Commit 6ce825c7 authored by Niels Möller's avatar Niels Möller
Browse files

(_nettle_generate_pocklington_prime): New

function. Rely on mpz_probab_prime_p (for lack of a trial division
function) for trial division.
(nettle_random_prime): Rewritten. Uses the prime table for the
smallest sizes, then trial division using a new set of tables, and
then Maurer's algorithm, calling the new
_nettle_generate_pocklington_prime for the final search.

Rev: nettle/ChangeLog:1.80
Rev: nettle/bignum-random-prime.c:1.4
Rev: nettle/bignum.h:1.5
parent 7c49b0dc
2010-05-26 Niels Mller <nisse@lysator.liu.se>
* bignum-random-prime.c (_nettle_generate_pocklington_prime): New
function. Rely on mpz_probab_prime_p (for lack of a trial division
function) for trial division.
(nettle_random_prime): Rewritten. Uses the prime table for the
smallest sizes, then trial division using a new set of tables, and
then Maurer's algorithm, calling the new
_nettle_generate_pocklington_prime for the final search.
2010-05-25 Niels Mller <nisse@lysator.liu.se> 2010-05-25 Niels Mller <nisse@lysator.liu.se>
* testsuite/dsa-test.c (test_main): Updated for dsa testing * testsuite/dsa-test.c (test_main): Updated for dsa testing
changes. changes.
* testsuite/dsa-keygen-test.c (test_main): Test dsa256. * testsuite/dsa-keygen-test.c (test_main): Test dsa256.
......
...@@ -45,73 +45,126 @@ ...@@ -45,73 +45,126 @@
#include "macros.h" #include "macros.h"
/* Use a table of p_2 = 3 to p_{172} = 1021, multiplied to 32-bit or /* Use a table of p_2 = 3 to p_{172} = 1021, used for sieving numbers
64-bit size. */ of up to 20 bits. */
struct sieve_element { #define NPRIMES 171
/* Product of some small primes. */ #define TRIAL_DIV_BITS 20
unsigned long prod; #define TRIAL_DIV_MASK ((1 << TRIAL_DIV_BITS) - 1)
/* Square of the smallest one. */
unsigned long p2; /* 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
}; };
static const struct sieve_element static const struct trial_div_info
sieve_table[] = { trial_div_table[NPRIMES] = {
{111546435, 9}, /* 3 -- 23 */ {699051,349525},{838861,209715},{748983,149796},{953251,95325},
{58642669, 841}, /* 29 -- 43 */ {806597,80659},{61681,61680},{772635,55188},{866215,45590},
{600662303, 2209}, /* 47 -- 67 */ {180789,36157},{1014751,33825},{793517,28339},{1023001,25575},
{33984931, 5041}, /* 71 -- 83 */ {48771,24385},{870095,22310},{217629,19784},{710899,17772},
{89809099, 7921}, /* 89 -- 103 */ {825109,17189},{281707,15650},{502135,14768},{258553,14364},
{167375713, 11449}, /* 107 -- 127 */ {464559,13273},{934875,12633},{1001449,11781},{172961,10810},
{371700317, 17161}, /* 131 -- 149 */ {176493,10381},{203607,10180},{568387,9799},{788837,9619},
{645328247, 22801}, /* 151 -- 167 */ {770193,9279},{1032063,8256},{544299,8004},{619961,7653},
{1070560157, 29929}, /* 173 -- 191 */ {550691,7543},{182973,7037},{229159,6944},{427445,6678},
{1596463769, 37249}, /* 193 -- 211 */ {701195,6432},{370455,6278},{90917,6061},{175739,5857},
{11592209, 49729}, /* 223 -- 229 */ {585117,5793},{225087,5489},{298817,5433},{228877,5322},
{13420567, 54289}, /* 233 -- 241 */ {442615,5269},{546651,4969},{244511,4702},{83147,4619},
{16965341, 63001}, /* 251 -- 263 */ {769261,4578},{841561,4500},{732687,4387},{978961,4350},
{20193023, 72361}, /* 269 -- 277 */ {133683,4177},{65281,4080},{629943,3986},{374213,3898},
{23300239, 78961}, /* 281 -- 293 */ {708079,3869},{280125,3785},{641833,3731},{618771,3705},
{29884301, 94249}, /* 307 -- 313 */ {930477,3578},{778747,3415},{623751,3371},{40201,3350},
{35360399, 100489}, /* 317 -- 337 */ {122389,3307},{950371,3167},{1042353,3111},{18131,3021},
{42749359, 120409}, /* 347 -- 353 */ {285429,3004},{549537,2970},{166487,2920},{294287,2857},
{49143869, 128881}, /* 359 -- 373 */ {919261,2811},{636339,2766},{900735,2737},{118605,2695},
{56466073, 143641}, /* 379 -- 389 */ {10565,2641},{188273,2614},{115369,2563},{735755,2502},
{65111573, 157609}, /* 397 -- 409 */ {458285,2490},{914767,2432},{370513,2421},{1027079,2388},
{76027969, 175561}, /* 419 -- 431 */ {629619,2366},{462401,2335},{649337,2294},{316165,2274},
{84208541, 187489}, /* 433 -- 443 */ {484655,2264},{65115,2245},{326175,2189},{1016279,2153},
{94593973, 201601}, /* 449 -- 461 */ {990915,2135},{556859,2101},{462791,2084},{844629,2060},
{103569859, 214369}, /* 463 -- 479 */ {404537,2012},{457123,2004},{577589,1938},{638347,1916},
{119319383, 237169}, /* 487 -- 499 */ {892325,1882},{182523,1862},{1002505,1842},{624371,1836},
{133390067, 253009}, /* 503 -- 521 */ {69057,1817},{210787,1786},{558769,1768},{395623,1750},
{154769821, 273529}, /* 523 -- 547 */ {992745,1744},{317855,1727},{384877,1710},{372185,1699},
{178433279, 310249}, /* 557 -- 569 */ {105027,1693},{423751,1661},{408961,1635},{908331,1630},
{193397129, 326041}, /* 571 -- 587 */ {74551,1620},{36933,1605},{617371,1591},{506045,1586},
{213479407, 351649}, /* 593 -- 601 */ {24929,1558},{529709,1548},{1042435,1535},{31867,1517},
{229580147, 368449}, /* 607 -- 617 */ {166037,1495},{928781,1478},{508975,1458},{4327,1442},
{250367549, 383161}, /* 619 -- 641 */ {779637,1430},{742091,1418},{258263,1411},{879631,1396},
{271661713, 413449}, /* 643 -- 653 */ {72029,1385},{728905,1377},{589057,1363},{348621,1356},
{293158127, 434281}, /* 659 -- 673 */ {671515,1332},{710453,1315},{84249,1296},{959363,1292},
{319512181, 458329}, /* 677 -- 691 */ {685853,1277},{467591,1274},{646643,1267},{683029,1264},
{357349471, 491401}, /* 701 -- 719 */ {439927,1249},{254461,1229},{660713,1223},{554195,1220},
{393806449, 528529}, /* 727 -- 739 */ {202911,1215},{753253,1195},{941457,1190},{776635,1187},
{422400701, 552049}, /* 743 -- 757 */ {509511,1182},{986147,1156},{768879,1151},{699431,1140},
{452366557, 579121}, /* 761 -- 773 */ {696417,1128},{86169,1119},{808997,1114},{25467,1107},
{507436351, 619369}, /* 787 -- 809 */ {201353,1100},{708087,1084},{1018339,1079},{341297,1073},
{547978913, 657721}, /* 811 -- 823 */ {434151,1066},{96287,1058},{950765,1051},{298257,1039},
{575204137, 683929}, /* 827 -- 839 */ {675933,1035},{167731,1029},{815445,1027},
{627947039, 727609}, /* 853 -- 859 */
{666785731, 744769}, /* 863 -- 881 */
{710381447, 779689}, /* 883 -- 907 */
{777767161, 829921}, /* 911 -- 929 */
{834985999, 877969}, /* 937 -- 947 */
{894826021, 908209}, /* 953 -- 971 */
{951747481, 954529}, /* 977 -- 991 */
{1019050649, 994009}, /* 997 -- 1013 */
{1040399, 1038361}, /* 1019 -- 1021 */
}; };
#define SIEVE_SIZE (sizeof(sieve_table) / sizeof(sieve_table[0])) /* 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
};
/* Combined Miller-Rabin test to the base a, and checking the /* Combined Miller-Rabin test to the base a, and checking the
conditions from Pocklington's theorem. */ conditions from Pocklington's theorem. */
...@@ -176,19 +229,14 @@ miller_rabin_pocklington(mpz_t n, mpz_t nm1, mpz_t nm1dq, mpz_t a) ...@@ -176,19 +229,14 @@ miller_rabin_pocklington(mpz_t n, mpz_t nm1, mpz_t nm1dq, mpz_t a)
return is_prime; return is_prime;
} }
/* Generate random prime of a given size. Maurer's algorithm (Alg.
6.42 Handbook of applied cryptography), but with ratio = 1/2 (like
the variant in fips186-3). FIXME: Force primes to start with two
one bits? */
/* The algorithm is based on the following special case of /* The algorithm is based on the following special case of
Pocklington's theorem: Pocklington's theorem:
Assume that n = 1 + r q, where q is a prime, q > sqrt(n) - 1. If we Assume that n = 1 + f q, where q is a prime, q > sqrt(n) - 1. If we
can find an a such that can find an a such that
a^{n-1} = 1 (mod n) a^{n-1} = 1 (mod n)
gcd(a^r - 1, n) = 1 gcd(a^f - 1, n) = 1
then n is prime. then n is prime.
...@@ -203,42 +251,98 @@ miller_rabin_pocklington(mpz_t n, mpz_t nm1, mpz_t nm1dq, mpz_t a) ...@@ -203,42 +251,98 @@ miller_rabin_pocklington(mpz_t n, mpz_t nm1, mpz_t nm1dq, mpz_t a)
If n is specified as k bits, we need q of size ceil(k/2) + 1 bits If n is specified as k bits, we need q of size ceil(k/2) + 1 bits
(or more) to make the theorem apply. (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. */
void void
nettle_random_prime(mpz_t p, unsigned bits, _nettle_generate_pocklington_prime (mpz_t p, unsigned bits,
void *ctx, nettle_random_func random) void *ctx, nettle_random_func random,
const mpz_t p0,
const mpz_t q,
const mpz_t p0q)
{ {
assert (bits >= 6); mpz_t i, r, pm1,a;
if (bits < 20)
assert (2*mpz_sizeinbase (p0, 2) > bits + 1);
mpz_init (i);
mpz_init (r);
mpz_init (pm1);
mpz_init (a);
/* i = floor (2^{bits-2} / p0q) */
mpz_init_set_ui (i, 1);
mpz_mul_2exp (i, i, bits-2);
mpz_fdiv_q (i, i, p0q);
for (;;)
{ {
unsigned long highbit; uint8_t buf[1];
uint8_t buf[3];
unsigned long x;
unsigned j;
/* Small cases: /* Generate r in the range i + 1 <= r <= 2*i */
nettle_mpz_random (r, ctx, random, i);
mpz_add (r, r, i);
mpz_add_ui (r, r, 1);
3 bits: 5 or 7 /* Set p = 2*r*p0q + 1 */
4 bits: 11, 13, 15 mpz_mul_2exp(r, r, 1);
5 bits: 17, 19, 23, 29, 31 mpz_mul (pm1, r, p0q);
mpz_add_ui (p, pm1, 1);
With 3 bits, no sieving is done, since candidates are smaller assert(mpz_sizeinbase(p, 2) == bits);
than 3^2 = 9 (and this is ok; all odd 3-bit numbers are
prime). /* 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_mul (r, r, q);
if (miller_rabin_pocklington(p, pm1, r, a))
break;
}
mpz_clear (i);
mpz_clear (r);
mpz_clear (pm1);
mpz_clear (a);
}
With 4 bits, sieving with the first value, 3*5*...*23 doesn't /* Generate random prime of a given size. Maurer's algorithm (Alg.
work, since this includes the primes 11 and 13 in the 6.42 Handbook of applied cryptography), but with ratio = 1/2 (like
interval. Of the odd numbers in the interval, 9, 11, 13, 15, the variant in fips186-3). FIXME: Force primes to start with two
only the factors of three need be discarded. one bits? */
void
nettle_random_prime(mpz_t p, unsigned bits,
void *ctx, nettle_random_func random)
{
assert (bits >= 3);
if (bits <= 10)
{
unsigned first;
unsigned choices;
uint8_t buf;
With 5 bits, we still sieve with only the first value, which random (ctx, sizeof(buf), &buf);
includes three of the primes in the interval. Of the odd
numbers in the interval, 17, 19, (21), 23, (25), (27), 29,
31, we need to discard multiples of 3 and 5 only.
With 6 bits, we sieve with only the first value (since 63 < first = prime_by_size[bits-3];
29^2), and there's no problem. 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); highbit = 1L << (bits - 1);
...@@ -248,69 +352,28 @@ nettle_random_prime(mpz_t p, unsigned bits, ...@@ -248,69 +352,28 @@ nettle_random_prime(mpz_t p, unsigned bits,
x &= (highbit - 1); x &= (highbit - 1);
x |= highbit | 1; x |= highbit | 1;
mpz_set_ui (p, x); for (j = 0; prime_square[j] <= x; j++)
for (j = 0; j < SIEVE_SIZE && x >= sieve_table[j].p2; j++) {
if (mpz_gcd_ui (NULL, p, sieve_table[j].prod) != 1) unsigned q = x * trial_div_table[j].inverse & TRIAL_DIV_MASK;
goto again; if (q <= trial_div_table[j].limit)
goto again;
}
mpz_set_ui (p, x);
} }
else else
{ {
mpz_t q, r, nm1, t, a, i; mpz_t q;
unsigned j;
mpz_init (q); mpz_init (q);
mpz_init (r);
mpz_init (nm1);
mpz_init (t);
mpz_init (a);
mpz_init (i);
/* Bit size ceil(k/2) + 1, slightly larger than used in Alg. 4.62 /* Bit size ceil(k/2) + 1, slightly larger than used in Alg. 4.62
in Handbook of Applied Cryptography (which seems to be in Handbook of Applied Cryptography (which seems to be
incorrect for odd k). */ incorrect for odd k). */
nettle_random_prime (q, (bits+3)/2, ctx, random); nettle_random_prime (q, (bits+3)/2, ctx, random);
/* i = floor (2^{bits-2} / q) */ _nettle_generate_pocklington_prime (p, bits, ctx, random,
mpz_init_set_ui (i, 1); q, NULL, q);
mpz_mul_2exp (i, i, bits-2);
mpz_fdiv_q (i, i, q);
for (;;)
{
uint8_t buf[1];
/* Generate r in the range i + 1 <= r <= 2*i */
nettle_mpz_random (r, ctx, random, i);
mpz_add (r, r, i);
mpz_add_ui (r, r, 1);
/* Set p = 2*r*q + 1 */
mpz_mul_2exp(r, r, 1);
mpz_mul (nm1, r, q);
mpz_add_ui (p, nm1, 1);
assert(mpz_sizeinbase(p, 2) == bits);
for (j = 0; j < SIEVE_SIZE; j++)
{
if (mpz_gcd_ui (NULL, p, sieve_table[j].prod) != 1)
goto composite;
}
random(ctx, sizeof(buf), buf);
mpz_set_ui (a, buf[0] + 2);
if (miller_rabin_pocklington(p, nm1, r, a))
break;
composite:
;
}
mpz_clear (q); mpz_clear (q);
mpz_clear (r);
mpz_clear (nm1);
mpz_clear (t);
mpz_clear (a);
mpz_clear (i);
} }
} }
...@@ -89,6 +89,13 @@ void ...@@ -89,6 +89,13 @@ void
nettle_random_prime(mpz_t p, unsigned bits, nettle_random_prime(mpz_t p, unsigned bits,
void *ctx, nettle_random_func random); void *ctx, nettle_random_func random);
void
_nettle_generate_pocklington_prime (mpz_t p, unsigned bits,
void *ctx, nettle_random_func random,
const mpz_t p0,
const mpz_t q,
const mpz_t p0q);
/* sexp parsing */ /* sexp parsing */
struct sexp_iterator; struct sexp_iterator;
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment