1 /* $NetBSD: moduli.c,v 1.2 2009/06/07 22:38:46 christos Exp $ */ 2 /* $OpenBSD: moduli.c,v 1.21 2008/06/26 09:19:40 djm Exp $ */ 3 /* 4 * Copyright 1994 Phil Karn <karn@qualcomm.com> 5 * Copyright 1996-1998, 2003 William Allen Simpson <wsimpson@greendragon.com> 6 * Copyright 2000 Niels Provos <provos@citi.umich.edu> 7 * All rights reserved. 8 * 9 * Redistribution and use in source and binary forms, with or without 10 * modification, are permitted provided that the following conditions 11 * are met: 12 * 1. Redistributions of source code must retain the above copyright 13 * notice, this list of conditions and the following disclaimer. 14 * 2. Redistributions in binary form must reproduce the above copyright 15 * notice, this list of conditions and the following disclaimer in the 16 * documentation and/or other materials provided with the distribution. 17 * 18 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 19 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 20 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 21 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 22 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 23 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 24 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 25 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 26 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 27 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 28 */ 29 30 /* 31 * Two-step process to generate safe primes for DHGEX 32 * 33 * Sieve candidates for "safe" primes, 34 * suitable for use as Diffie-Hellman moduli; 35 * that is, where q = (p-1)/2 is also prime. 36 * 37 * First step: generate candidate primes (memory intensive) 38 * Second step: test primes' safety (processor intensive) 39 */ 40 #include "includes.h" 41 __RCSID("$NetBSD: moduli.c,v 1.2 2009/06/07 22:38:46 christos Exp $"); 42 43 #include <sys/types.h> 44 45 #include <openssl/bn.h> 46 #include <openssl/dh.h> 47 48 #include <stdio.h> 49 #include <stdlib.h> 50 #include <string.h> 51 #include <stdarg.h> 52 #include <time.h> 53 54 #include "xmalloc.h" 55 #include "dh.h" 56 #include "log.h" 57 58 /* 59 * File output defines 60 */ 61 62 /* need line long enough for largest moduli plus headers */ 63 #define QLINESIZE (100+8192) 64 65 /* 66 * Size: decimal. 67 * Specifies the number of the most significant bit (0 to M). 68 * WARNING: internally, usually 1 to N. 69 */ 70 #define QSIZE_MINIMUM (511) 71 72 /* 73 * Prime sieving defines 74 */ 75 76 /* Constant: assuming 8 bit bytes and 32 bit words */ 77 #define SHIFT_BIT (3) 78 #define SHIFT_BYTE (2) 79 #define SHIFT_WORD (SHIFT_BIT+SHIFT_BYTE) 80 #define SHIFT_MEGABYTE (20) 81 #define SHIFT_MEGAWORD (SHIFT_MEGABYTE-SHIFT_BYTE) 82 83 /* 84 * Using virtual memory can cause thrashing. This should be the largest 85 * number that is supported without a large amount of disk activity -- 86 * that would increase the run time from hours to days or weeks! 87 */ 88 #define LARGE_MINIMUM (8UL) /* megabytes */ 89 90 /* 91 * Do not increase this number beyond the unsigned integer bit size. 92 * Due to a multiple of 4, it must be LESS than 128 (yielding 2**30 bits). 93 */ 94 #define LARGE_MAXIMUM (127UL) /* megabytes */ 95 96 /* 97 * Constant: when used with 32-bit integers, the largest sieve prime 98 * has to be less than 2**32. 99 */ 100 #define SMALL_MAXIMUM (0xffffffffUL) 101 102 /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */ 103 #define TINY_NUMBER (1UL<<16) 104 105 /* Ensure enough bit space for testing 2*q. */ 106 #define TEST_MAXIMUM (1UL<<16) 107 #define TEST_MINIMUM (QSIZE_MINIMUM + 1) 108 /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */ 109 #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */ 110 111 /* bit operations on 32-bit words */ 112 #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31))) 113 #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31))) 114 #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31))) 115 116 /* 117 * Prime testing defines 118 */ 119 120 /* Minimum number of primality tests to perform */ 121 #define TRIAL_MINIMUM (4) 122 123 /* 124 * Sieving data (XXX - move to struct) 125 */ 126 127 /* sieve 2**16 */ 128 static u_int32_t *TinySieve, tinybits; 129 130 /* sieve 2**30 in 2**16 parts */ 131 static u_int32_t *SmallSieve, smallbits, smallbase; 132 133 /* sieve relative to the initial value */ 134 static u_int32_t *LargeSieve, largewords, largetries, largenumbers; 135 static u_int32_t largebits, largememory; /* megabytes */ 136 static BIGNUM *largebase; 137 138 int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *); 139 int prime_test(FILE *, FILE *, u_int32_t, u_int32_t); 140 141 /* 142 * print moduli out in consistent form, 143 */ 144 static int 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) 147 { 148 struct tm *gtm; 149 time_t time_now; 150 int res; 151 152 time(&time_now); 153 gtm = gmtime(&time_now); 154 155 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); 159 160 if (res < 0) 161 return (-1); 162 163 if (BN_print_fp(ofile, omodulus) < 1) 164 return (-1); 165 166 res = fprintf(ofile, "\n"); 167 fflush(ofile); 168 169 return (res > 0 ? 0 : -1); 170 } 171 172 173 /* 174 ** Sieve p's and q's with small factors 175 */ 176 static void 177 sieve_large(u_int32_t s) 178 { 179 u_int32_t r, u; 180 181 debug3("sieve_large %u", s); 182 largetries++; 183 /* r = largebase mod s */ 184 r = BN_mod_word(largebase, s); 185 if (r == 0) 186 u = 0; /* s divides into largebase exactly */ 187 else 188 u = s - r; /* largebase+u is first entry divisible by s */ 189 190 if (u < largebits * 2) { 191 /* 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 194 * increments of 2*s 195 */ 196 if (u & 0x1) 197 u += s; /* Make largebase+u odd, and u even */ 198 199 /* Mark all multiples of 2*s */ 200 for (u /= 2; u < largebits; u += s) 201 BIT_SET(LargeSieve, u); 202 } 203 204 /* r = p mod s */ 205 r = (2 * r + 1) % s; 206 if (r == 0) 207 u = 0; /* s divides p exactly */ 208 else 209 u = s - r; /* p+u is first entry divisible by s */ 210 211 if (u < largebits * 4) { 212 /* 213 * The sieve omits p's divisible by 4, so ensure that 214 * largebase+u is not. Then, step through the sieve in 215 * increments of 4*s 216 */ 217 while (u & 0x3) { 218 if (SMALL_MAXIMUM - u < s) 219 return; 220 u += s; 221 } 222 223 /* Mark all multiples of 4*s */ 224 for (u /= 4; u < largebits; u += s) 225 BIT_SET(LargeSieve, u); 226 } 227 } 228 229 /* 230 * list candidates for Sophie-Germain primes (where q = (p-1)/2) 231 * to standard output. 232 * The list is checked against small known primes (less than 2**30). 233 */ 234 int 235 gen_candidates(FILE *out, u_int32_t memory, u_int32_t power, BIGNUM *start) 236 { 237 BIGNUM *q; 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; 242 u_int32_t i; 243 int ret = 0; 244 245 largememory = memory; 246 247 if (memory != 0 && 248 (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) { 249 error("Invalid memory amount (min %ld, max %ld)", 250 LARGE_MINIMUM, LARGE_MAXIMUM); 251 return (-1); 252 } 253 254 /* 255 * Set power to the length in bits of the prime to be generated. 256 * This is changed to 1 less than the desired safe prime moduli p. 257 */ 258 if (power > TEST_MAXIMUM) { 259 error("Too many bits: %u > %lu", power, TEST_MAXIMUM); 260 return (-1); 261 } else if (power < TEST_MINIMUM) { 262 error("Too few bits: %u < %u", power, TEST_MINIMUM); 263 return (-1); 264 } 265 power--; /* decrement before squaring */ 266 267 /* 268 * The density of ordinary primes is on the order of 1/bits, so the 269 * density of safe primes should be about (1/bits)**2. Set test range 270 * to something well above bits**2 to be reasonably sure (but not 271 * guaranteed) of catching at least one safe prime. 272 */ 273 largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER)); 274 275 /* 276 * Need idea of how much memory is available. We don't have to use all 277 * of it. 278 */ 279 if (largememory > LARGE_MAXIMUM) { 280 logit("Limited memory: %u MB; limit %lu MB", 281 largememory, LARGE_MAXIMUM); 282 largememory = LARGE_MAXIMUM; 283 } 284 285 if (largewords <= (largememory << SHIFT_MEGAWORD)) { 286 logit("Increased memory: %u MB; need %u bytes", 287 largememory, (largewords << SHIFT_BYTE)); 288 largewords = (largememory << SHIFT_MEGAWORD); 289 } else if (largememory > 0) { 290 logit("Decreased memory: %u MB; want %u bytes", 291 largememory, (largewords << SHIFT_BYTE)); 292 largewords = (largememory << SHIFT_MEGAWORD); 293 } 294 295 TinySieve = xcalloc(tinywords, sizeof(u_int32_t)); 296 tinybits = tinywords << SHIFT_WORD; 297 298 SmallSieve = xcalloc(smallwords, sizeof(u_int32_t)); 299 smallbits = smallwords << SHIFT_WORD; 300 301 /* 302 * dynamically determine available memory 303 */ 304 while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL) 305 largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */ 306 307 largebits = largewords << SHIFT_WORD; 308 largenumbers = largebits * 2; /* even numbers excluded */ 309 310 /* validation check: count the number of primes tried */ 311 largetries = 0; 312 if ((q = BN_new()) == NULL) 313 fatal("BN_new failed"); 314 315 /* 316 * Generate random starting point for subprime search, or use 317 * specified parameter. 318 */ 319 if ((largebase = BN_new()) == NULL) 320 fatal("BN_new failed"); 321 if (start == NULL) { 322 if (BN_rand(largebase, power, 1, 1) == 0) 323 fatal("BN_rand failed"); 324 } else { 325 if (BN_copy(largebase, start) == NULL) 326 fatal("BN_copy: failed"); 327 } 328 329 /* ensure odd */ 330 if (BN_set_bit(largebase, 0) == 0) 331 fatal("BN_set_bit: failed"); 332 333 time(&time_start); 334 335 logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start), 336 largenumbers, power); 337 debug2("start point: 0x%s", BN_bn2hex(largebase)); 338 339 /* 340 * TinySieve 341 */ 342 for (i = 0; i < tinybits; i++) { 343 if (BIT_TEST(TinySieve, i)) 344 continue; /* 2*i+3 is composite */ 345 346 /* The next tiny prime */ 347 t = 2 * i + 3; 348 349 /* Mark all multiples of t */ 350 for (j = i + t; j < tinybits; j += t) 351 BIT_SET(TinySieve, j); 352 353 sieve_large(t); 354 } 355 356 /* 357 * Start the small block search at the next possible prime. To avoid 358 * fencepost errors, the last pass is skipped. 359 */ 360 for (smallbase = TINY_NUMBER + 3; 361 smallbase < (SMALL_MAXIMUM - TINY_NUMBER); 362 smallbase += TINY_NUMBER) { 363 for (i = 0; i < tinybits; i++) { 364 if (BIT_TEST(TinySieve, i)) 365 continue; /* 2*i+3 is composite */ 366 367 /* The next tiny prime */ 368 t = 2 * i + 3; 369 r = smallbase % t; 370 371 if (r == 0) { 372 s = 0; /* t divides into smallbase exactly */ 373 } else { 374 /* smallbase+s is first entry divisible by t */ 375 s = t - r; 376 } 377 378 /* 379 * The sieve omits even numbers, so ensure that 380 * smallbase+s is odd. Then, step through the sieve 381 * in increments of 2*t 382 */ 383 if (s & 1) 384 s += t; /* Make smallbase+s odd, and s even */ 385 386 /* Mark all multiples of 2*t */ 387 for (s /= 2; s < smallbits; s += t) 388 BIT_SET(SmallSieve, s); 389 } 390 391 /* 392 * SmallSieve 393 */ 394 for (i = 0; i < smallbits; i++) { 395 if (BIT_TEST(SmallSieve, i)) 396 continue; /* 2*i+smallbase is composite */ 397 398 /* The next small prime */ 399 sieve_large((2 * i) + smallbase); 400 } 401 402 memset(SmallSieve, 0, smallwords << SHIFT_BYTE); 403 } 404 405 time(&time_stop); 406 407 logit("%.24s Sieved with %u small primes in %ld seconds", 408 ctime(&time_stop), largetries, (long) (time_stop - time_start)); 409 410 for (j = r = 0; j < largebits; j++) { 411 if (BIT_TEST(LargeSieve, j)) 412 continue; /* Definitely composite, skip */ 413 414 debug2("test q = largebase+%u", 2 * j); 415 if (BN_set_word(q, 2 * j) == 0) 416 fatal("BN_set_word failed"); 417 if (BN_add(q, q, largebase) == 0) 418 fatal("BN_add failed"); 419 if (qfileout(out, MODULI_TYPE_SOPHIE_GERMAIN, 420 MODULI_TESTS_SIEVE, largetries, 421 (power - 1) /* MSB */, (0), q) == -1) { 422 ret = -1; 423 break; 424 } 425 426 r++; /* count q */ 427 } 428 429 time(&time_stop); 430 431 xfree(LargeSieve); 432 xfree(SmallSieve); 433 xfree(TinySieve); 434 435 logit("%.24s Found %u candidates", ctime(&time_stop), r); 436 437 return (ret); 438 } 439 440 /* 441 * perform a Miller-Rabin primality test 442 * on the list of candidates 443 * (checking both q and p) 444 * The result is a list of so-call "safe" primes 445 */ 446 int 447 prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted) 448 { 449 BIGNUM *q, *p, *a; 450 BN_CTX *ctx; 451 char *cp, *lp; 452 u_int32_t count_in = 0, count_out = 0, count_possible = 0; 453 u_int32_t generator_known, in_tests, in_tries, in_type, in_size; 454 time_t time_start, time_stop; 455 int res; 456 457 if (trials < TRIAL_MINIMUM) { 458 error("Minimum primality trials is %d", TRIAL_MINIMUM); 459 return (-1); 460 } 461 462 time(&time_start); 463 464 if ((p = BN_new()) == NULL) 465 fatal("BN_new failed"); 466 if ((q = BN_new()) == NULL) 467 fatal("BN_new failed"); 468 if ((ctx = BN_CTX_new()) == NULL) 469 fatal("BN_CTX_new failed"); 470 471 debug2("%.24s Final %u Miller-Rabin trials (%x generator)", 472 ctime(&time_start), trials, generator_wanted); 473 474 res = 0; 475 lp = xmalloc(QLINESIZE + 1); 476 while (fgets(lp, QLINESIZE + 1, in) != NULL) { 477 count_in++; 478 if (strlen(lp) < 14 || *lp == '!' || *lp == '#') { 479 debug2("%10u: comment or short line", count_in); 480 continue; 481 } 482 483 /* XXX - fragile parser */ 484 /* time */ 485 cp = &lp[14]; /* (skip) */ 486 487 /* type */ 488 in_type = strtoul(cp, &cp, 10); 489 490 /* tests */ 491 in_tests = strtoul(cp, &cp, 10); 492 493 if (in_tests & MODULI_TESTS_COMPOSITE) { 494 debug2("%10u: known composite", count_in); 495 continue; 496 } 497 498 /* tries */ 499 in_tries = strtoul(cp, &cp, 10); 500 501 /* size (most significant bit) */ 502 in_size = strtoul(cp, &cp, 10); 503 504 /* generator (hex) */ 505 generator_known = strtoul(cp, &cp, 16); 506 507 /* Skip white space */ 508 cp += strspn(cp, " "); 509 510 /* modulus (hex) */ 511 switch (in_type) { 512 case MODULI_TYPE_SOPHIE_GERMAIN: 513 debug2("%10u: (%u) Sophie-Germain", count_in, in_type); 514 a = q; 515 if (BN_hex2bn(&a, cp) == 0) 516 fatal("BN_hex2bn failed"); 517 /* p = 2*q + 1 */ 518 if (BN_lshift(p, q, 1) == 0) 519 fatal("BN_lshift failed"); 520 if (BN_add_word(p, 1) == 0) 521 fatal("BN_add_word failed"); 522 in_size += 1; 523 generator_known = 0; 524 break; 525 case MODULI_TYPE_UNSTRUCTURED: 526 case MODULI_TYPE_SAFE: 527 case MODULI_TYPE_SCHNORR: 528 case MODULI_TYPE_STRONG: 529 case MODULI_TYPE_UNKNOWN: 530 debug2("%10u: (%u)", count_in, in_type); 531 a = p; 532 if (BN_hex2bn(&a, cp) == 0) 533 fatal("BN_hex2bn failed"); 534 /* q = (p-1) / 2 */ 535 if (BN_rshift(q, p, 1) == 0) 536 fatal("BN_rshift failed"); 537 break; 538 default: 539 debug2("Unknown prime type"); 540 break; 541 } 542 543 /* 544 * due to earlier inconsistencies in interpretation, check 545 * the proposed bit size. 546 */ 547 if ((u_int32_t)BN_num_bits(p) != (in_size + 1)) { 548 debug2("%10u: bit size %u mismatch", count_in, in_size); 549 continue; 550 } 551 if (in_size < QSIZE_MINIMUM) { 552 debug2("%10u: bit size %u too short", count_in, in_size); 553 continue; 554 } 555 556 if (in_tests & MODULI_TESTS_MILLER_RABIN) 557 in_tries += trials; 558 else 559 in_tries = trials; 560 561 /* 562 * guess unknown generator 563 */ 564 if (generator_known == 0) { 565 if (BN_mod_word(p, 24) == 11) 566 generator_known = 2; 567 else if (BN_mod_word(p, 12) == 5) 568 generator_known = 3; 569 else { 570 u_int32_t r = BN_mod_word(p, 10); 571 572 if (r == 3 || r == 7) 573 generator_known = 5; 574 } 575 } 576 /* 577 * skip tests when desired generator doesn't match 578 */ 579 if (generator_wanted > 0 && 580 generator_wanted != generator_known) { 581 debug2("%10u: generator %d != %d", 582 count_in, generator_known, generator_wanted); 583 continue; 584 } 585 586 /* 587 * Primes with no known generator are useless for DH, so 588 * skip those. 589 */ 590 if (generator_known == 0) { 591 debug2("%10u: no known generator", count_in); 592 continue; 593 } 594 595 count_possible++; 596 597 /* 598 * The (1/4)^N performance bound on Miller-Rabin is 599 * extremely pessimistic, so don't spend a lot of time 600 * really verifying that q is prime until after we know 601 * that p is also prime. A single pass will weed out the 602 * vast majority of composite q's. 603 */ 604 if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) { 605 debug("%10u: q failed first possible prime test", 606 count_in); 607 continue; 608 } 609 610 /* 611 * q is possibly prime, so go ahead and really make sure 612 * that p is prime. If it is, then we can go back and do 613 * the same for q. If p is composite, chances are that 614 * will show up on the first Rabin-Miller iteration so it 615 * doesn't hurt to specify a high iteration count. 616 */ 617 if (!BN_is_prime(p, trials, NULL, ctx, NULL)) { 618 debug("%10u: p is not prime", count_in); 619 continue; 620 } 621 debug("%10u: p is almost certainly prime", count_in); 622 623 /* recheck q more rigorously */ 624 if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) { 625 debug("%10u: q is not prime", count_in); 626 continue; 627 } 628 debug("%10u: q is almost certainly prime", count_in); 629 630 if (qfileout(out, MODULI_TYPE_SAFE, 631 in_tests | MODULI_TESTS_MILLER_RABIN, 632 in_tries, in_size, generator_known, p)) { 633 res = -1; 634 break; 635 } 636 637 count_out++; 638 } 639 640 time(&time_stop); 641 xfree(lp); 642 BN_free(p); 643 BN_free(q); 644 BN_CTX_free(ctx); 645 646 logit("%.24s Found %u safe primes of %u candidates in %ld seconds", 647 ctime(&time_stop), count_out, count_possible, 648 (long) (time_stop - time_start)); 649 650 return (res); 651 } 652