1 /* $OpenBSD: moduli.c,v 1.1 2003/07/28 09:49:56 djm Exp $ */
3 * Copyright 1994 Phil Karn <karn@qualcomm.com>
4 * Copyright 1996-1998, 2003 William Allen Simpson <wsimpson@greendragon.com>
5 * Copyright 2000 Niels Provos <provos@citi.umich.edu>
8 * Redistribution and use in source and binary forms, with or without
9 * modification, are permitted provided that the following conditions
11 * 1. Redistributions of source code must retain the above copyright
12 * notice, this list of conditions and the following disclaimer.
13 * 2. Redistributions in binary form must reproduce the above copyright
14 * notice, this list of conditions and the following disclaimer in the
15 * documentation and/or other materials provided with the distribution.
17 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
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19 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
20 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
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22 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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30 * Two-step process to generate safe primes for DHGEX
32 * Sieve candidates for "safe" primes,
33 * suitable for use as Diffie-Hellman moduli;
34 * that is, where q = (p-1)/2 is also prime.
36 * First step: generate candidate primes (memory intensive)
37 * Second step: test primes' safety (processor intensive)
45 #include <openssl/bn.h>
52 /* define DEBUG_LARGE 1 */
53 /* define DEBUG_SMALL 1 */
54 /* define DEBUG_TEST 1 */
60 /* need line long enough for largest moduli plus headers */
61 #define QLINESIZE (100+8192)
64 * Specifies the internal structure of the prime modulus.
66 #define QTYPE_UNKNOWN (0)
67 #define QTYPE_UNSTRUCTURED (1)
68 #define QTYPE_SAFE (2)
69 #define QTYPE_SCHNOOR (3)
70 #define QTYPE_SOPHIE_GERMAINE (4)
71 #define QTYPE_STRONG (5)
73 /* Tests: decimal (bit field).
74 * Specifies the methods used in checking for primality.
75 * Usually, more than one test is used.
77 #define QTEST_UNTESTED (0x00)
78 #define QTEST_COMPOSITE (0x01)
79 #define QTEST_SIEVE (0x02)
80 #define QTEST_MILLER_RABIN (0x04)
81 #define QTEST_JACOBI (0x08)
82 #define QTEST_ELLIPTIC (0x10)
85 * Specifies the number of the most significant bit (0 to M).
86 ** WARNING: internally, usually 1 to N.
88 #define QSIZE_MINIMUM (511)
91 * Prime sieving defines
94 /* Constant: assuming 8 bit bytes and 32 bit words */
96 #define SHIFT_BYTE (2)
97 #define SHIFT_WORD (SHIFT_BIT+SHIFT_BYTE)
98 #define SHIFT_MEGABYTE (20)
99 #define SHIFT_MEGAWORD (SHIFT_MEGABYTE-SHIFT_BYTE)
102 * Constant: when used with 32-bit integers, the largest sieve prime
103 * has to be less than 2**32.
105 #define SMALL_MAXIMUM (0xffffffffUL)
107 /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */
108 #define TINY_NUMBER (1UL<<16)
110 /* Ensure enough bit space for testing 2*q. */
111 #define TEST_MAXIMUM (1UL<<16)
112 #define TEST_MINIMUM (QSIZE_MINIMUM + 1)
113 /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */
114 #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */
116 /* bit operations on 32-bit words */
117 #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31)))
118 #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31)))
119 #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31)))
122 * Prime testing defines
126 * Sieving data (XXX - move to struct)
130 static u_int32_t *TinySieve, tinybits;
132 /* sieve 2**30 in 2**16 parts */
133 static u_int32_t *SmallSieve, smallbits, smallbase;
135 /* sieve relative to the initial value */
136 static u_int32_t *LargeSieve, largewords, largetries, largenumbers;
137 static u_int32_t largebits, largememory; /* megabytes */
138 static BIGNUM *largebase;
142 * print moduli out in consistent form,
145 qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries,
146 u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus)
153 gtm = gmtime(&time_now);
155 res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ",
156 gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday,
157 gtm->tm_hour, gtm->tm_min, gtm->tm_sec,
158 otype, otests, otries, osize, ogenerator);
163 if (BN_print_fp(ofile, omodulus) < 1)
166 res = fprintf(ofile, "\n");
169 return (res > 0 ? 0 : -1);
174 ** Sieve p's and q's with small factors
177 sieve_large(u_int32_t s)
181 debug2("sieve_large %u", s);
183 /* r = largebase mod s */
184 r = BN_mod_word(largebase, s);
186 u = 0; /* s divides into largebase exactly */
188 u = s - r; /* largebase+u is first entry divisible by s */
190 if (u < largebits * 2) {
192 * The sieve omits p's and q's divisible by 2, so ensure that
193 * largebase+u is odd. Then, step through the sieve in
197 u += s; /* Make largebase+u odd, and u even */
199 /* Mark all multiples of 2*s */
200 for (u /= 2; u < largebits; u += s)
201 BIT_SET(LargeSieve, u);
207 u = 0; /* s divides p exactly */
209 u = s - r; /* p+u is first entry divisible by s */
211 if (u < largebits * 4) {
213 * The sieve omits p's divisible by 4, so ensure that
214 * largebase+u is not. Then, step through the sieve in
218 if (SMALL_MAXIMUM - u < s)
223 /* Mark all multiples of 4*s */
224 for (u /= 4; u < largebits; u += s)
225 BIT_SET(LargeSieve, u);
230 * list candidates for Sophie-Germaine primes (where q = (p-1)/2)
231 * to standard output.
232 * The list is checked against small known primes (less than 2**30).
235 gen_candidates(FILE *out, int memory, int power, BIGNUM *start)
238 u_int32_t j, r, s, t;
239 u_int32_t smallwords = TINY_NUMBER >> 6;
240 u_int32_t tinywords = TINY_NUMBER >> 6;
241 time_t time_start, time_stop;
244 largememory = memory;
247 * Set power to the length in bits of the prime to be generated.
248 * This is changed to 1 less than the desired safe prime moduli p.
250 if (power > TEST_MAXIMUM) {
251 error("Too many bits: %u > %lu", power, TEST_MAXIMUM);
253 } else if (power < TEST_MINIMUM) {
254 error("Too few bits: %u < %u", power, TEST_MINIMUM);
257 power--; /* decrement before squaring */
260 * The density of ordinary primes is on the order of 1/bits, so the
261 * density of safe primes should be about (1/bits)**2. Set test range
262 * to something well above bits**2 to be reasonably sure (but not
263 * guaranteed) of catching at least one safe prime.
265 largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER));
268 * Need idea of how much memory is available. We don't have to use all
271 if (largememory > LARGE_MAXIMUM) {
272 logit("Limited memory: %u MB; limit %lu MB",
273 largememory, LARGE_MAXIMUM);
274 largememory = LARGE_MAXIMUM;
277 if (largewords <= (largememory << SHIFT_MEGAWORD)) {
278 logit("Increased memory: %u MB; need %u bytes",
279 largememory, (largewords << SHIFT_BYTE));
280 largewords = (largememory << SHIFT_MEGAWORD);
281 } else if (largememory > 0) {
282 logit("Decreased memory: %u MB; want %u bytes",
283 largememory, (largewords << SHIFT_BYTE));
284 largewords = (largememory << SHIFT_MEGAWORD);
287 TinySieve = calloc(tinywords, sizeof(u_int32_t));
288 if (TinySieve == NULL) {
289 error("Insufficient memory for tiny sieve: need %u bytes",
290 tinywords << SHIFT_BYTE);
293 tinybits = tinywords << SHIFT_WORD;
295 SmallSieve = calloc(smallwords, sizeof(u_int32_t));
296 if (SmallSieve == NULL) {
297 error("Insufficient memory for small sieve: need %u bytes",
298 smallwords << SHIFT_BYTE);
302 smallbits = smallwords << SHIFT_WORD;
305 * dynamically determine available memory
307 while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL)
308 largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */
310 largebits = largewords << SHIFT_WORD;
311 largenumbers = largebits * 2; /* even numbers excluded */
313 /* validation check: count the number of primes tried */
318 * Generate random starting point for subprime search, or use
319 * specified parameter.
321 largebase = BN_new();
323 BN_rand(largebase, power, 1, 1);
325 BN_copy(largebase, start);
328 BN_set_bit(largebase, 0);
332 logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start),
333 largenumbers, power);
334 debug2("start point: 0x%s", BN_bn2hex(largebase));
339 for (i = 0; i < tinybits; i++) {
340 if (BIT_TEST(TinySieve, i))
341 continue; /* 2*i+3 is composite */
343 /* The next tiny prime */
346 /* Mark all multiples of t */
347 for (j = i + t; j < tinybits; j += t)
348 BIT_SET(TinySieve, j);
354 * Start the small block search at the next possible prime. To avoid
355 * fencepost errors, the last pass is skipped.
357 for (smallbase = TINY_NUMBER + 3;
358 smallbase < (SMALL_MAXIMUM - TINY_NUMBER);
359 smallbase += TINY_NUMBER) {
360 for (i = 0; i < tinybits; i++) {
361 if (BIT_TEST(TinySieve, i))
362 continue; /* 2*i+3 is composite */
364 /* The next tiny prime */
369 s = 0; /* t divides into smallbase exactly */
371 /* smallbase+s is first entry divisible by t */
376 * The sieve omits even numbers, so ensure that
377 * smallbase+s is odd. Then, step through the sieve
378 * in increments of 2*t
381 s += t; /* Make smallbase+s odd, and s even */
383 /* Mark all multiples of 2*t */
384 for (s /= 2; s < smallbits; s += t)
385 BIT_SET(SmallSieve, s);
391 for (i = 0; i < smallbits; i++) {
392 if (BIT_TEST(SmallSieve, i))
393 continue; /* 2*i+smallbase is composite */
395 /* The next small prime */
396 sieve_large((2 * i) + smallbase);
399 memset(SmallSieve, 0, smallwords << SHIFT_BYTE);
404 logit("%.24s Sieved with %u small primes in %ld seconds",
405 ctime(&time_stop), largetries, (long) (time_stop - time_start));
407 for (j = r = 0; j < largebits; j++) {
408 if (BIT_TEST(LargeSieve, j))
409 continue; /* Definitely composite, skip */
411 debug2("test q = largebase+%u", 2 * j);
412 BN_set_word(q, 2 * j);
413 BN_add(q, q, largebase);
414 if (qfileout(out, QTYPE_SOPHIE_GERMAINE, QTEST_SIEVE,
415 largetries, (power - 1) /* MSB */, (0), q) == -1) {
429 logit("%.24s Found %u candidates", ctime(&time_stop), r);
435 * perform a Miller-Rabin primality test
436 * on the list of candidates
437 * (checking both q and p)
438 * The result is a list of so-call "safe" primes
441 prime_test(FILE *in, FILE *out, u_int32_t trials,
442 u_int32_t generator_wanted)
447 u_int32_t count_in = 0, count_out = 0, count_possible = 0;
448 u_int32_t generator_known, in_tests, in_tries, in_type, in_size;
449 time_t time_start, time_stop;
458 debug2("%.24s Final %u Miller-Rabin trials (%x generator)",
459 ctime(&time_start), trials, generator_wanted);
462 lp = xmalloc(QLINESIZE + 1);
463 while (fgets(lp, QLINESIZE, in) != NULL) {
467 if (ll < 14 || *lp == '!' || *lp == '#') {
468 debug2("%10u: comment or short line", count_in);
472 /* XXX - fragile parser */
474 cp = &lp[14]; /* (skip) */
477 in_type = strtoul(cp, &cp, 10);
480 in_tests = strtoul(cp, &cp, 10);
482 if (in_tests & QTEST_COMPOSITE) {
483 debug2("%10u: known composite", count_in);
487 in_tries = strtoul(cp, &cp, 10);
489 /* size (most significant bit) */
490 in_size = strtoul(cp, &cp, 10);
492 /* generator (hex) */
493 generator_known = strtoul(cp, &cp, 16);
495 /* Skip white space */
496 cp += strspn(cp, " ");
500 case QTYPE_SOPHIE_GERMAINE:
501 debug2("%10u: (%u) Sophie-Germaine", count_in, in_type);
511 debug2("%10u: (%u)", count_in, in_type);
520 * due to earlier inconsistencies in interpretation, check
521 * the proposed bit size.
523 if (BN_num_bits(p) != (in_size + 1)) {
524 debug2("%10u: bit size %u mismatch", count_in, in_size);
527 if (in_size < QSIZE_MINIMUM) {
528 debug2("%10u: bit size %u too short", count_in, in_size);
532 if (in_tests & QTEST_MILLER_RABIN)
537 * guess unknown generator
539 if (generator_known == 0) {
540 if (BN_mod_word(p, 24) == 11)
542 else if (BN_mod_word(p, 12) == 5)
545 u_int32_t r = BN_mod_word(p, 10);
547 if (r == 3 || r == 7) {
553 * skip tests when desired generator doesn't match
555 if (generator_wanted > 0 &&
556 generator_wanted != generator_known) {
557 debug2("%10u: generator %d != %d",
558 count_in, generator_known, generator_wanted);
565 * The (1/4)^N performance bound on Miller-Rabin is
566 * extremely pessimistic, so don't spend a lot of time
567 * really verifying that q is prime until after we know
568 * that p is also prime. A single pass will weed out the
569 * vast majority of composite q's.
571 if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) {
572 debug2("%10u: q failed first possible prime test",
578 * q is possibly prime, so go ahead and really make sure
579 * that p is prime. If it is, then we can go back and do
580 * the same for q. If p is composite, chances are that
581 * will show up on the first Rabin-Miller iteration so it
582 * doesn't hurt to specify a high iteration count.
584 if (!BN_is_prime(p, trials, NULL, ctx, NULL)) {
585 debug2("%10u: p is not prime", count_in);
588 debug("%10u: p is almost certainly prime", count_in);
590 /* recheck q more rigorously */
591 if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) {
592 debug("%10u: q is not prime", count_in);
595 debug("%10u: q is almost certainly prime", count_in);
597 if (qfileout(out, QTYPE_SAFE, (in_tests | QTEST_MILLER_RABIN),
598 in_tries, in_size, generator_known, p)) {
612 logit("%.24s Found %u safe primes of %u candidates in %ld seconds",
613 ctime(&time_stop), count_out, count_possible,
614 (long) (time_stop - time_start));