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/* bignum-random-prime.c
*
* Generation of random provable primes.
*/
/* nettle, low-level cryptographics library
*
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*
* The nettle 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 nettle 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 nettle 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.
*/
#if HAVE_CONFIG_H
# include "config.h"
#endif
#ifndef RANDOM_PRIME_VERBOSE
#define RANDOM_PRIME_VERBOSE 0
#endif
#include <assert.h>
#include <stdlib.h>
#if RANDOM_PRIME_VERBOSE
#include <stdio.h>
#define VERBOSE(x) (fputs((x), stderr))
#else
#define VERBOSE(x)
#endif
#include "bignum.h"
#include "macros.h"
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/* Use a table of p_2 = 3 to p_{172} = 1021, used for sieving numbers
of up to 20 bits. */
#define NPRIMES 171
#define TRIAL_DIV_BITS 20
#define TRIAL_DIV_MASK ((1 << TRIAL_DIV_BITS) - 1)
/* A 20-bit number x is divisible by p iff
((x * inverse) & TRIAL_DIV_MASK) <= limit
*/
struct trial_div_info {
uint32_t inverse; /* p^{-1} (mod 2^20) */
uint32_t limit; /* floor( (2^20 - 1) / p) */
};
static const uint16_t
primes[NPRIMES] = {
3,5,7,11,13,17,19,23,
29,31,37,41,43,47,53,59,
61,67,71,73,79,83,89,97,
101,103,107,109,113,127,131,137,
139,149,151,157,163,167,173,179,
181,191,193,197,199,211,223,227,
229,233,239,241,251,257,263,269,
271,277,281,283,293,307,311,313,
317,331,337,347,349,353,359,367,
373,379,383,389,397,401,409,419,
421,431,433,439,443,449,457,461,
463,467,479,487,491,499,503,509,
521,523,541,547,557,563,569,571,
577,587,593,599,601,607,613,617,
619,631,641,643,647,653,659,661,
673,677,683,691,701,709,719,727,
733,739,743,751,757,761,769,773,
787,797,809,811,821,823,827,829,
839,853,857,859,863,877,881,883,
887,907,911,919,929,937,941,947,
953,967,971,977,983,991,997,1009,
1013,1019,1021,
};
static const uint32_t
prime_square[NPRIMES+1] = {
9,25,49,121,169,289,361,529,
841,961,1369,1681,1849,2209,2809,3481,
3721,4489,5041,5329,6241,6889,7921,9409,
10201,10609,11449,11881,12769,16129,17161,18769,
19321,22201,22801,24649,26569,27889,29929,32041,
32761,36481,37249,38809,39601,44521,49729,51529,
52441,54289,57121,58081,63001,66049,69169,72361,
73441,76729,78961,80089,85849,94249,96721,97969,
100489,109561,113569,120409,121801,124609,128881,134689,
139129,143641,146689,151321,157609,160801,167281,175561,
177241,185761,187489,192721,196249,201601,208849,212521,
214369,218089,229441,237169,241081,249001,253009,259081,
271441,273529,292681,299209,310249,316969,323761,326041,
332929,344569,351649,358801,361201,368449,375769,380689,
383161,398161,410881,413449,418609,426409,434281,436921,
452929,458329,466489,477481,491401,502681,516961,528529,
537289,546121,552049,564001,573049,579121,591361,597529,
619369,635209,654481,657721,674041,677329,683929,687241,
703921,727609,734449,737881,744769,769129,776161,779689,
786769,822649,829921,844561,863041,877969,885481,896809,
908209,935089,942841,954529,966289,982081,994009,1018081,
1026169,1038361,1042441,1L<<20
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static const struct trial_div_info
trial_div_table[NPRIMES] = {
{699051,349525},{838861,209715},{748983,149796},{953251,95325},
{806597,80659},{61681,61680},{772635,55188},{866215,45590},
{180789,36157},{1014751,33825},{793517,28339},{1023001,25575},
{48771,24385},{870095,22310},{217629,19784},{710899,17772},
{825109,17189},{281707,15650},{502135,14768},{258553,14364},
{464559,13273},{934875,12633},{1001449,11781},{172961,10810},
{176493,10381},{203607,10180},{568387,9799},{788837,9619},
{770193,9279},{1032063,8256},{544299,8004},{619961,7653},
{550691,7543},{182973,7037},{229159,6944},{427445,6678},
{701195,6432},{370455,6278},{90917,6061},{175739,5857},
{585117,5793},{225087,5489},{298817,5433},{228877,5322},
{442615,5269},{546651,4969},{244511,4702},{83147,4619},
{769261,4578},{841561,4500},{732687,4387},{978961,4350},
{133683,4177},{65281,4080},{629943,3986},{374213,3898},
{708079,3869},{280125,3785},{641833,3731},{618771,3705},
{930477,3578},{778747,3415},{623751,3371},{40201,3350},
{122389,3307},{950371,3167},{1042353,3111},{18131,3021},
{285429,3004},{549537,2970},{166487,2920},{294287,2857},
{919261,2811},{636339,2766},{900735,2737},{118605,2695},
{10565,2641},{188273,2614},{115369,2563},{735755,2502},
{458285,2490},{914767,2432},{370513,2421},{1027079,2388},
{629619,2366},{462401,2335},{649337,2294},{316165,2274},
{484655,2264},{65115,2245},{326175,2189},{1016279,2153},
{990915,2135},{556859,2101},{462791,2084},{844629,2060},
{404537,2012},{457123,2004},{577589,1938},{638347,1916},
{892325,1882},{182523,1862},{1002505,1842},{624371,1836},
{69057,1817},{210787,1786},{558769,1768},{395623,1750},
{992745,1744},{317855,1727},{384877,1710},{372185,1699},
{105027,1693},{423751,1661},{408961,1635},{908331,1630},
{74551,1620},{36933,1605},{617371,1591},{506045,1586},
{24929,1558},{529709,1548},{1042435,1535},{31867,1517},
{166037,1495},{928781,1478},{508975,1458},{4327,1442},
{779637,1430},{742091,1418},{258263,1411},{879631,1396},
{72029,1385},{728905,1377},{589057,1363},{348621,1356},
{671515,1332},{710453,1315},{84249,1296},{959363,1292},
{685853,1277},{467591,1274},{646643,1267},{683029,1264},
{439927,1249},{254461,1229},{660713,1223},{554195,1220},
{202911,1215},{753253,1195},{941457,1190},{776635,1187},
{509511,1182},{986147,1156},{768879,1151},{699431,1140},
{696417,1128},{86169,1119},{808997,1114},{25467,1107},
{201353,1100},{708087,1084},{1018339,1079},{341297,1073},
{434151,1066},{96287,1058},{950765,1051},{298257,1039},
{675933,1035},{167731,1029},{815445,1027},
/* Element j gives the index of the first prime of size 3+j bits */
static uint8_t
prime_by_size[9] = {
1,3,5,10,17,30,53,96,171
};
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/* Combined Miller-Rabin test to the base a, and checking the
conditions from Pocklington's theorem. */
static int
miller_rabin_pocklington(mpz_t n, mpz_t nm1, mpz_t nm1dq, mpz_t a)
{
mpz_t r;
mpz_t y;
int is_prime = 0;
/* Avoid the mp_bitcnt_t type for compatibility with older GMP
versions. */
unsigned k;
unsigned j;
VERBOSE(".");
if (mpz_even_p(n) || mpz_cmp_ui(n, 3) < 0)
return 0;
mpz_init(r);
mpz_init(y);
k = mpz_scan1(nm1, 0);
assert(k > 0);
mpz_fdiv_q_2exp (r, nm1, k);
mpz_powm(y, a, r, n);
if (mpz_cmp_ui(y, 1) == 0 || mpz_cmp(y, nm1) == 0)
for (j = 1; j < k; j++)
{
mpz_powm_ui (y, y, 2, n);
if (mpz_cmp_ui (y, 1) == 0)
break;
if (mpz_cmp (y, nm1) == 0)
{
passed_miller_rabin:
/* We know that a^{n-1} = 1 (mod n)
Remains to check that gcd(a^{(n-1)/q} - 1, n) == 1 */
VERBOSE("x");
mpz_powm(y, a, nm1dq, n);
mpz_sub_ui(y, y, 1);
mpz_gcd(y, y, n);
is_prime = mpz_cmp_ui (y, 1) == 0;
VERBOSE(is_prime ? "\n" : "");
break;
}
}
mpz_clear(r);
mpz_clear(y);
return is_prime;
}
/* The algorithm is based on the following special case of
Pocklington's theorem:
Assume that n = 1 + f q, where q is a prime, q > sqrt(n) - 1. If we
can find an a such that
a^{n-1} = 1 (mod n)
then n is prime.
Proof: Assume that n is composite, with smallest prime factor p <=
sqrt(n). Since q is prime, and q > sqrt(n) - 1 >= p - 1, q and p-1
are coprime, so that we can define u = q^{-1} (mod (p-1)). The
assumption a^{n-1} = 1 (mod n) implies that also a^{n-1} = 1 (mod
p). Since p is prime, we have a^{(p-1)} = 1 (mod p). Now, r =
(n-1)/q = (n-1) u (mod (p-1)), and it follows that a^r = a^{(n-1)
u} = 1 (mod p). Then p is a common factor of a^r - 1 and n. This
contradicts gcd(a^r - 1, n) = 1, and concludes the proof.
If n is specified as k bits, we need q of size ceil(k/2) + 1 bits
(or more) to make the theorem apply.
*/
/* Generate a prime number p of size bits with 2 p0q dividing (p-1).
p0 must be of size >= ceil(bits/2) + 1. The extra factor q can be
omitted. If top_bits_set is one, the top most two bits are one,
suitable for RSA primes. */
_nettle_generate_pocklington_prime (mpz_t p, mpz_t r,
unsigned bits, int top_bits_set,
void *ctx, nettle_random_func random,
const mpz_t p0,
const mpz_t q,
const mpz_t p0q)
assert (2*mpz_sizeinbase (p0, 2) > bits + 1);
mpz_init (r_min);
mpz_init (r_range);
mpz_init (pm1);
mpz_init (a);
if (top_bits_set)
{
/* i = floor (2^{bits-3} / p0q), then 3I + 3 <= r <= 4I, with I
- 2 possible values. */
mpz_set_ui (r_min, 1);
mpz_mul_2exp (r_min, r_min, bits-3);
mpz_fdiv_q (r_min, r_min, p0q);
mpz_sub_ui (r_range, r_min, 2);
mpz_mul_ui (r_min, r_min, 3);
mpz_add_ui (r_min, r_min, 3);
}
else
{
/* i = floor (2^{bits-2} / p0q), I + 1 <= r <= 2I */
mpz_set_ui (r_range, 1);
mpz_mul_2exp (r_range, r_range, bits-2);
mpz_fdiv_q (r_range, r_range, p0q);
mpz_add_ui (r_min, r_range, 1);
}
nettle_mpz_random (r, ctx, random, r_range);
mpz_add (r, r, r_min);
/* Set p = 2*r*p0q + 1 */
mpz_mul_2exp(r, r, 1);
mpz_mul (pm1, r, p0q);
mpz_add_ui (p, pm1, 1);
assert(mpz_sizeinbase(p, 2) == bits);
/* Should use GMP trial division interface when that
materializes, we don't need any testing beyond trial
division. */
if (!mpz_probab_prime_p (p, 1))
continue;
random(ctx, sizeof(buf), buf);
mpz_set_ui (a, buf[0] + 2);
if (q)
{
mpz_t e;
int is_prime;
mpz_init (e);
mpz_mul (e, r, q);
is_prime = miller_rabin_pocklington(p, pm1, e, a);
mpz_clear (e);
if (is_prime)
break;
}
else if (miller_rabin_pocklington(p, pm1, r, a))
mpz_clear (r_min);
mpz_clear (r_range);
mpz_clear (pm1);
mpz_clear (a);
}
/* Generate random prime of a given size. Maurer's algorithm (Alg.
6.42 Handbook of applied cryptography), but with ratio = 1/2 (like
nettle_random_prime(mpz_t p, unsigned bits, int top_bits_set,
void *random_ctx, nettle_random_func random,
void *progress_ctx, nettle_progress_func progress)
{
assert (bits >= 3);
if (bits <= 10)
{
unsigned first;
unsigned choices;
uint8_t buf;
assert (!top_bits_set);
random (random_ctx, sizeof(buf), &buf);
first = prime_by_size[bits-3];
choices = prime_by_size[bits-2] - first;
mpz_set_ui (p, primes[first + buf % choices]);
}
else if (bits <= 20)
{
unsigned long highbit;
uint8_t buf[3];
unsigned long x;
unsigned j;
highbit = 1L << (bits - 1);
again:
random (random_ctx, sizeof(buf), buf);
x = READ_UINT24(buf);
x &= (highbit - 1);
x |= highbit | 1;
for (j = 0; prime_square[j] <= x; j++)
{
unsigned q = x * trial_div_table[j].inverse & TRIAL_DIV_MASK;
if (q <= trial_div_table[j].limit)
goto again;
}
mpz_set_ui (p, x);
}
else
{
mpz_t q, r;
mpz_init (q);
mpz_init (r);
/* Bit size ceil(k/2) + 1, slightly larger than used in Alg. 4.62
in Handbook of Applied Cryptography (which seems to be
incorrect for odd k). */
nettle_random_prime (q, (bits+3)/2, 0, random_ctx, random,
progress_ctx, progress);
_nettle_generate_pocklington_prime (p, r, bits, top_bits_set,
random_ctx, random,
if (progress)
progress (progress_ctx, 'x');
mpz_clear (q);
mpz_clear (r);