1 /* $OpenBSD: moduli.c,v 1.5 2003/12/22 09:16:57 djm Exp $ */ 2 /* 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> 6 * All rights reserved. 7 * 8 * Redistribution and use in source and binary forms, with or without 9 * modification, are permitted provided that the following conditions 10 * are met: 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. 16 * 17 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 18 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 19 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 20 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 21 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 22 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 23 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 24 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 25 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 26 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 27 */ 28 29 /* 30 * Two-step process to generate safe primes for DHGEX 31 * 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. 35 * 36 * First step: generate candidate primes (memory intensive) 37 * Second step: test primes' safety (processor intensive) 38 */ 39 40 #include "includes.h" 41 #include "moduli.h" 42 #include "xmalloc.h" 43 #include "log.h" 44 45 #include <openssl/bn.h> 46 47 /* 48 * File output defines 49 */ 50 51 /* need line long enough for largest moduli plus headers */ 52 #define QLINESIZE (100+8192) 53 54 /* Type: decimal. 55 * Specifies the internal structure of the prime modulus. 56 */ 57 #define QTYPE_UNKNOWN (0) 58 #define QTYPE_UNSTRUCTURED (1) 59 #define QTYPE_SAFE (2) 60 #define QTYPE_SCHNOOR (3) 61 #define QTYPE_SOPHIE_GERMAINE (4) 62 #define QTYPE_STRONG (5) 63 64 /* Tests: decimal (bit field). 65 * Specifies the methods used in checking for primality. 66 * Usually, more than one test is used. 67 */ 68 #define QTEST_UNTESTED (0x00) 69 #define QTEST_COMPOSITE (0x01) 70 #define QTEST_SIEVE (0x02) 71 #define QTEST_MILLER_RABIN (0x04) 72 #define QTEST_JACOBI (0x08) 73 #define QTEST_ELLIPTIC (0x10) 74 75 /* 76 * Size: decimal. 77 * Specifies the number of the most significant bit (0 to M). 78 * WARNING: internally, usually 1 to N. 79 */ 80 #define QSIZE_MINIMUM (511) 81 82 /* 83 * Prime sieving defines 84 */ 85 86 /* Constant: assuming 8 bit bytes and 32 bit words */ 87 #define SHIFT_BIT (3) 88 #define SHIFT_BYTE (2) 89 #define SHIFT_WORD (SHIFT_BIT+SHIFT_BYTE) 90 #define SHIFT_MEGABYTE (20) 91 #define SHIFT_MEGAWORD (SHIFT_MEGABYTE-SHIFT_BYTE) 92 93 /* 94 * Constant: when used with 32-bit integers, the largest sieve prime 95 * has to be less than 2**32. 96 */ 97 #define SMALL_MAXIMUM (0xffffffffUL) 98 99 /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */ 100 #define TINY_NUMBER (1UL<<16) 101 102 /* Ensure enough bit space for testing 2*q. */ 103 #define TEST_MAXIMUM (1UL<<16) 104 #define TEST_MINIMUM (QSIZE_MINIMUM + 1) 105 /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */ 106 #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */ 107 108 /* bit operations on 32-bit words */ 109 #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31))) 110 #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31))) 111 #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31))) 112 113 /* 114 * Prime testing defines 115 */ 116 117 /* 118 * Sieving data (XXX - move to struct) 119 */ 120 121 /* sieve 2**16 */ 122 static u_int32_t *TinySieve, tinybits; 123 124 /* sieve 2**30 in 2**16 parts */ 125 static u_int32_t *SmallSieve, smallbits, smallbase; 126 127 /* sieve relative to the initial value */ 128 static u_int32_t *LargeSieve, largewords, largetries, largenumbers; 129 static u_int32_t largebits, largememory; /* megabytes */ 130 static BIGNUM *largebase; 131 132 133 /* 134 * print moduli out in consistent form, 135 */ 136 static int 137 qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries, 138 u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus) 139 { 140 struct tm *gtm; 141 time_t time_now; 142 int res; 143 144 time(&time_now); 145 gtm = gmtime(&time_now); 146 147 res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ", 148 gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday, 149 gtm->tm_hour, gtm->tm_min, gtm->tm_sec, 150 otype, otests, otries, osize, ogenerator); 151 152 if (res < 0) 153 return (-1); 154 155 if (BN_print_fp(ofile, omodulus) < 1) 156 return (-1); 157 158 res = fprintf(ofile, "\n"); 159 fflush(ofile); 160 161 return (res > 0 ? 0 : -1); 162 } 163 164 165 /* 166 ** Sieve p's and q's with small factors 167 */ 168 static void 169 sieve_large(u_int32_t s) 170 { 171 u_int32_t r, u; 172 173 debug3("sieve_large %u", s); 174 largetries++; 175 /* r = largebase mod s */ 176 r = BN_mod_word(largebase, s); 177 if (r == 0) 178 u = 0; /* s divides into largebase exactly */ 179 else 180 u = s - r; /* largebase+u is first entry divisible by s */ 181 182 if (u < largebits * 2) { 183 /* 184 * The sieve omits p's and q's divisible by 2, so ensure that 185 * largebase+u is odd. Then, step through the sieve in 186 * increments of 2*s 187 */ 188 if (u & 0x1) 189 u += s; /* Make largebase+u odd, and u even */ 190 191 /* Mark all multiples of 2*s */ 192 for (u /= 2; u < largebits; u += s) 193 BIT_SET(LargeSieve, u); 194 } 195 196 /* r = p mod s */ 197 r = (2 * r + 1) % s; 198 if (r == 0) 199 u = 0; /* s divides p exactly */ 200 else 201 u = s - r; /* p+u is first entry divisible by s */ 202 203 if (u < largebits * 4) { 204 /* 205 * The sieve omits p's divisible by 4, so ensure that 206 * largebase+u is not. Then, step through the sieve in 207 * increments of 4*s 208 */ 209 while (u & 0x3) { 210 if (SMALL_MAXIMUM - u < s) 211 return; 212 u += s; 213 } 214 215 /* Mark all multiples of 4*s */ 216 for (u /= 4; u < largebits; u += s) 217 BIT_SET(LargeSieve, u); 218 } 219 } 220 221 /* 222 * list candidates for Sophie-Germaine primes (where q = (p-1)/2) 223 * to standard output. 224 * The list is checked against small known primes (less than 2**30). 225 */ 226 int 227 gen_candidates(FILE *out, int memory, int power, BIGNUM *start) 228 { 229 BIGNUM *q; 230 u_int32_t j, r, s, t; 231 u_int32_t smallwords = TINY_NUMBER >> 6; 232 u_int32_t tinywords = TINY_NUMBER >> 6; 233 time_t time_start, time_stop; 234 int i, ret = 0; 235 236 largememory = memory; 237 238 /* 239 * Set power to the length in bits of the prime to be generated. 240 * This is changed to 1 less than the desired safe prime moduli p. 241 */ 242 if (power > TEST_MAXIMUM) { 243 error("Too many bits: %u > %lu", power, TEST_MAXIMUM); 244 return (-1); 245 } else if (power < TEST_MINIMUM) { 246 error("Too few bits: %u < %u", power, TEST_MINIMUM); 247 return (-1); 248 } 249 power--; /* decrement before squaring */ 250 251 /* 252 * The density of ordinary primes is on the order of 1/bits, so the 253 * density of safe primes should be about (1/bits)**2. Set test range 254 * to something well above bits**2 to be reasonably sure (but not 255 * guaranteed) of catching at least one safe prime. 256 */ 257 largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER)); 258 259 /* 260 * Need idea of how much memory is available. We don't have to use all 261 * of it. 262 */ 263 if (largememory > LARGE_MAXIMUM) { 264 logit("Limited memory: %u MB; limit %lu MB", 265 largememory, LARGE_MAXIMUM); 266 largememory = LARGE_MAXIMUM; 267 } 268 269 if (largewords <= (largememory << SHIFT_MEGAWORD)) { 270 logit("Increased memory: %u MB; need %u bytes", 271 largememory, (largewords << SHIFT_BYTE)); 272 largewords = (largememory << SHIFT_MEGAWORD); 273 } else if (largememory > 0) { 274 logit("Decreased memory: %u MB; want %u bytes", 275 largememory, (largewords << SHIFT_BYTE)); 276 largewords = (largememory << SHIFT_MEGAWORD); 277 } 278 279 TinySieve = calloc(tinywords, sizeof(u_int32_t)); 280 if (TinySieve == NULL) { 281 error("Insufficient memory for tiny sieve: need %u bytes", 282 tinywords << SHIFT_BYTE); 283 exit(1); 284 } 285 tinybits = tinywords << SHIFT_WORD; 286 287 SmallSieve = calloc(smallwords, sizeof(u_int32_t)); 288 if (SmallSieve == NULL) { 289 error("Insufficient memory for small sieve: need %u bytes", 290 smallwords << SHIFT_BYTE); 291 xfree(TinySieve); 292 exit(1); 293 } 294 smallbits = smallwords << SHIFT_WORD; 295 296 /* 297 * dynamically determine available memory 298 */ 299 while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL) 300 largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */ 301 302 largebits = largewords << SHIFT_WORD; 303 largenumbers = largebits * 2; /* even numbers excluded */ 304 305 /* validation check: count the number of primes tried */ 306 largetries = 0; 307 q = BN_new(); 308 309 /* 310 * Generate random starting point for subprime search, or use 311 * specified parameter. 312 */ 313 largebase = BN_new(); 314 if (start == NULL) 315 BN_rand(largebase, power, 1, 1); 316 else 317 BN_copy(largebase, start); 318 319 /* ensure odd */ 320 BN_set_bit(largebase, 0); 321 322 time(&time_start); 323 324 logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start), 325 largenumbers, power); 326 debug2("start point: 0x%s", BN_bn2hex(largebase)); 327 328 /* 329 * TinySieve 330 */ 331 for (i = 0; i < tinybits; i++) { 332 if (BIT_TEST(TinySieve, i)) 333 continue; /* 2*i+3 is composite */ 334 335 /* The next tiny prime */ 336 t = 2 * i + 3; 337 338 /* Mark all multiples of t */ 339 for (j = i + t; j < tinybits; j += t) 340 BIT_SET(TinySieve, j); 341 342 sieve_large(t); 343 } 344 345 /* 346 * Start the small block search at the next possible prime. To avoid 347 * fencepost errors, the last pass is skipped. 348 */ 349 for (smallbase = TINY_NUMBER + 3; 350 smallbase < (SMALL_MAXIMUM - TINY_NUMBER); 351 smallbase += TINY_NUMBER) { 352 for (i = 0; i < tinybits; i++) { 353 if (BIT_TEST(TinySieve, i)) 354 continue; /* 2*i+3 is composite */ 355 356 /* The next tiny prime */ 357 t = 2 * i + 3; 358 r = smallbase % t; 359 360 if (r == 0) { 361 s = 0; /* t divides into smallbase exactly */ 362 } else { 363 /* smallbase+s is first entry divisible by t */ 364 s = t - r; 365 } 366 367 /* 368 * The sieve omits even numbers, so ensure that 369 * smallbase+s is odd. Then, step through the sieve 370 * in increments of 2*t 371 */ 372 if (s & 1) 373 s += t; /* Make smallbase+s odd, and s even */ 374 375 /* Mark all multiples of 2*t */ 376 for (s /= 2; s < smallbits; s += t) 377 BIT_SET(SmallSieve, s); 378 } 379 380 /* 381 * SmallSieve 382 */ 383 for (i = 0; i < smallbits; i++) { 384 if (BIT_TEST(SmallSieve, i)) 385 continue; /* 2*i+smallbase is composite */ 386 387 /* The next small prime */ 388 sieve_large((2 * i) + smallbase); 389 } 390 391 memset(SmallSieve, 0, smallwords << SHIFT_BYTE); 392 } 393 394 time(&time_stop); 395 396 logit("%.24s Sieved with %u small primes in %ld seconds", 397 ctime(&time_stop), largetries, (long) (time_stop - time_start)); 398 399 for (j = r = 0; j < largebits; j++) { 400 if (BIT_TEST(LargeSieve, j)) 401 continue; /* Definitely composite, skip */ 402 403 debug2("test q = largebase+%u", 2 * j); 404 BN_set_word(q, 2 * j); 405 BN_add(q, q, largebase); 406 if (qfileout(out, QTYPE_SOPHIE_GERMAINE, QTEST_SIEVE, 407 largetries, (power - 1) /* MSB */, (0), q) == -1) { 408 ret = -1; 409 break; 410 } 411 412 r++; /* count q */ 413 } 414 415 time(&time_stop); 416 417 xfree(LargeSieve); 418 xfree(SmallSieve); 419 xfree(TinySieve); 420 421 logit("%.24s Found %u candidates", ctime(&time_stop), r); 422 423 return (ret); 424 } 425 426 /* 427 * perform a Miller-Rabin primality test 428 * on the list of candidates 429 * (checking both q and p) 430 * The result is a list of so-call "safe" primes 431 */ 432 int 433 prime_test(FILE *in, FILE *out, u_int32_t trials, 434 u_int32_t generator_wanted) 435 { 436 BIGNUM *q, *p, *a; 437 BN_CTX *ctx; 438 char *cp, *lp; 439 u_int32_t count_in = 0, count_out = 0, count_possible = 0; 440 u_int32_t generator_known, in_tests, in_tries, in_type, in_size; 441 time_t time_start, time_stop; 442 int res; 443 444 time(&time_start); 445 446 p = BN_new(); 447 q = BN_new(); 448 ctx = BN_CTX_new(); 449 450 debug2("%.24s Final %u Miller-Rabin trials (%x generator)", 451 ctime(&time_start), trials, generator_wanted); 452 453 res = 0; 454 lp = xmalloc(QLINESIZE + 1); 455 while (fgets(lp, QLINESIZE, in) != NULL) { 456 int ll = strlen(lp); 457 458 count_in++; 459 if (ll < 14 || *lp == '!' || *lp == '#') { 460 debug2("%10u: comment or short line", count_in); 461 continue; 462 } 463 464 /* XXX - fragile parser */ 465 /* time */ 466 cp = &lp[14]; /* (skip) */ 467 468 /* type */ 469 in_type = strtoul(cp, &cp, 10); 470 471 /* tests */ 472 in_tests = strtoul(cp, &cp, 10); 473 474 if (in_tests & QTEST_COMPOSITE) { 475 debug2("%10u: known composite", count_in); 476 continue; 477 } 478 479 /* tries */ 480 in_tries = strtoul(cp, &cp, 10); 481 482 /* size (most significant bit) */ 483 in_size = strtoul(cp, &cp, 10); 484 485 /* generator (hex) */ 486 generator_known = strtoul(cp, &cp, 16); 487 488 /* Skip white space */ 489 cp += strspn(cp, " "); 490 491 /* modulus (hex) */ 492 switch (in_type) { 493 case QTYPE_SOPHIE_GERMAINE: 494 debug2("%10u: (%u) Sophie-Germaine", count_in, in_type); 495 a = q; 496 BN_hex2bn(&a, cp); 497 /* p = 2*q + 1 */ 498 BN_lshift(p, q, 1); 499 BN_add_word(p, 1); 500 in_size += 1; 501 generator_known = 0; 502 break; 503 case QTYPE_UNSTRUCTURED: 504 case QTYPE_SAFE: 505 case QTYPE_SCHNOOR: 506 case QTYPE_STRONG: 507 case QTYPE_UNKNOWN: 508 debug2("%10u: (%u)", count_in, in_type); 509 a = p; 510 BN_hex2bn(&a, cp); 511 /* q = (p-1) / 2 */ 512 BN_rshift(q, p, 1); 513 break; 514 default: 515 debug2("Unknown prime type"); 516 break; 517 } 518 519 /* 520 * due to earlier inconsistencies in interpretation, check 521 * the proposed bit size. 522 */ 523 if (BN_num_bits(p) != (in_size + 1)) { 524 debug2("%10u: bit size %u mismatch", count_in, in_size); 525 continue; 526 } 527 if (in_size < QSIZE_MINIMUM) { 528 debug2("%10u: bit size %u too short", count_in, in_size); 529 continue; 530 } 531 532 if (in_tests & QTEST_MILLER_RABIN) 533 in_tries += trials; 534 else 535 in_tries = trials; 536 537 /* 538 * guess unknown generator 539 */ 540 if (generator_known == 0) { 541 if (BN_mod_word(p, 24) == 11) 542 generator_known = 2; 543 else if (BN_mod_word(p, 12) == 5) 544 generator_known = 3; 545 else { 546 u_int32_t r = BN_mod_word(p, 10); 547 548 if (r == 3 || r == 7) 549 generator_known = 5; 550 } 551 } 552 /* 553 * skip tests when desired generator doesn't match 554 */ 555 if (generator_wanted > 0 && 556 generator_wanted != generator_known) { 557 debug2("%10u: generator %d != %d", 558 count_in, generator_known, generator_wanted); 559 continue; 560 } 561 562 /* 563 * Primes with no known generator are useless for DH, so 564 * skip those. 565 */ 566 if (generator_known == 0) { 567 debug2("%10u: no known generator", count_in); 568 continue; 569 } 570 571 count_possible++; 572 573 /* 574 * The (1/4)^N performance bound on Miller-Rabin is 575 * extremely pessimistic, so don't spend a lot of time 576 * really verifying that q is prime until after we know 577 * that p is also prime. A single pass will weed out the 578 * vast majority of composite q's. 579 */ 580 if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) { 581 debug("%10u: q failed first possible prime test", 582 count_in); 583 continue; 584 } 585 586 /* 587 * q is possibly prime, so go ahead and really make sure 588 * that p is prime. If it is, then we can go back and do 589 * the same for q. If p is composite, chances are that 590 * will show up on the first Rabin-Miller iteration so it 591 * doesn't hurt to specify a high iteration count. 592 */ 593 if (!BN_is_prime(p, trials, NULL, ctx, NULL)) { 594 debug("%10u: p is not prime", count_in); 595 continue; 596 } 597 debug("%10u: p is almost certainly prime", count_in); 598 599 /* recheck q more rigorously */ 600 if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) { 601 debug("%10u: q is not prime", count_in); 602 continue; 603 } 604 debug("%10u: q is almost certainly prime", count_in); 605 606 if (qfileout(out, QTYPE_SAFE, (in_tests | QTEST_MILLER_RABIN), 607 in_tries, in_size, generator_known, p)) { 608 res = -1; 609 break; 610 } 611 612 count_out++; 613 } 614 615 time(&time_stop); 616 xfree(lp); 617 BN_free(p); 618 BN_free(q); 619 BN_CTX_free(ctx); 620 621 logit("%.24s Found %u safe primes of %u candidates in %ld seconds", 622 ctime(&time_stop), count_out, count_possible, 623 (long) (time_stop - time_start)); 624 625 return (res); 626 } 627