1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
|
/* 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, 1997, 1998, 1999, 2000, 2001 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
mpz_probab_prime_p (mpz_srcptr n, int reps)
{
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;
#if defined (PP)
/* Check if n has small factors. */
#if defined (PP_INVERTED)
r = MPN_MOD_OR_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);
#endif
if (r % 3 == 0
#if BITS_PER_MP_LIMB >= 4
|| r % 5 == 0
#endif
#if BITS_PER_MP_LIMB >= 8
|| r % 7 == 0
#endif
#if BITS_PER_MP_LIMB >= 16
|| r % 11 == 0 || r % 13 == 0
#endif
#if BITS_PER_MP_LIMB >= 32
|| r % 17 == 0 || r % 19 == 0 || r % 23 == 0 || r % 29 == 0
#endif
#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;
}
#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) / 30; ln2 = ln2 * ln2;
for (q = PP_FIRST_OMITTED; 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
isprime (unsigned long int t)
{
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
mpz_millerrabin (mpz_srcptr n, int reps)
{
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)
{
TMP_FREE (marker);
return 0;
}
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
millerrabin (mpz_srcptr n, mpz_srcptr nm1, 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 || 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;
}
|
