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 <sys/types.h> 41 42 #include <openssl/bn.h> 43 44 #include <stdio.h> 45 #include <stdlib.h> 46 #include <string.h> 47 #include <stdarg.h> 48 #include <time.h> 49 50 #include "xmalloc.h" 51 #include "log.h" 52 53 /* 54 * File output defines 55 */ 56 57 /* need line long enough for largest moduli plus headers */ 58 #define QLINESIZE (100+8192) 59 60 /* Type: decimal. 61 * Specifies the internal structure of the prime modulus. 62 */ 63 #define QTYPE_UNKNOWN (0) 64 #define QTYPE_UNSTRUCTURED (1) 65 #define QTYPE_SAFE (2) 66 #define QTYPE_SCHNORR (3) 67 #define QTYPE_SOPHIE_GERMAIN (4) 68 #define QTYPE_STRONG (5) 69 70 /* Tests: decimal (bit field). 71 * Specifies the methods used in checking for primality. 72 * Usually, more than one test is used. 73 */ 74 #define QTEST_UNTESTED (0x00) 75 #define QTEST_COMPOSITE (0x01) 76 #define QTEST_SIEVE (0x02) 77 #define QTEST_MILLER_RABIN (0x04) 78 #define QTEST_JACOBI (0x08) 79 #define QTEST_ELLIPTIC (0x10) 80 81 /* 82 * Size: decimal. 83 * Specifies the number of the most significant bit (0 to M). 84 * WARNING: internally, usually 1 to N. 85 */ 86 #define QSIZE_MINIMUM (511) 87 88 /* 89 * Prime sieving defines 90 */ 91 92 /* Constant: assuming 8 bit bytes and 32 bit words */ 93 #define SHIFT_BIT (3) 94 #define SHIFT_BYTE (2) 95 #define SHIFT_WORD (SHIFT_BIT+SHIFT_BYTE) 96 #define SHIFT_MEGABYTE (20) 97 #define SHIFT_MEGAWORD (SHIFT_MEGABYTE-SHIFT_BYTE) 98 99 /* 100 * Using virtual memory can cause thrashing. This should be the largest 101 * number that is supported without a large amount of disk activity -- 102 * that would increase the run time from hours to days or weeks! 103 */ 104 #define LARGE_MINIMUM (8UL) /* megabytes */ 105 106 /* 107 * Do not increase this number beyond the unsigned integer bit size. 108 * Due to a multiple of 4, it must be LESS than 128 (yielding 2**30 bits). 109 */ 110 #define LARGE_MAXIMUM (127UL) /* megabytes */ 111 112 /* 113 * Constant: when used with 32-bit integers, the largest sieve prime 114 * has to be less than 2**32. 115 */ 116 #define SMALL_MAXIMUM (0xffffffffUL) 117 118 /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */ 119 #define TINY_NUMBER (1UL<<16) 120 121 /* Ensure enough bit space for testing 2*q. */ 122 #define TEST_MAXIMUM (1UL<<16) 123 #define TEST_MINIMUM (QSIZE_MINIMUM + 1) 124 /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */ 125 #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */ 126 127 /* bit operations on 32-bit words */ 128 #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31))) 129 #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31))) 130 #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31))) 131 132 /* 133 * Prime testing defines 134 */ 135 136 /* Minimum number of primality tests to perform */ 137 #define TRIAL_MINIMUM (4) 138 139 /* 140 * Sieving data (XXX - move to struct) 141 */ 142 143 /* sieve 2**16 */ 144 static u_int32_t *TinySieve, tinybits; 145 146 /* sieve 2**30 in 2**16 parts */ 147 static u_int32_t *SmallSieve, smallbits, smallbase; 148 149 /* sieve relative to the initial value */ 150 static u_int32_t *LargeSieve, largewords, largetries, largenumbers; 151 static u_int32_t largebits, largememory; /* megabytes */ 152 static BIGNUM *largebase; 153 154 int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *); 155 int prime_test(FILE *, FILE *, u_int32_t, u_int32_t); 156 157 /* 158 * print moduli out in consistent form, 159 */ 160 static int 161 qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries, 162 u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus) 163 { 164 struct tm *gtm; 165 time_t time_now; 166 int res; 167 168 time(&time_now); 169 gtm = gmtime(&time_now); 170 171 res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ", 172 gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday, 173 gtm->tm_hour, gtm->tm_min, gtm->tm_sec, 174 otype, otests, otries, osize, ogenerator); 175 176 if (res < 0) 177 return (-1); 178 179 if (BN_print_fp(ofile, omodulus) < 1) 180 return (-1); 181 182 res = fprintf(ofile, "\n"); 183 fflush(ofile); 184 185 return (res > 0 ? 0 : -1); 186 } 187 188 189 /* 190 ** Sieve p's and q's with small factors 191 */ 192 static void 193 sieve_large(u_int32_t s) 194 { 195 u_int32_t r, u; 196 197 debug3("sieve_large %u", s); 198 largetries++; 199 /* r = largebase mod s */ 200 r = BN_mod_word(largebase, s); 201 if (r == 0) 202 u = 0; /* s divides into largebase exactly */ 203 else 204 u = s - r; /* largebase+u is first entry divisible by s */ 205 206 if (u < largebits * 2) { 207 /* 208 * The sieve omits p's and q's divisible by 2, so ensure that 209 * largebase+u is odd. Then, step through the sieve in 210 * increments of 2*s 211 */ 212 if (u & 0x1) 213 u += s; /* Make largebase+u odd, and u even */ 214 215 /* Mark all multiples of 2*s */ 216 for (u /= 2; u < largebits; u += s) 217 BIT_SET(LargeSieve, u); 218 } 219 220 /* r = p mod s */ 221 r = (2 * r + 1) % s; 222 if (r == 0) 223 u = 0; /* s divides p exactly */ 224 else 225 u = s - r; /* p+u is first entry divisible by s */ 226 227 if (u < largebits * 4) { 228 /* 229 * The sieve omits p's divisible by 4, so ensure that 230 * largebase+u is not. Then, step through the sieve in 231 * increments of 4*s 232 */ 233 while (u & 0x3) { 234 if (SMALL_MAXIMUM - u < s) 235 return; 236 u += s; 237 } 238 239 /* Mark all multiples of 4*s */ 240 for (u /= 4; u < largebits; u += s) 241 BIT_SET(LargeSieve, u); 242 } 243 } 244 245 /* 246 * list candidates for Sophie-Germain primes (where q = (p-1)/2) 247 * to standard output. 248 * The list is checked against small known primes (less than 2**30). 249 */ 250 int 251 gen_candidates(FILE *out, u_int32_t memory, u_int32_t power, BIGNUM *start) 252 { 253 BIGNUM *q; 254 u_int32_t j, r, s, t; 255 u_int32_t smallwords = TINY_NUMBER >> 6; 256 u_int32_t tinywords = TINY_NUMBER >> 6; 257 time_t time_start, time_stop; 258 u_int32_t i; 259 int ret = 0; 260 261 largememory = memory; 262 263 if (memory != 0 && 264 (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) { 265 error("Invalid memory amount (min %ld, max %ld)", 266 LARGE_MINIMUM, LARGE_MAXIMUM); 267 return (-1); 268 } 269 270 /* 271 * Set power to the length in bits of the prime to be generated. 272 * This is changed to 1 less than the desired safe prime moduli p. 273 */ 274 if (power > TEST_MAXIMUM) { 275 error("Too many bits: %u > %lu", power, TEST_MAXIMUM); 276 return (-1); 277 } else if (power < TEST_MINIMUM) { 278 error("Too few bits: %u < %u", power, TEST_MINIMUM); 279 return (-1); 280 } 281 power--; /* decrement before squaring */ 282 283 /* 284 * The density of ordinary primes is on the order of 1/bits, so the 285 * density of safe primes should be about (1/bits)**2. Set test range 286 * to something well above bits**2 to be reasonably sure (but not 287 * guaranteed) of catching at least one safe prime. 288 */ 289 largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER)); 290 291 /* 292 * Need idea of how much memory is available. We don't have to use all 293 * of it. 294 */ 295 if (largememory > LARGE_MAXIMUM) { 296 logit("Limited memory: %u MB; limit %lu MB", 297 largememory, LARGE_MAXIMUM); 298 largememory = LARGE_MAXIMUM; 299 } 300 301 if (largewords <= (largememory << SHIFT_MEGAWORD)) { 302 logit("Increased memory: %u MB; need %u bytes", 303 largememory, (largewords << SHIFT_BYTE)); 304 largewords = (largememory << SHIFT_MEGAWORD); 305 } else if (largememory > 0) { 306 logit("Decreased memory: %u MB; want %u bytes", 307 largememory, (largewords << SHIFT_BYTE)); 308 largewords = (largememory << SHIFT_MEGAWORD); 309 } 310 311 TinySieve = xcalloc(tinywords, sizeof(u_int32_t)); 312 tinybits = tinywords << SHIFT_WORD; 313 314 SmallSieve = xcalloc(smallwords, sizeof(u_int32_t)); 315 smallbits = smallwords << SHIFT_WORD; 316 317 /* 318 * dynamically determine available memory 319 */ 320 while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL) 321 largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */ 322 323 largebits = largewords << SHIFT_WORD; 324 largenumbers = largebits * 2; /* even numbers excluded */ 325 326 /* validation check: count the number of primes tried */ 327 largetries = 0; 328 if ((q = BN_new()) == NULL) 329 fatal("BN_new failed"); 330 331 /* 332 * Generate random starting point for subprime search, or use 333 * specified parameter. 334 */ 335 if ((largebase = BN_new()) == NULL) 336 fatal("BN_new failed"); 337 if (start == NULL) { 338 if (BN_rand(largebase, power, 1, 1) == 0) 339 fatal("BN_rand failed"); 340 } else { 341 if (BN_copy(largebase, start) == NULL) 342 fatal("BN_copy: failed"); 343 } 344 345 /* ensure odd */ 346 if (BN_set_bit(largebase, 0) == 0) 347 fatal("BN_set_bit: failed"); 348 349 time(&time_start); 350 351 logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start), 352 largenumbers, power); 353 debug2("start point: 0x%s", BN_bn2hex(largebase)); 354 355 /* 356 * TinySieve 357 */ 358 for (i = 0; i < tinybits; i++) { 359 if (BIT_TEST(TinySieve, i)) 360 continue; /* 2*i+3 is composite */ 361 362 /* The next tiny prime */ 363 t = 2 * i + 3; 364 365 /* Mark all multiples of t */ 366 for (j = i + t; j < tinybits; j += t) 367 BIT_SET(TinySieve, j); 368 369 sieve_large(t); 370 } 371 372 /* 373 * Start the small block search at the next possible prime. To avoid 374 * fencepost errors, the last pass is skipped. 375 */ 376 for (smallbase = TINY_NUMBER + 3; 377 smallbase < (SMALL_MAXIMUM - TINY_NUMBER); 378 smallbase += TINY_NUMBER) { 379 for (i = 0; i < tinybits; i++) { 380 if (BIT_TEST(TinySieve, i)) 381 continue; /* 2*i+3 is composite */ 382 383 /* The next tiny prime */ 384 t = 2 * i + 3; 385 r = smallbase % t; 386 387 if (r == 0) { 388 s = 0; /* t divides into smallbase exactly */ 389 } else { 390 /* smallbase+s is first entry divisible by t */ 391 s = t - r; 392 } 393 394 /* 395 * The sieve omits even numbers, so ensure that 396 * smallbase+s is odd. Then, step through the sieve 397 * in increments of 2*t 398 */ 399 if (s & 1) 400 s += t; /* Make smallbase+s odd, and s even */ 401 402 /* Mark all multiples of 2*t */ 403 for (s /= 2; s < smallbits; s += t) 404 BIT_SET(SmallSieve, s); 405 } 406 407 /* 408 * SmallSieve 409 */ 410 for (i = 0; i < smallbits; i++) { 411 if (BIT_TEST(SmallSieve, i)) 412 continue; /* 2*i+smallbase is composite */ 413 414 /* The next small prime */ 415 sieve_large((2 * i) + smallbase); 416 } 417 418 memset(SmallSieve, 0, smallwords << SHIFT_BYTE); 419 } 420 421 time(&time_stop); 422 423 logit("%.24s Sieved with %u small primes in %ld seconds", 424 ctime(&time_stop), largetries, (long) (time_stop - time_start)); 425 426 for (j = r = 0; j < largebits; j++) { 427 if (BIT_TEST(LargeSieve, j)) 428 continue; /* Definitely composite, skip */ 429 430 debug2("test q = largebase+%u", 2 * j); 431 if (BN_set_word(q, 2 * j) == 0) 432 fatal("BN_set_word failed"); 433 if (BN_add(q, q, largebase) == 0) 434 fatal("BN_add failed"); 435 if (qfileout(out, QTYPE_SOPHIE_GERMAIN, QTEST_SIEVE, 436 largetries, (power - 1) /* MSB */, (0), q) == -1) { 437 ret = -1; 438 break; 439 } 440 441 r++; /* count q */ 442 } 443 444 time(&time_stop); 445 446 xfree(LargeSieve); 447 xfree(SmallSieve); 448 xfree(TinySieve); 449 450 logit("%.24s Found %u candidates", ctime(&time_stop), r); 451 452 return (ret); 453 } 454 455 /* 456 * perform a Miller-Rabin primality test 457 * on the list of candidates 458 * (checking both q and p) 459 * The result is a list of so-call "safe" primes 460 */ 461 int 462 prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted) 463 { 464 BIGNUM *q, *p, *a; 465 BN_CTX *ctx; 466 char *cp, *lp; 467 u_int32_t count_in = 0, count_out = 0, count_possible = 0; 468 u_int32_t generator_known, in_tests, in_tries, in_type, in_size; 469 time_t time_start, time_stop; 470 int res; 471 472 if (trials < TRIAL_MINIMUM) { 473 error("Minimum primality trials is %d", TRIAL_MINIMUM); 474 return (-1); 475 } 476 477 time(&time_start); 478 479 if ((p = BN_new()) == NULL) 480 fatal("BN_new failed"); 481 if ((q = BN_new()) == NULL) 482 fatal("BN_new failed"); 483 if ((ctx = BN_CTX_new()) == NULL) 484 fatal("BN_CTX_new failed"); 485 486 debug2("%.24s Final %u Miller-Rabin trials (%x generator)", 487 ctime(&time_start), trials, generator_wanted); 488 489 res = 0; 490 lp = xmalloc(QLINESIZE + 1); 491 while (fgets(lp, QLINESIZE, in) != NULL) { 492 int ll = strlen(lp); 493 494 count_in++; 495 if (ll < 14 || *lp == '!' || *lp == '#') { 496 debug2("%10u: comment or short line", count_in); 497 continue; 498 } 499 500 /* XXX - fragile parser */ 501 /* time */ 502 cp = &lp[14]; /* (skip) */ 503 504 /* type */ 505 in_type = strtoul(cp, &cp, 10); 506 507 /* tests */ 508 in_tests = strtoul(cp, &cp, 10); 509 510 if (in_tests & QTEST_COMPOSITE) { 511 debug2("%10u: known composite", count_in); 512 continue; 513 } 514 515 /* tries */ 516 in_tries = strtoul(cp, &cp, 10); 517 518 /* size (most significant bit) */ 519 in_size = strtoul(cp, &cp, 10); 520 521 /* generator (hex) */ 522 generator_known = strtoul(cp, &cp, 16); 523 524 /* Skip white space */ 525 cp += strspn(cp, " "); 526 527 /* modulus (hex) */ 528 switch (in_type) { 529 case QTYPE_SOPHIE_GERMAIN: 530 debug2("%10u: (%u) Sophie-Germain", count_in, in_type); 531 a = q; 532 if (BN_hex2bn(&a, cp) == 0) 533 fatal("BN_hex2bn failed"); 534 /* p = 2*q + 1 */ 535 if (BN_lshift(p, q, 1) == 0) 536 fatal("BN_lshift failed"); 537 if (BN_add_word(p, 1) == 0) 538 fatal("BN_add_word failed"); 539 in_size += 1; 540 generator_known = 0; 541 break; 542 case QTYPE_UNSTRUCTURED: 543 case QTYPE_SAFE: 544 case QTYPE_SCHNORR: 545 case QTYPE_STRONG: 546 case QTYPE_UNKNOWN: 547 debug2("%10u: (%u)", count_in, in_type); 548 a = p; 549 if (BN_hex2bn(&a, cp) == 0) 550 fatal("BN_hex2bn failed"); 551 /* q = (p-1) / 2 */ 552 if (BN_rshift(q, p, 1) == 0) 553 fatal("BN_rshift failed"); 554 break; 555 default: 556 debug2("Unknown prime type"); 557 break; 558 } 559 560 /* 561 * due to earlier inconsistencies in interpretation, check 562 * the proposed bit size. 563 */ 564 if ((u_int32_t)BN_num_bits(p) != (in_size + 1)) { 565 debug2("%10u: bit size %u mismatch", count_in, in_size); 566 continue; 567 } 568 if (in_size < QSIZE_MINIMUM) { 569 debug2("%10u: bit size %u too short", count_in, in_size); 570 continue; 571 } 572 573 if (in_tests & QTEST_MILLER_RABIN) 574 in_tries += trials; 575 else 576 in_tries = trials; 577 578 /* 579 * guess unknown generator 580 */ 581 if (generator_known == 0) { 582 if (BN_mod_word(p, 24) == 11) 583 generator_known = 2; 584 else if (BN_mod_word(p, 12) == 5) 585 generator_known = 3; 586 else { 587 u_int32_t r = BN_mod_word(p, 10); 588 589 if (r == 3 || r == 7) 590 generator_known = 5; 591 } 592 } 593 /* 594 * skip tests when desired generator doesn't match 595 */ 596 if (generator_wanted > 0 && 597 generator_wanted != generator_known) { 598 debug2("%10u: generator %d != %d", 599 count_in, generator_known, generator_wanted); 600 continue; 601 } 602 603 /* 604 * Primes with no known generator are useless for DH, so 605 * skip those. 606 */ 607 if (generator_known == 0) { 608 debug2("%10u: no known generator", count_in); 609 continue; 610 } 611 612 count_possible++; 613 614 /* 615 * The (1/4)^N performance bound on Miller-Rabin is 616 * extremely pessimistic, so don't spend a lot of time 617 * really verifying that q is prime until after we know 618 * that p is also prime. A single pass will weed out the 619 * vast majority of composite q's. 620 */ 621 if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) { 622 debug("%10u: q failed first possible prime test", 623 count_in); 624 continue; 625 } 626 627 /* 628 * q is possibly prime, so go ahead and really make sure 629 * that p is prime. If it is, then we can go back and do 630 * the same for q. If p is composite, chances are that 631 * will show up on the first Rabin-Miller iteration so it 632 * doesn't hurt to specify a high iteration count. 633 */ 634 if (!BN_is_prime(p, trials, NULL, ctx, NULL)) { 635 debug("%10u: p is not prime", count_in); 636 continue; 637 } 638 debug("%10u: p is almost certainly prime", count_in); 639 640 /* recheck q more rigorously */ 641 if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) { 642 debug("%10u: q is not prime", count_in); 643 continue; 644 } 645 debug("%10u: q is almost certainly prime", count_in); 646 647 if (qfileout(out, QTYPE_SAFE, (in_tests | QTEST_MILLER_RABIN), 648 in_tries, in_size, generator_known, p)) { 649 res = -1; 650 break; 651 } 652 653 count_out++; 654 } 655 656 time(&time_stop); 657 xfree(lp); 658 BN_free(p); 659 BN_free(q); 660 BN_CTX_free(ctx); 661 662 logit("%.24s Found %u safe primes of %u candidates in %ld seconds", 663 ctime(&time_stop), count_out, count_possible, 664 (long) (time_stop - time_start)); 665 666 return (res); 667 } 668