1 /* $OpenBSD: moduli.c,v 1.9 2004/07/11 17:48:47 deraadt 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 "xmalloc.h" 42 #include "log.h" 43 44 #include <openssl/bn.h> 45 46 /* 47 * File output defines 48 */ 49 50 /* need line long enough for largest moduli plus headers */ 51 #define QLINESIZE (100+8192) 52 53 /* Type: decimal. 54 * Specifies the internal structure of the prime modulus. 55 */ 56 #define QTYPE_UNKNOWN (0) 57 #define QTYPE_UNSTRUCTURED (1) 58 #define QTYPE_SAFE (2) 59 #define QTYPE_SCHNOOR (3) 60 #define QTYPE_SOPHIE_GERMAIN (4) 61 #define QTYPE_STRONG (5) 62 63 /* Tests: decimal (bit field). 64 * Specifies the methods used in checking for primality. 65 * Usually, more than one test is used. 66 */ 67 #define QTEST_UNTESTED (0x00) 68 #define QTEST_COMPOSITE (0x01) 69 #define QTEST_SIEVE (0x02) 70 #define QTEST_MILLER_RABIN (0x04) 71 #define QTEST_JACOBI (0x08) 72 #define QTEST_ELLIPTIC (0x10) 73 74 /* 75 * Size: decimal. 76 * Specifies the number of the most significant bit (0 to M). 77 * WARNING: internally, usually 1 to N. 78 */ 79 #define QSIZE_MINIMUM (511) 80 81 /* 82 * Prime sieving defines 83 */ 84 85 /* Constant: assuming 8 bit bytes and 32 bit words */ 86 #define SHIFT_BIT (3) 87 #define SHIFT_BYTE (2) 88 #define SHIFT_WORD (SHIFT_BIT+SHIFT_BYTE) 89 #define SHIFT_MEGABYTE (20) 90 #define SHIFT_MEGAWORD (SHIFT_MEGABYTE-SHIFT_BYTE) 91 92 /* 93 * Using virtual memory can cause thrashing. This should be the largest 94 * number that is supported without a large amount of disk activity -- 95 * that would increase the run time from hours to days or weeks! 96 */ 97 #define LARGE_MINIMUM (8UL) /* megabytes */ 98 99 /* 100 * Do not increase this number beyond the unsigned integer bit size. 101 * Due to a multiple of 4, it must be LESS than 128 (yielding 2**30 bits). 102 */ 103 #define LARGE_MAXIMUM (127UL) /* megabytes */ 104 105 /* 106 * Constant: when used with 32-bit integers, the largest sieve prime 107 * has to be less than 2**32. 108 */ 109 #define SMALL_MAXIMUM (0xffffffffUL) 110 111 /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */ 112 #define TINY_NUMBER (1UL<<16) 113 114 /* Ensure enough bit space for testing 2*q. */ 115 #define TEST_MAXIMUM (1UL<<16) 116 #define TEST_MINIMUM (QSIZE_MINIMUM + 1) 117 /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */ 118 #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */ 119 120 /* bit operations on 32-bit words */ 121 #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31))) 122 #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31))) 123 #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31))) 124 125 /* 126 * Prime testing defines 127 */ 128 129 /* Minimum number of primality tests to perform */ 130 #define TRIAL_MINIMUM (4) 131 132 /* 133 * Sieving data (XXX - move to struct) 134 */ 135 136 /* sieve 2**16 */ 137 static u_int32_t *TinySieve, tinybits; 138 139 /* sieve 2**30 in 2**16 parts */ 140 static u_int32_t *SmallSieve, smallbits, smallbase; 141 142 /* sieve relative to the initial value */ 143 static u_int32_t *LargeSieve, largewords, largetries, largenumbers; 144 static u_int32_t largebits, largememory; /* megabytes */ 145 static BIGNUM *largebase; 146 147 int gen_candidates(FILE *, int, int, BIGNUM *); 148 int prime_test(FILE *, FILE *, u_int32_t, u_int32_t); 149 150 /* 151 * print moduli out in consistent form, 152 */ 153 static int 154 qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries, 155 u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus) 156 { 157 struct tm *gtm; 158 time_t time_now; 159 int res; 160 161 time(&time_now); 162 gtm = gmtime(&time_now); 163 164 res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ", 165 gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday, 166 gtm->tm_hour, gtm->tm_min, gtm->tm_sec, 167 otype, otests, otries, osize, ogenerator); 168 169 if (res < 0) 170 return (-1); 171 172 if (BN_print_fp(ofile, omodulus) < 1) 173 return (-1); 174 175 res = fprintf(ofile, "\n"); 176 fflush(ofile); 177 178 return (res > 0 ? 0 : -1); 179 } 180 181 182 /* 183 ** Sieve p's and q's with small factors 184 */ 185 static void 186 sieve_large(u_int32_t s) 187 { 188 u_int32_t r, u; 189 190 debug3("sieve_large %u", s); 191 largetries++; 192 /* r = largebase mod s */ 193 r = BN_mod_word(largebase, s); 194 if (r == 0) 195 u = 0; /* s divides into largebase exactly */ 196 else 197 u = s - r; /* largebase+u is first entry divisible by s */ 198 199 if (u < largebits * 2) { 200 /* 201 * The sieve omits p's and q's divisible by 2, so ensure that 202 * largebase+u is odd. Then, step through the sieve in 203 * increments of 2*s 204 */ 205 if (u & 0x1) 206 u += s; /* Make largebase+u odd, and u even */ 207 208 /* Mark all multiples of 2*s */ 209 for (u /= 2; u < largebits; u += s) 210 BIT_SET(LargeSieve, u); 211 } 212 213 /* r = p mod s */ 214 r = (2 * r + 1) % s; 215 if (r == 0) 216 u = 0; /* s divides p exactly */ 217 else 218 u = s - r; /* p+u is first entry divisible by s */ 219 220 if (u < largebits * 4) { 221 /* 222 * The sieve omits p's divisible by 4, so ensure that 223 * largebase+u is not. Then, step through the sieve in 224 * increments of 4*s 225 */ 226 while (u & 0x3) { 227 if (SMALL_MAXIMUM - u < s) 228 return; 229 u += s; 230 } 231 232 /* Mark all multiples of 4*s */ 233 for (u /= 4; u < largebits; u += s) 234 BIT_SET(LargeSieve, u); 235 } 236 } 237 238 /* 239 * list candidates for Sophie-Germain primes (where q = (p-1)/2) 240 * to standard output. 241 * The list is checked against small known primes (less than 2**30). 242 */ 243 int 244 gen_candidates(FILE *out, int memory, int power, BIGNUM *start) 245 { 246 BIGNUM *q; 247 u_int32_t j, r, s, t; 248 u_int32_t smallwords = TINY_NUMBER >> 6; 249 u_int32_t tinywords = TINY_NUMBER >> 6; 250 time_t time_start, time_stop; 251 int i, ret = 0; 252 253 largememory = memory; 254 255 if (memory != 0 && 256 (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) { 257 error("Invalid memory amount (min %ld, max %ld)", 258 LARGE_MINIMUM, LARGE_MAXIMUM); 259 return (-1); 260 } 261 262 /* 263 * Set power to the length in bits of the prime to be generated. 264 * This is changed to 1 less than the desired safe prime moduli p. 265 */ 266 if (power > TEST_MAXIMUM) { 267 error("Too many bits: %u > %lu", power, TEST_MAXIMUM); 268 return (-1); 269 } else if (power < TEST_MINIMUM) { 270 error("Too few bits: %u < %u", power, TEST_MINIMUM); 271 return (-1); 272 } 273 power--; /* decrement before squaring */ 274 275 /* 276 * The density of ordinary primes is on the order of 1/bits, so the 277 * density of safe primes should be about (1/bits)**2. Set test range 278 * to something well above bits**2 to be reasonably sure (but not 279 * guaranteed) of catching at least one safe prime. 280 */ 281 largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER)); 282 283 /* 284 * Need idea of how much memory is available. We don't have to use all 285 * of it. 286 */ 287 if (largememory > LARGE_MAXIMUM) { 288 logit("Limited memory: %u MB; limit %lu MB", 289 largememory, LARGE_MAXIMUM); 290 largememory = LARGE_MAXIMUM; 291 } 292 293 if (largewords <= (largememory << SHIFT_MEGAWORD)) { 294 logit("Increased memory: %u MB; need %u bytes", 295 largememory, (largewords << SHIFT_BYTE)); 296 largewords = (largememory << SHIFT_MEGAWORD); 297 } else if (largememory > 0) { 298 logit("Decreased memory: %u MB; want %u bytes", 299 largememory, (largewords << SHIFT_BYTE)); 300 largewords = (largememory << SHIFT_MEGAWORD); 301 } 302 303 TinySieve = calloc(tinywords, sizeof(u_int32_t)); 304 if (TinySieve == NULL) { 305 error("Insufficient memory for tiny sieve: need %u bytes", 306 tinywords << SHIFT_BYTE); 307 exit(1); 308 } 309 tinybits = tinywords << SHIFT_WORD; 310 311 SmallSieve = calloc(smallwords, sizeof(u_int32_t)); 312 if (SmallSieve == NULL) { 313 error("Insufficient memory for small sieve: need %u bytes", 314 smallwords << SHIFT_BYTE); 315 xfree(TinySieve); 316 exit(1); 317 } 318 smallbits = smallwords << SHIFT_WORD; 319 320 /* 321 * dynamically determine available memory 322 */ 323 while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL) 324 largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */ 325 326 largebits = largewords << SHIFT_WORD; 327 largenumbers = largebits * 2; /* even numbers excluded */ 328 329 /* validation check: count the number of primes tried */ 330 largetries = 0; 331 q = BN_new(); 332 333 /* 334 * Generate random starting point for subprime search, or use 335 * specified parameter. 336 */ 337 largebase = BN_new(); 338 if (start == NULL) 339 BN_rand(largebase, power, 1, 1); 340 else 341 BN_copy(largebase, start); 342 343 /* ensure odd */ 344 BN_set_bit(largebase, 0); 345 346 time(&time_start); 347 348 logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start), 349 largenumbers, power); 350 debug2("start point: 0x%s", BN_bn2hex(largebase)); 351 352 /* 353 * TinySieve 354 */ 355 for (i = 0; i < tinybits; i++) { 356 if (BIT_TEST(TinySieve, i)) 357 continue; /* 2*i+3 is composite */ 358 359 /* The next tiny prime */ 360 t = 2 * i + 3; 361 362 /* Mark all multiples of t */ 363 for (j = i + t; j < tinybits; j += t) 364 BIT_SET(TinySieve, j); 365 366 sieve_large(t); 367 } 368 369 /* 370 * Start the small block search at the next possible prime. To avoid 371 * fencepost errors, the last pass is skipped. 372 */ 373 for (smallbase = TINY_NUMBER + 3; 374 smallbase < (SMALL_MAXIMUM - TINY_NUMBER); 375 smallbase += TINY_NUMBER) { 376 for (i = 0; i < tinybits; i++) { 377 if (BIT_TEST(TinySieve, i)) 378 continue; /* 2*i+3 is composite */ 379 380 /* The next tiny prime */ 381 t = 2 * i + 3; 382 r = smallbase % t; 383 384 if (r == 0) { 385 s = 0; /* t divides into smallbase exactly */ 386 } else { 387 /* smallbase+s is first entry divisible by t */ 388 s = t - r; 389 } 390 391 /* 392 * The sieve omits even numbers, so ensure that 393 * smallbase+s is odd. Then, step through the sieve 394 * in increments of 2*t 395 */ 396 if (s & 1) 397 s += t; /* Make smallbase+s odd, and s even */ 398 399 /* Mark all multiples of 2*t */ 400 for (s /= 2; s < smallbits; s += t) 401 BIT_SET(SmallSieve, s); 402 } 403 404 /* 405 * SmallSieve 406 */ 407 for (i = 0; i < smallbits; i++) { 408 if (BIT_TEST(SmallSieve, i)) 409 continue; /* 2*i+smallbase is composite */ 410 411 /* The next small prime */ 412 sieve_large((2 * i) + smallbase); 413 } 414 415 memset(SmallSieve, 0, smallwords << SHIFT_BYTE); 416 } 417 418 time(&time_stop); 419 420 logit("%.24s Sieved with %u small primes in %ld seconds", 421 ctime(&time_stop), largetries, (long) (time_stop - time_start)); 422 423 for (j = r = 0; j < largebits; j++) { 424 if (BIT_TEST(LargeSieve, j)) 425 continue; /* Definitely composite, skip */ 426 427 debug2("test q = largebase+%u", 2 * j); 428 BN_set_word(q, 2 * j); 429 BN_add(q, q, largebase); 430 if (qfileout(out, QTYPE_SOPHIE_GERMAIN, QTEST_SIEVE, 431 largetries, (power - 1) /* MSB */, (0), q) == -1) { 432 ret = -1; 433 break; 434 } 435 436 r++; /* count q */ 437 } 438 439 time(&time_stop); 440 441 xfree(LargeSieve); 442 xfree(SmallSieve); 443 xfree(TinySieve); 444 445 logit("%.24s Found %u candidates", ctime(&time_stop), r); 446 447 return (ret); 448 } 449 450 /* 451 * perform a Miller-Rabin primality test 452 * on the list of candidates 453 * (checking both q and p) 454 * The result is a list of so-call "safe" primes 455 */ 456 int 457 prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted) 458 { 459 BIGNUM *q, *p, *a; 460 BN_CTX *ctx; 461 char *cp, *lp; 462 u_int32_t count_in = 0, count_out = 0, count_possible = 0; 463 u_int32_t generator_known, in_tests, in_tries, in_type, in_size; 464 time_t time_start, time_stop; 465 int res; 466 467 if (trials < TRIAL_MINIMUM) { 468 error("Minimum primality trials is %d", TRIAL_MINIMUM); 469 return (-1); 470 } 471 472 time(&time_start); 473 474 p = BN_new(); 475 q = BN_new(); 476 ctx = BN_CTX_new(); 477 478 debug2("%.24s Final %u Miller-Rabin trials (%x generator)", 479 ctime(&time_start), trials, generator_wanted); 480 481 res = 0; 482 lp = xmalloc(QLINESIZE + 1); 483 while (fgets(lp, QLINESIZE, in) != NULL) { 484 int ll = strlen(lp); 485 486 count_in++; 487 if (ll < 14 || *lp == '!' || *lp == '#') { 488 debug2("%10u: comment or short line", count_in); 489 continue; 490 } 491 492 /* XXX - fragile parser */ 493 /* time */ 494 cp = &lp[14]; /* (skip) */ 495 496 /* type */ 497 in_type = strtoul(cp, &cp, 10); 498 499 /* tests */ 500 in_tests = strtoul(cp, &cp, 10); 501 502 if (in_tests & QTEST_COMPOSITE) { 503 debug2("%10u: known composite", count_in); 504 continue; 505 } 506 507 /* tries */ 508 in_tries = strtoul(cp, &cp, 10); 509 510 /* size (most significant bit) */ 511 in_size = strtoul(cp, &cp, 10); 512 513 /* generator (hex) */ 514 generator_known = strtoul(cp, &cp, 16); 515 516 /* Skip white space */ 517 cp += strspn(cp, " "); 518 519 /* modulus (hex) */ 520 switch (in_type) { 521 case QTYPE_SOPHIE_GERMAIN: 522 debug2("%10u: (%u) Sophie-Germain", count_in, in_type); 523 a = q; 524 BN_hex2bn(&a, cp); 525 /* p = 2*q + 1 */ 526 BN_lshift(p, q, 1); 527 BN_add_word(p, 1); 528 in_size += 1; 529 generator_known = 0; 530 break; 531 case QTYPE_UNSTRUCTURED: 532 case QTYPE_SAFE: 533 case QTYPE_SCHNOOR: 534 case QTYPE_STRONG: 535 case QTYPE_UNKNOWN: 536 debug2("%10u: (%u)", count_in, in_type); 537 a = p; 538 BN_hex2bn(&a, cp); 539 /* q = (p-1) / 2 */ 540 BN_rshift(q, p, 1); 541 break; 542 default: 543 debug2("Unknown prime type"); 544 break; 545 } 546 547 /* 548 * due to earlier inconsistencies in interpretation, check 549 * the proposed bit size. 550 */ 551 if (BN_num_bits(p) != (in_size + 1)) { 552 debug2("%10u: bit size %u mismatch", count_in, in_size); 553 continue; 554 } 555 if (in_size < QSIZE_MINIMUM) { 556 debug2("%10u: bit size %u too short", count_in, in_size); 557 continue; 558 } 559 560 if (in_tests & QTEST_MILLER_RABIN) 561 in_tries += trials; 562 else 563 in_tries = trials; 564 565 /* 566 * guess unknown generator 567 */ 568 if (generator_known == 0) { 569 if (BN_mod_word(p, 24) == 11) 570 generator_known = 2; 571 else if (BN_mod_word(p, 12) == 5) 572 generator_known = 3; 573 else { 574 u_int32_t r = BN_mod_word(p, 10); 575 576 if (r == 3 || r == 7) 577 generator_known = 5; 578 } 579 } 580 /* 581 * skip tests when desired generator doesn't match 582 */ 583 if (generator_wanted > 0 && 584 generator_wanted != generator_known) { 585 debug2("%10u: generator %d != %d", 586 count_in, generator_known, generator_wanted); 587 continue; 588 } 589 590 /* 591 * Primes with no known generator are useless for DH, so 592 * skip those. 593 */ 594 if (generator_known == 0) { 595 debug2("%10u: no known generator", count_in); 596 continue; 597 } 598 599 count_possible++; 600 601 /* 602 * The (1/4)^N performance bound on Miller-Rabin is 603 * extremely pessimistic, so don't spend a lot of time 604 * really verifying that q is prime until after we know 605 * that p is also prime. A single pass will weed out the 606 * vast majority of composite q's. 607 */ 608 if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) { 609 debug("%10u: q failed first possible prime test", 610 count_in); 611 continue; 612 } 613 614 /* 615 * q is possibly prime, so go ahead and really make sure 616 * that p is prime. If it is, then we can go back and do 617 * the same for q. If p is composite, chances are that 618 * will show up on the first Rabin-Miller iteration so it 619 * doesn't hurt to specify a high iteration count. 620 */ 621 if (!BN_is_prime(p, trials, NULL, ctx, NULL)) { 622 debug("%10u: p is not prime", count_in); 623 continue; 624 } 625 debug("%10u: p is almost certainly prime", count_in); 626 627 /* recheck q more rigorously */ 628 if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) { 629 debug("%10u: q is not prime", count_in); 630 continue; 631 } 632 debug("%10u: q is almost certainly prime", count_in); 633 634 if (qfileout(out, QTYPE_SAFE, (in_tests | QTEST_MILLER_RABIN), 635 in_tries, in_size, generator_known, p)) { 636 res = -1; 637 break; 638 } 639 640 count_out++; 641 } 642 643 time(&time_stop); 644 xfree(lp); 645 BN_free(p); 646 BN_free(q); 647 BN_CTX_free(ctx); 648 649 logit("%.24s Found %u safe primes of %u candidates in %ld seconds", 650 ctime(&time_stop), count_out, count_possible, 651 (long) (time_stop - time_start)); 652 653 return (res); 654 } 655