1 /* $OpenBSD: moduli.c,v 1.21 2008/06/26 09:19:40 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 42 #include <sys/types.h> 43 44 #include <openssl/bn.h> 45 #include <openssl/dh.h> 46 47 #include <stdio.h> 48 #include <stdlib.h> 49 #include <string.h> 50 #include <stdarg.h> 51 #include <time.h> 52 53 #include "xmalloc.h" 54 #include "dh.h" 55 #include "log.h" 56 57 /* 58 * File output defines 59 */ 60 61 /* need line long enough for largest moduli plus headers */ 62 #define QLINESIZE (100+8192) 63 64 /* 65 * Size: decimal. 66 * Specifies the number of the most significant bit (0 to M). 67 * WARNING: internally, usually 1 to N. 68 */ 69 #define QSIZE_MINIMUM (511) 70 71 /* 72 * Prime sieving defines 73 */ 74 75 /* Constant: assuming 8 bit bytes and 32 bit words */ 76 #define SHIFT_BIT (3) 77 #define SHIFT_BYTE (2) 78 #define SHIFT_WORD (SHIFT_BIT+SHIFT_BYTE) 79 #define SHIFT_MEGABYTE (20) 80 #define SHIFT_MEGAWORD (SHIFT_MEGABYTE-SHIFT_BYTE) 81 82 /* 83 * Using virtual memory can cause thrashing. This should be the largest 84 * number that is supported without a large amount of disk activity -- 85 * that would increase the run time from hours to days or weeks! 86 */ 87 #define LARGE_MINIMUM (8UL) /* megabytes */ 88 89 /* 90 * Do not increase this number beyond the unsigned integer bit size. 91 * Due to a multiple of 4, it must be LESS than 128 (yielding 2**30 bits). 92 */ 93 #define LARGE_MAXIMUM (127UL) /* megabytes */ 94 95 /* 96 * Constant: when used with 32-bit integers, the largest sieve prime 97 * has to be less than 2**32. 98 */ 99 #define SMALL_MAXIMUM (0xffffffffUL) 100 101 /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */ 102 #define TINY_NUMBER (1UL<<16) 103 104 /* Ensure enough bit space for testing 2*q. */ 105 #define TEST_MAXIMUM (1UL<<16) 106 #define TEST_MINIMUM (QSIZE_MINIMUM + 1) 107 /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */ 108 #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */ 109 110 /* bit operations on 32-bit words */ 111 #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31))) 112 #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31))) 113 #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31))) 114 115 /* 116 * Prime testing defines 117 */ 118 119 /* Minimum number of primality tests to perform */ 120 #define TRIAL_MINIMUM (4) 121 122 /* 123 * Sieving data (XXX - move to struct) 124 */ 125 126 /* sieve 2**16 */ 127 static u_int32_t *TinySieve, tinybits; 128 129 /* sieve 2**30 in 2**16 parts */ 130 static u_int32_t *SmallSieve, smallbits, smallbase; 131 132 /* sieve relative to the initial value */ 133 static u_int32_t *LargeSieve, largewords, largetries, largenumbers; 134 static u_int32_t largebits, largememory; /* megabytes */ 135 static BIGNUM *largebase; 136 137 int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *); 138 int prime_test(FILE *, FILE *, u_int32_t, u_int32_t); 139 140 /* 141 * print moduli out in consistent form, 142 */ 143 static int 144 qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries, 145 u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus) 146 { 147 struct tm *gtm; 148 time_t time_now; 149 int res; 150 151 time(&time_now); 152 gtm = gmtime(&time_now); 153 154 res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ", 155 gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday, 156 gtm->tm_hour, gtm->tm_min, gtm->tm_sec, 157 otype, otests, otries, osize, ogenerator); 158 159 if (res < 0) 160 return (-1); 161 162 if (BN_print_fp(ofile, omodulus) < 1) 163 return (-1); 164 165 res = fprintf(ofile, "\n"); 166 fflush(ofile); 167 168 return (res > 0 ? 0 : -1); 169 } 170 171 172 /* 173 ** Sieve p's and q's with small factors 174 */ 175 static void 176 sieve_large(u_int32_t s) 177 { 178 u_int32_t r, u; 179 180 debug3("sieve_large %u", s); 181 largetries++; 182 /* r = largebase mod s */ 183 r = BN_mod_word(largebase, s); 184 if (r == 0) 185 u = 0; /* s divides into largebase exactly */ 186 else 187 u = s - r; /* largebase+u is first entry divisible by s */ 188 189 if (u < largebits * 2) { 190 /* 191 * The sieve omits p's and q's divisible by 2, so ensure that 192 * largebase+u is odd. Then, step through the sieve in 193 * increments of 2*s 194 */ 195 if (u & 0x1) 196 u += s; /* Make largebase+u odd, and u even */ 197 198 /* Mark all multiples of 2*s */ 199 for (u /= 2; u < largebits; u += s) 200 BIT_SET(LargeSieve, u); 201 } 202 203 /* r = p mod s */ 204 r = (2 * r + 1) % s; 205 if (r == 0) 206 u = 0; /* s divides p exactly */ 207 else 208 u = s - r; /* p+u is first entry divisible by s */ 209 210 if (u < largebits * 4) { 211 /* 212 * The sieve omits p's divisible by 4, so ensure that 213 * largebase+u is not. Then, step through the sieve in 214 * increments of 4*s 215 */ 216 while (u & 0x3) { 217 if (SMALL_MAXIMUM - u < s) 218 return; 219 u += s; 220 } 221 222 /* Mark all multiples of 4*s */ 223 for (u /= 4; u < largebits; u += s) 224 BIT_SET(LargeSieve, u); 225 } 226 } 227 228 /* 229 * list candidates for Sophie-Germain primes (where q = (p-1)/2) 230 * to standard output. 231 * The list is checked against small known primes (less than 2**30). 232 */ 233 int 234 gen_candidates(FILE *out, u_int32_t memory, u_int32_t power, BIGNUM *start) 235 { 236 BIGNUM *q; 237 u_int32_t j, r, s, t; 238 u_int32_t smallwords = TINY_NUMBER >> 6; 239 u_int32_t tinywords = TINY_NUMBER >> 6; 240 time_t time_start, time_stop; 241 u_int32_t i; 242 int ret = 0; 243 244 largememory = memory; 245 246 if (memory != 0 && 247 (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) { 248 error("Invalid memory amount (min %ld, max %ld)", 249 LARGE_MINIMUM, LARGE_MAXIMUM); 250 return (-1); 251 } 252 253 /* 254 * Set power to the length in bits of the prime to be generated. 255 * This is changed to 1 less than the desired safe prime moduli p. 256 */ 257 if (power > TEST_MAXIMUM) { 258 error("Too many bits: %u > %lu", power, TEST_MAXIMUM); 259 return (-1); 260 } else if (power < TEST_MINIMUM) { 261 error("Too few bits: %u < %u", power, TEST_MINIMUM); 262 return (-1); 263 } 264 power--; /* decrement before squaring */ 265 266 /* 267 * The density of ordinary primes is on the order of 1/bits, so the 268 * density of safe primes should be about (1/bits)**2. Set test range 269 * to something well above bits**2 to be reasonably sure (but not 270 * guaranteed) of catching at least one safe prime. 271 */ 272 largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER)); 273 274 /* 275 * Need idea of how much memory is available. We don't have to use all 276 * of it. 277 */ 278 if (largememory > LARGE_MAXIMUM) { 279 logit("Limited memory: %u MB; limit %lu MB", 280 largememory, LARGE_MAXIMUM); 281 largememory = LARGE_MAXIMUM; 282 } 283 284 if (largewords <= (largememory << SHIFT_MEGAWORD)) { 285 logit("Increased memory: %u MB; need %u bytes", 286 largememory, (largewords << SHIFT_BYTE)); 287 largewords = (largememory << SHIFT_MEGAWORD); 288 } else if (largememory > 0) { 289 logit("Decreased memory: %u MB; want %u bytes", 290 largememory, (largewords << SHIFT_BYTE)); 291 largewords = (largememory << SHIFT_MEGAWORD); 292 } 293 294 TinySieve = xcalloc(tinywords, sizeof(u_int32_t)); 295 tinybits = tinywords << SHIFT_WORD; 296 297 SmallSieve = xcalloc(smallwords, sizeof(u_int32_t)); 298 smallbits = smallwords << SHIFT_WORD; 299 300 /* 301 * dynamically determine available memory 302 */ 303 while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL) 304 largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */ 305 306 largebits = largewords << SHIFT_WORD; 307 largenumbers = largebits * 2; /* even numbers excluded */ 308 309 /* validation check: count the number of primes tried */ 310 largetries = 0; 311 if ((q = BN_new()) == NULL) 312 fatal("BN_new failed"); 313 314 /* 315 * Generate random starting point for subprime search, or use 316 * specified parameter. 317 */ 318 if ((largebase = BN_new()) == NULL) 319 fatal("BN_new failed"); 320 if (start == NULL) { 321 if (BN_rand(largebase, power, 1, 1) == 0) 322 fatal("BN_rand failed"); 323 } else { 324 if (BN_copy(largebase, start) == NULL) 325 fatal("BN_copy: failed"); 326 } 327 328 /* ensure odd */ 329 if (BN_set_bit(largebase, 0) == 0) 330 fatal("BN_set_bit: failed"); 331 332 time(&time_start); 333 334 logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start), 335 largenumbers, power); 336 debug2("start point: 0x%s", BN_bn2hex(largebase)); 337 338 /* 339 * TinySieve 340 */ 341 for (i = 0; i < tinybits; i++) { 342 if (BIT_TEST(TinySieve, i)) 343 continue; /* 2*i+3 is composite */ 344 345 /* The next tiny prime */ 346 t = 2 * i + 3; 347 348 /* Mark all multiples of t */ 349 for (j = i + t; j < tinybits; j += t) 350 BIT_SET(TinySieve, j); 351 352 sieve_large(t); 353 } 354 355 /* 356 * Start the small block search at the next possible prime. To avoid 357 * fencepost errors, the last pass is skipped. 358 */ 359 for (smallbase = TINY_NUMBER + 3; 360 smallbase < (SMALL_MAXIMUM - TINY_NUMBER); 361 smallbase += TINY_NUMBER) { 362 for (i = 0; i < tinybits; i++) { 363 if (BIT_TEST(TinySieve, i)) 364 continue; /* 2*i+3 is composite */ 365 366 /* The next tiny prime */ 367 t = 2 * i + 3; 368 r = smallbase % t; 369 370 if (r == 0) { 371 s = 0; /* t divides into smallbase exactly */ 372 } else { 373 /* smallbase+s is first entry divisible by t */ 374 s = t - r; 375 } 376 377 /* 378 * The sieve omits even numbers, so ensure that 379 * smallbase+s is odd. Then, step through the sieve 380 * in increments of 2*t 381 */ 382 if (s & 1) 383 s += t; /* Make smallbase+s odd, and s even */ 384 385 /* Mark all multiples of 2*t */ 386 for (s /= 2; s < smallbits; s += t) 387 BIT_SET(SmallSieve, s); 388 } 389 390 /* 391 * SmallSieve 392 */ 393 for (i = 0; i < smallbits; i++) { 394 if (BIT_TEST(SmallSieve, i)) 395 continue; /* 2*i+smallbase is composite */ 396 397 /* The next small prime */ 398 sieve_large((2 * i) + smallbase); 399 } 400 401 memset(SmallSieve, 0, smallwords << SHIFT_BYTE); 402 } 403 404 time(&time_stop); 405 406 logit("%.24s Sieved with %u small primes in %ld seconds", 407 ctime(&time_stop), largetries, (long) (time_stop - time_start)); 408 409 for (j = r = 0; j < largebits; j++) { 410 if (BIT_TEST(LargeSieve, j)) 411 continue; /* Definitely composite, skip */ 412 413 debug2("test q = largebase+%u", 2 * j); 414 if (BN_set_word(q, 2 * j) == 0) 415 fatal("BN_set_word failed"); 416 if (BN_add(q, q, largebase) == 0) 417 fatal("BN_add failed"); 418 if (qfileout(out, MODULI_TYPE_SOPHIE_GERMAIN, 419 MODULI_TESTS_SIEVE, largetries, 420 (power - 1) /* MSB */, (0), q) == -1) { 421 ret = -1; 422 break; 423 } 424 425 r++; /* count q */ 426 } 427 428 time(&time_stop); 429 430 xfree(LargeSieve); 431 xfree(SmallSieve); 432 xfree(TinySieve); 433 434 logit("%.24s Found %u candidates", ctime(&time_stop), r); 435 436 return (ret); 437 } 438 439 /* 440 * perform a Miller-Rabin primality test 441 * on the list of candidates 442 * (checking both q and p) 443 * The result is a list of so-call "safe" primes 444 */ 445 int 446 prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted) 447 { 448 BIGNUM *q, *p, *a; 449 BN_CTX *ctx; 450 char *cp, *lp; 451 u_int32_t count_in = 0, count_out = 0, count_possible = 0; 452 u_int32_t generator_known, in_tests, in_tries, in_type, in_size; 453 time_t time_start, time_stop; 454 int res; 455 456 if (trials < TRIAL_MINIMUM) { 457 error("Minimum primality trials is %d", TRIAL_MINIMUM); 458 return (-1); 459 } 460 461 time(&time_start); 462 463 if ((p = BN_new()) == NULL) 464 fatal("BN_new failed"); 465 if ((q = BN_new()) == NULL) 466 fatal("BN_new failed"); 467 if ((ctx = BN_CTX_new()) == NULL) 468 fatal("BN_CTX_new failed"); 469 470 debug2("%.24s Final %u Miller-Rabin trials (%x generator)", 471 ctime(&time_start), trials, generator_wanted); 472 473 res = 0; 474 lp = xmalloc(QLINESIZE + 1); 475 while (fgets(lp, QLINESIZE + 1, in) != NULL) { 476 count_in++; 477 if (strlen(lp) < 14 || *lp == '!' || *lp == '#') { 478 debug2("%10u: comment or short line", count_in); 479 continue; 480 } 481 482 /* XXX - fragile parser */ 483 /* time */ 484 cp = &lp[14]; /* (skip) */ 485 486 /* type */ 487 in_type = strtoul(cp, &cp, 10); 488 489 /* tests */ 490 in_tests = strtoul(cp, &cp, 10); 491 492 if (in_tests & MODULI_TESTS_COMPOSITE) { 493 debug2("%10u: known composite", count_in); 494 continue; 495 } 496 497 /* tries */ 498 in_tries = strtoul(cp, &cp, 10); 499 500 /* size (most significant bit) */ 501 in_size = strtoul(cp, &cp, 10); 502 503 /* generator (hex) */ 504 generator_known = strtoul(cp, &cp, 16); 505 506 /* Skip white space */ 507 cp += strspn(cp, " "); 508 509 /* modulus (hex) */ 510 switch (in_type) { 511 case MODULI_TYPE_SOPHIE_GERMAIN: 512 debug2("%10u: (%u) Sophie-Germain", count_in, in_type); 513 a = q; 514 if (BN_hex2bn(&a, cp) == 0) 515 fatal("BN_hex2bn failed"); 516 /* p = 2*q + 1 */ 517 if (BN_lshift(p, q, 1) == 0) 518 fatal("BN_lshift failed"); 519 if (BN_add_word(p, 1) == 0) 520 fatal("BN_add_word failed"); 521 in_size += 1; 522 generator_known = 0; 523 break; 524 case MODULI_TYPE_UNSTRUCTURED: 525 case MODULI_TYPE_SAFE: 526 case MODULI_TYPE_SCHNORR: 527 case MODULI_TYPE_STRONG: 528 case MODULI_TYPE_UNKNOWN: 529 debug2("%10u: (%u)", count_in, in_type); 530 a = p; 531 if (BN_hex2bn(&a, cp) == 0) 532 fatal("BN_hex2bn failed"); 533 /* q = (p-1) / 2 */ 534 if (BN_rshift(q, p, 1) == 0) 535 fatal("BN_rshift failed"); 536 break; 537 default: 538 debug2("Unknown prime type"); 539 break; 540 } 541 542 /* 543 * due to earlier inconsistencies in interpretation, check 544 * the proposed bit size. 545 */ 546 if ((u_int32_t)BN_num_bits(p) != (in_size + 1)) { 547 debug2("%10u: bit size %u mismatch", count_in, in_size); 548 continue; 549 } 550 if (in_size < QSIZE_MINIMUM) { 551 debug2("%10u: bit size %u too short", count_in, in_size); 552 continue; 553 } 554 555 if (in_tests & MODULI_TESTS_MILLER_RABIN) 556 in_tries += trials; 557 else 558 in_tries = trials; 559 560 /* 561 * guess unknown generator 562 */ 563 if (generator_known == 0) { 564 if (BN_mod_word(p, 24) == 11) 565 generator_known = 2; 566 else if (BN_mod_word(p, 12) == 5) 567 generator_known = 3; 568 else { 569 u_int32_t r = BN_mod_word(p, 10); 570 571 if (r == 3 || r == 7) 572 generator_known = 5; 573 } 574 } 575 /* 576 * skip tests when desired generator doesn't match 577 */ 578 if (generator_wanted > 0 && 579 generator_wanted != generator_known) { 580 debug2("%10u: generator %d != %d", 581 count_in, generator_known, generator_wanted); 582 continue; 583 } 584 585 /* 586 * Primes with no known generator are useless for DH, so 587 * skip those. 588 */ 589 if (generator_known == 0) { 590 debug2("%10u: no known generator", count_in); 591 continue; 592 } 593 594 count_possible++; 595 596 /* 597 * The (1/4)^N performance bound on Miller-Rabin is 598 * extremely pessimistic, so don't spend a lot of time 599 * really verifying that q is prime until after we know 600 * that p is also prime. A single pass will weed out the 601 * vast majority of composite q's. 602 */ 603 if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) { 604 debug("%10u: q failed first possible prime test", 605 count_in); 606 continue; 607 } 608 609 /* 610 * q is possibly prime, so go ahead and really make sure 611 * that p is prime. If it is, then we can go back and do 612 * the same for q. If p is composite, chances are that 613 * will show up on the first Rabin-Miller iteration so it 614 * doesn't hurt to specify a high iteration count. 615 */ 616 if (!BN_is_prime(p, trials, NULL, ctx, NULL)) { 617 debug("%10u: p is not prime", count_in); 618 continue; 619 } 620 debug("%10u: p is almost certainly prime", count_in); 621 622 /* recheck q more rigorously */ 623 if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) { 624 debug("%10u: q is not prime", count_in); 625 continue; 626 } 627 debug("%10u: q is almost certainly prime", count_in); 628 629 if (qfileout(out, MODULI_TYPE_SAFE, 630 in_tests | MODULI_TESTS_MILLER_RABIN, 631 in_tries, in_size, generator_known, p)) { 632 res = -1; 633 break; 634 } 635 636 count_out++; 637 } 638 639 time(&time_stop); 640 xfree(lp); 641 BN_free(p); 642 BN_free(q); 643 BN_CTX_free(ctx); 644 645 logit("%.24s Found %u safe primes of %u candidates in %ld seconds", 646 ctime(&time_stop), count_out, count_possible, 647 (long) (time_stop - time_start)); 648 649 return (res); 650 } 651