return 0;
133 }
134 p = q;
135 nprimes = 0;
136 }
137 else
138 {
139 p = p0;
140 }
141 primes[nprimes++] = q;
142 }
143 }
144 }
145
146 /* Perform a number of Miller-Rabin tests. */
147 return mpz_millerrabin (n, reps);
148 }
149
150 static int
151 isprime (unsigned long int t)
152 {
153 unsigned long int q, r, d;
154
155 ASSERT (t >= 3 && (t & 1) != 0);
156
157 d = 3;
158 do {
159 q = t / d;
160 r = t - q * d;
161 if (q < d)
162 return 1;
163 d += 2;
164 } while (r != 0);
165 return 0;
166 }