1 /* $OpenBSD: moduli.c,v 1.19 2006/11/06 21:25:28 markus 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 46 #include <stdio.h> 47 #include <stdlib.h> 48 #include <string.h> 49 #include <stdarg.h> 50 #include <time.h> 51 52 #include "xmalloc.h" 53 #include "log.h" 54 55 /* 56 * File output defines 57 */ 58 59 /* need line long enough for largest moduli plus headers */ 60 #define QLINESIZE (100+8192) 61 62 /* Type: decimal. 63 * Specifies the internal structure of the prime modulus. 64 */ 65 #define QTYPE_UNKNOWN (0) 66 #define QTYPE_UNSTRUCTURED (1) 67 #define QTYPE_SAFE (2) 68 #define QTYPE_SCHNORR (3) 69 #define QTYPE_SOPHIE_GERMAIN (4) 70 #define QTYPE_STRONG (5) 71 72 /* Tests: decimal (bit field). 73 * Specifies the methods used in checking for primality. 74 * Usually, more than one test is used. 75 */ 76 #define QTEST_UNTESTED (0x00) 77 #define QTEST_COMPOSITE (0x01) 78 #define QTEST_SIEVE (0x02) 79 #define QTEST_MILLER_RABIN (0x04) 80 #define QTEST_JACOBI (0x08) 81 #define QTEST_ELLIPTIC (0x10) 82 83 /* 84 * Size: decimal. 85 * Specifies the number of the most significant bit (0 to M). 86 * WARNING: internally, usually 1 to N. 87 */ 88 #define QSIZE_MINIMUM (511) 89 90 /* 91 * Prime sieving defines 92 */ 93 94 /* Constant: assuming 8 bit bytes and 32 bit words */ 95 #define SHIFT_BIT (3) 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) 100 101 /* 102 * Using virtual memory can cause thrashing. This should be the largest 103 * number that is supported without a large amount of disk activity -- 104 * that would increase the run time from hours to days or weeks! 105 */ 106 #define LARGE_MINIMUM (8UL) /* megabytes */ 107 108 /* 109 * Do not increase this number beyond the unsigned integer bit size. 110 * Due to a multiple of 4, it must be LESS than 128 (yielding 2**30 bits). 111 */ 112 #define LARGE_MAXIMUM (127UL) /* megabytes */ 113 114 /* 115 * Constant: when used with 32-bit integers, the largest sieve prime 116 * has to be less than 2**32. 117 */ 118 #define SMALL_MAXIMUM (0xffffffffUL) 119 120 /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */ 121 #define TINY_NUMBER (1UL<<16) 122 123 /* Ensure enough bit space for testing 2*q. */ 124 #define TEST_MAXIMUM (1UL<<16) 125 #define TEST_MINIMUM (QSIZE_MINIMUM + 1) 126 /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */ 127 #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */ 128 129 /* bit operations on 32-bit words */ 130 #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31))) 131 #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31))) 132 #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31))) 133 134 /* 135 * Prime testing defines 136 */ 137 138 /* Minimum number of primality tests to perform */ 139 #define TRIAL_MINIMUM (4) 140 141 /* 142 * Sieving data (XXX - move to struct) 143 */ 144 145 /* sieve 2**16 */ 146 static u_int32_t *TinySieve, tinybits; 147 148 /* sieve 2**30 in 2**16 parts */ 149 static u_int32_t *SmallSieve, smallbits, smallbase; 150 151 /* sieve relative to the initial value */ 152 static u_int32_t *LargeSieve, largewords, largetries, largenumbers; 153 static u_int32_t largebits, largememory; /* megabytes */ 154 static BIGNUM *largebase; 155 156 int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *); 157 int prime_test(FILE *, FILE *, u_int32_t, u_int32_t); 158 159 /* 160 * print moduli out in consistent form, 161 */ 162 static int 163 qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries, 164 u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus) 165 { 166 struct tm *gtm; 167 time_t time_now; 168 int res; 169 170 time(&time_now); 171 gtm = gmtime(&time_now); 172 173 res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ", 174 gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday, 175 gtm->tm_hour, gtm->tm_min, gtm->tm_sec, 176 otype, otests, otries, osize, ogenerator); 177 178 if (res < 0) 179 return (-1); 180 181 if (BN_print_fp(ofile, omodulus) < 1) 182 return (-1); 183 184 res = fprintf(ofile, "\n"); 185 fflush(ofile); 186 187 return (res > 0 ? 0 : -1); 188 } 189 190 191 /* 192 ** Sieve p's and q's with small factors 193 */ 194 static void 195 sieve_large(u_int32_t s) 196 { 197 u_int32_t r, u; 198 199 debug3("sieve_large %u", s); 200 largetries++; 201 /* r = largebase mod s */ 202 r = BN_mod_word(largebase, s); 203 if (r == 0) 204 u = 0; /* s divides into largebase exactly */ 205 else 206 u = s - r; /* largebase+u is first entry divisible by s */ 207 208 if (u < largebits * 2) { 209 /* 210 * The sieve omits p's and q's divisible by 2, so ensure that 211 * largebase+u is odd. Then, step through the sieve in 212 * increments of 2*s 213 */ 214 if (u & 0x1) 215 u += s; /* Make largebase+u odd, and u even */ 216 217 /* Mark all multiples of 2*s */ 218 for (u /= 2; u < largebits; u += s) 219 BIT_SET(LargeSieve, u); 220 } 221 222 /* r = p mod s */ 223 r = (2 * r + 1) % s; 224 if (r == 0) 225 u = 0; /* s divides p exactly */ 226 else 227 u = s - r; /* p+u is first entry divisible by s */ 228 229 if (u < largebits * 4) { 230 /* 231 * The sieve omits p's divisible by 4, so ensure that 232 * largebase+u is not. Then, step through the sieve in 233 * increments of 4*s 234 */ 235 while (u & 0x3) { 236 if (SMALL_MAXIMUM - u < s) 237 return; 238 u += s; 239 } 240 241 /* Mark all multiples of 4*s */ 242 for (u /= 4; u < largebits; u += s) 243 BIT_SET(LargeSieve, u); 244 } 245 } 246 247 /* 248 * list candidates for Sophie-Germain primes (where q = (p-1)/2) 249 * to standard output. 250 * The list is checked against small known primes (less than 2**30). 251 */ 252 int 253 gen_candidates(FILE *out, u_int32_t memory, u_int32_t power, BIGNUM *start) 254 { 255 BIGNUM *q; 256 u_int32_t j, r, s, t; 257 u_int32_t smallwords = TINY_NUMBER >> 6; 258 u_int32_t tinywords = TINY_NUMBER >> 6; 259 time_t time_start, time_stop; 260 u_int32_t i; 261 int ret = 0; 262 263 largememory = memory; 264 265 if (memory != 0 && 266 (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) { 267 error("Invalid memory amount (min %ld, max %ld)", 268 LARGE_MINIMUM, LARGE_MAXIMUM); 269 return (-1); 270 } 271 272 /* 273 * Set power to the length in bits of the prime to be generated. 274 * This is changed to 1 less than the desired safe prime moduli p. 275 */ 276 if (power > TEST_MAXIMUM) { 277 error("Too many bits: %u > %lu", power, TEST_MAXIMUM); 278 return (-1); 279 } else if (power < TEST_MINIMUM) { 280 error("Too few bits: %u < %u", power, TEST_MINIMUM); 281 return (-1); 282 } 283 power--; /* decrement before squaring */ 284 285 /* 286 * The density of ordinary primes is on the order of 1/bits, so the 287 * density of safe primes should be about (1/bits)**2. Set test range 288 * to something well above bits**2 to be reasonably sure (but not 289 * guaranteed) of catching at least one safe prime. 290 */ 291 largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER)); 292 293 /* 294 * Need idea of how much memory is available. We don't have to use all 295 * of it. 296 */ 297 if (largememory > LARGE_MAXIMUM) { 298 logit("Limited memory: %u MB; limit %lu MB", 299 largememory, LARGE_MAXIMUM); 300 largememory = LARGE_MAXIMUM; 301 } 302 303 if (largewords <= (largememory << SHIFT_MEGAWORD)) { 304 logit("Increased memory: %u MB; need %u bytes", 305 largememory, (largewords << SHIFT_BYTE)); 306 largewords = (largememory << SHIFT_MEGAWORD); 307 } else if (largememory > 0) { 308 logit("Decreased memory: %u MB; want %u bytes", 309 largememory, (largewords << SHIFT_BYTE)); 310 largewords = (largememory << SHIFT_MEGAWORD); 311 } 312 313 TinySieve = xcalloc(tinywords, sizeof(u_int32_t)); 314 tinybits = tinywords << SHIFT_WORD; 315 316 SmallSieve = xcalloc(smallwords, sizeof(u_int32_t)); 317 smallbits = smallwords << SHIFT_WORD; 318 319 /* 320 * dynamically determine available memory 321 */ 322 while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL) 323 largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */ 324 325 largebits = largewords << SHIFT_WORD; 326 largenumbers = largebits * 2; /* even numbers excluded */ 327 328 /* validation check: count the number of primes tried */ 329 largetries = 0; 330 if ((q = BN_new()) == NULL) 331 fatal("BN_new failed"); 332 333 /* 334 * Generate random starting point for subprime search, or use 335 * specified parameter. 336 */ 337 if ((largebase = BN_new()) == NULL) 338 fatal("BN_new failed"); 339 if (start == NULL) { 340 if (BN_rand(largebase, power, 1, 1) == 0) 341 fatal("BN_rand failed"); 342 } else { 343 if (BN_copy(largebase, start) == NULL) 344 fatal("BN_copy: failed"); 345 } 346 347 /* ensure odd */ 348 if (BN_set_bit(largebase, 0) == 0) 349 fatal("BN_set_bit: failed"); 350 351 time(&time_start); 352 353 logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start), 354 largenumbers, power); 355 debug2("start point: 0x%s", BN_bn2hex(largebase)); 356 357 /* 358 * TinySieve 359 */ 360 for (i = 0; i < tinybits; i++) { 361 if (BIT_TEST(TinySieve, i)) 362 continue; /* 2*i+3 is composite */ 363 364 /* The next tiny prime */ 365 t = 2 * i + 3; 366 367 /* Mark all multiples of t */ 368 for (j = i + t; j < tinybits; j += t) 369 BIT_SET(TinySieve, j); 370 371 sieve_large(t); 372 } 373 374 /* 375 * Start the small block search at the next possible prime. To avoid 376 * fencepost errors, the last pass is skipped. 377 */ 378 for (smallbase = TINY_NUMBER + 3; 379 smallbase < (SMALL_MAXIMUM - TINY_NUMBER); 380 smallbase += TINY_NUMBER) { 381 for (i = 0; i < tinybits; i++) { 382 if (BIT_TEST(TinySieve, i)) 383 continue; /* 2*i+3 is composite */ 384 385 /* The next tiny prime */ 386 t = 2 * i + 3; 387 r = smallbase % t; 388 389 if (r == 0) { 390 s = 0; /* t divides into smallbase exactly */ 391 } else { 392 /* smallbase+s is first entry divisible by t */ 393 s = t - r; 394 } 395 396 /* 397 * The sieve omits even numbers, so ensure that 398 * smallbase+s is odd. Then, step through the sieve 399 * in increments of 2*t 400 */ 401 if (s & 1) 402 s += t; /* Make smallbase+s odd, and s even */ 403 404 /* Mark all multiples of 2*t */ 405 for (s /= 2; s < smallbits; s += t) 406 BIT_SET(SmallSieve, s); 407 } 408 409 /* 410 * SmallSieve 411 */ 412 for (i = 0; i < smallbits; i++) { 413 if (BIT_TEST(SmallSieve, i)) 414 continue; /* 2*i+smallbase is composite */ 415 416 /* The next small prime */ 417 sieve_large((2 * i) + smallbase); 418 } 419 420 memset(SmallSieve, 0, smallwords << SHIFT_BYTE); 421 } 422 423 time(&time_stop); 424 425 logit("%.24s Sieved with %u small primes in %ld seconds", 426 ctime(&time_stop), largetries, (long) (time_stop - time_start)); 427 428 for (j = r = 0; j < largebits; j++) { 429 if (BIT_TEST(LargeSieve, j)) 430 continue; /* Definitely composite, skip */ 431 432 debug2("test q = largebase+%u", 2 * j); 433 if (BN_set_word(q, 2 * j) == 0) 434 fatal("BN_set_word failed"); 435 if (BN_add(q, q, largebase) == 0) 436 fatal("BN_add failed"); 437 if (qfileout(out, QTYPE_SOPHIE_GERMAIN, QTEST_SIEVE, 438 largetries, (power - 1) /* MSB */, (0), q) == -1) { 439 ret = -1; 440 break; 441 } 442 443 r++; /* count q */ 444 } 445 446 time(&time_stop); 447 448 xfree(LargeSieve); 449 xfree(SmallSieve); 450 xfree(TinySieve); 451 452 logit("%.24s Found %u candidates", ctime(&time_stop), r); 453 454 return (ret); 455 } 456 457 /* 458 * perform a Miller-Rabin primality test 459 * on the list of candidates 460 * (checking both q and p) 461 * The result is a list of so-call "safe" primes 462 */ 463 int 464 prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted) 465 { 466 BIGNUM *q, *p, *a; 467 BN_CTX *ctx; 468 char *cp, *lp; 469 u_int32_t count_in = 0, count_out = 0, count_possible = 0; 470 u_int32_t generator_known, in_tests, in_tries, in_type, in_size; 471 time_t time_start, time_stop; 472 int res; 473 474 if (trials < TRIAL_MINIMUM) { 475 error("Minimum primality trials is %d", TRIAL_MINIMUM); 476 return (-1); 477 } 478 479 time(&time_start); 480 481 if ((p = BN_new()) == NULL) 482 fatal("BN_new failed"); 483 if ((q = BN_new()) == NULL) 484 fatal("BN_new failed"); 485 if ((ctx = BN_CTX_new()) == NULL) 486 fatal("BN_CTX_new failed"); 487 488 debug2("%.24s Final %u Miller-Rabin trials (%x generator)", 489 ctime(&time_start), trials, generator_wanted); 490 491 res = 0; 492 lp = xmalloc(QLINESIZE + 1); 493 while (fgets(lp, QLINESIZE, in) != NULL) { 494 int ll = strlen(lp); 495 496 count_in++; 497 if (ll < 14 || *lp == '!' || *lp == '#') { 498 debug2("%10u: comment or short line", count_in); 499 continue; 500 } 501 502 /* XXX - fragile parser */ 503 /* time */ 504 cp = &lp[14]; /* (skip) */ 505 506 /* type */ 507 in_type = strtoul(cp, &cp, 10); 508 509 /* tests */ 510 in_tests = strtoul(cp, &cp, 10); 511 512 if (in_tests & QTEST_COMPOSITE) { 513 debug2("%10u: known composite", count_in); 514 continue; 515 } 516 517 /* tries */ 518 in_tries = strtoul(cp, &cp, 10); 519 520 /* size (most significant bit) */ 521 in_size = strtoul(cp, &cp, 10); 522 523 /* generator (hex) */ 524 generator_known = strtoul(cp, &cp, 16); 525 526 /* Skip white space */ 527 cp += strspn(cp, " "); 528 529 /* modulus (hex) */ 530 switch (in_type) { 531 case QTYPE_SOPHIE_GERMAIN: 532 debug2("%10u: (%u) Sophie-Germain", count_in, in_type); 533 a = q; 534 if (BN_hex2bn(&a, cp) == 0) 535 fatal("BN_hex2bn failed"); 536 /* p = 2*q + 1 */ 537 if (BN_lshift(p, q, 1) == 0) 538 fatal("BN_lshift failed"); 539 if (BN_add_word(p, 1) == 0) 540 fatal("BN_add_word failed"); 541 in_size += 1; 542 generator_known = 0; 543 break; 544 case QTYPE_UNSTRUCTURED: 545 case QTYPE_SAFE: 546 case QTYPE_SCHNORR: 547 case QTYPE_STRONG: 548 case QTYPE_UNKNOWN: 549 debug2("%10u: (%u)", count_in, in_type); 550 a = p; 551 if (BN_hex2bn(&a, cp) == 0) 552 fatal("BN_hex2bn failed"); 553 /* q = (p-1) / 2 */ 554 if (BN_rshift(q, p, 1) == 0) 555 fatal("BN_rshift failed"); 556 break; 557 default: 558 debug2("Unknown prime type"); 559 break; 560 } 561 562 /* 563 * due to earlier inconsistencies in interpretation, check 564 * the proposed bit size. 565 */ 566 if ((u_int32_t)BN_num_bits(p) != (in_size + 1)) { 567 debug2("%10u: bit size %u mismatch", count_in, in_size); 568 continue; 569 } 570 if (in_size < QSIZE_MINIMUM) { 571 debug2("%10u: bit size %u too short", count_in, in_size); 572 continue; 573 } 574 575 if (in_tests & QTEST_MILLER_RABIN) 576 in_tries += trials; 577 else 578 in_tries = trials; 579 580 /* 581 * guess unknown generator 582 */ 583 if (generator_known == 0) { 584 if (BN_mod_word(p, 24) == 11) 585 generator_known = 2; 586 else if (BN_mod_word(p, 12) == 5) 587 generator_known = 3; 588 else { 589 u_int32_t r = BN_mod_word(p, 10); 590 591 if (r == 3 || r == 7) 592 generator_known = 5; 593 } 594 } 595 /* 596 * skip tests when desired generator doesn't match 597 */ 598 if (generator_wanted > 0 && 599 generator_wanted != generator_known) { 600 debug2("%10u: generator %d != %d", 601 count_in, generator_known, generator_wanted); 602 continue; 603 } 604 605 /* 606 * Primes with no known generator are useless for DH, so 607 * skip those. 608 */ 609 if (generator_known == 0) { 610 debug2("%10u: no known generator", count_in); 611 continue; 612 } 613 614 count_possible++; 615 616 /* 617 * The (1/4)^N performance bound on Miller-Rabin is 618 * extremely pessimistic, so don't spend a lot of time 619 * really verifying that q is prime until after we know 620 * that p is also prime. A single pass will weed out the 621 * vast majority of composite q's. 622 */ 623 if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) { 624 debug("%10u: q failed first possible prime test", 625 count_in); 626 continue; 627 } 628 629 /* 630 * q is possibly prime, so go ahead and really make sure 631 * that p is prime. If it is, then we can go back and do 632 * the same for q. If p is composite, chances are that 633 * will show up on the first Rabin-Miller iteration so it 634 * doesn't hurt to specify a high iteration count. 635 */ 636 if (!BN_is_prime(p, trials, NULL, ctx, NULL)) { 637 debug("%10u: p is not prime", count_in); 638 continue; 639 } 640 debug("%10u: p is almost certainly prime", count_in); 641 642 /* recheck q more rigorously */ 643 if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) { 644 debug("%10u: q is not prime", count_in); 645 continue; 646 } 647 debug("%10u: q is almost certainly prime", count_in); 648 649 if (qfileout(out, QTYPE_SAFE, (in_tests | QTEST_MILLER_RABIN), 650 in_tries, in_size, generator_known, p)) { 651 res = -1; 652 break; 653 } 654 655 count_out++; 656 } 657 658 time(&time_stop); 659 xfree(lp); 660 BN_free(p); 661 BN_free(q); 662 BN_CTX_free(ctx); 663 664 logit("%.24s Found %u safe primes of %u candidates in %ld seconds", 665 ctime(&time_stop), count_out, count_possible, 666 (long) (time_stop - time_start)); 667 668 return (res); 669 } 670