1 /* $OpenBSD: moduli.c,v 1.1 2003/07/28 09:49:56 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 #include "moduli.h" 42 #include "xmalloc.h" 43 #include "log.h" 44 45 #include <openssl/bn.h> 46 47 48 /* 49 * Debugging defines 50 */ 51 52 /* define DEBUG_LARGE 1 */ 53 /* define DEBUG_SMALL 1 */ 54 /* define DEBUG_TEST 1 */ 55 56 /* 57 * File output defines 58 */ 59 60 /* need line long enough for largest moduli plus headers */ 61 #define QLINESIZE (100+8192) 62 63 /* Type: decimal. 64 * Specifies the internal structure of the prime modulus. 65 */ 66 #define QTYPE_UNKNOWN (0) 67 #define QTYPE_UNSTRUCTURED (1) 68 #define QTYPE_SAFE (2) 69 #define QTYPE_SCHNOOR (3) 70 #define QTYPE_SOPHIE_GERMAINE (4) 71 #define QTYPE_STRONG (5) 72 73 /* Tests: decimal (bit field). 74 * Specifies the methods used in checking for primality. 75 * Usually, more than one test is used. 76 */ 77 #define QTEST_UNTESTED (0x00) 78 #define QTEST_COMPOSITE (0x01) 79 #define QTEST_SIEVE (0x02) 80 #define QTEST_MILLER_RABIN (0x04) 81 #define QTEST_JACOBI (0x08) 82 #define QTEST_ELLIPTIC (0x10) 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 * Constant: when used with 32-bit integers, the largest sieve prime 103 * has to be less than 2**32. 104 */ 105 #define SMALL_MAXIMUM (0xffffffffUL) 106 107 /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */ 108 #define TINY_NUMBER (1UL<<16) 109 110 /* Ensure enough bit space for testing 2*q. */ 111 #define TEST_MAXIMUM (1UL<<16) 112 #define TEST_MINIMUM (QSIZE_MINIMUM + 1) 113 /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */ 114 #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */ 115 116 /* bit operations on 32-bit words */ 117 #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31))) 118 #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31))) 119 #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31))) 120 121 /* 122 * Prime testing defines 123 */ 124 125 /* 126 * Sieving data (XXX - move to struct) 127 */ 128 129 /* sieve 2**16 */ 130 static u_int32_t *TinySieve, tinybits; 131 132 /* sieve 2**30 in 2**16 parts */ 133 static u_int32_t *SmallSieve, smallbits, smallbase; 134 135 /* sieve relative to the initial value */ 136 static u_int32_t *LargeSieve, largewords, largetries, largenumbers; 137 static u_int32_t largebits, largememory; /* megabytes */ 138 static BIGNUM *largebase; 139 140 141 /* 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 debug2("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-Germaine 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, int memory, int 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 int i, ret = 0; 243 244 largememory = memory; 245 246 /* 247 * Set power to the length in bits of the prime to be generated. 248 * This is changed to 1 less than the desired safe prime moduli p. 249 */ 250 if (power > TEST_MAXIMUM) { 251 error("Too many bits: %u > %lu", power, TEST_MAXIMUM); 252 return (-1); 253 } else if (power < TEST_MINIMUM) { 254 error("Too few bits: %u < %u", power, TEST_MINIMUM); 255 return (-1); 256 } 257 power--; /* decrement before squaring */ 258 259 /* 260 * The density of ordinary primes is on the order of 1/bits, so the 261 * density of safe primes should be about (1/bits)**2. Set test range 262 * to something well above bits**2 to be reasonably sure (but not 263 * guaranteed) of catching at least one safe prime. 264 */ 265 largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER)); 266 267 /* 268 * Need idea of how much memory is available. We don't have to use all 269 * of it. 270 */ 271 if (largememory > LARGE_MAXIMUM) { 272 logit("Limited memory: %u MB; limit %lu MB", 273 largememory, LARGE_MAXIMUM); 274 largememory = LARGE_MAXIMUM; 275 } 276 277 if (largewords <= (largememory << SHIFT_MEGAWORD)) { 278 logit("Increased memory: %u MB; need %u bytes", 279 largememory, (largewords << SHIFT_BYTE)); 280 largewords = (largememory << SHIFT_MEGAWORD); 281 } else if (largememory > 0) { 282 logit("Decreased memory: %u MB; want %u bytes", 283 largememory, (largewords << SHIFT_BYTE)); 284 largewords = (largememory << SHIFT_MEGAWORD); 285 } 286 287 TinySieve = calloc(tinywords, sizeof(u_int32_t)); 288 if (TinySieve == NULL) { 289 error("Insufficient memory for tiny sieve: need %u bytes", 290 tinywords << SHIFT_BYTE); 291 exit(1); 292 } 293 tinybits = tinywords << SHIFT_WORD; 294 295 SmallSieve = calloc(smallwords, sizeof(u_int32_t)); 296 if (SmallSieve == NULL) { 297 error("Insufficient memory for small sieve: need %u bytes", 298 smallwords << SHIFT_BYTE); 299 xfree(TinySieve); 300 exit(1); 301 } 302 smallbits = smallwords << SHIFT_WORD; 303 304 /* 305 * dynamically determine available memory 306 */ 307 while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL) 308 largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */ 309 310 largebits = largewords << SHIFT_WORD; 311 largenumbers = largebits * 2; /* even numbers excluded */ 312 313 /* validation check: count the number of primes tried */ 314 largetries = 0; 315 q = BN_new(); 316 317 /* 318 * Generate random starting point for subprime search, or use 319 * specified parameter. 320 */ 321 largebase = BN_new(); 322 if (start == NULL) 323 BN_rand(largebase, power, 1, 1); 324 else 325 BN_copy(largebase, start); 326 327 /* ensure odd */ 328 BN_set_bit(largebase, 0); 329 330 time(&time_start); 331 332 logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start), 333 largenumbers, power); 334 debug2("start point: 0x%s", BN_bn2hex(largebase)); 335 336 /* 337 * TinySieve 338 */ 339 for (i = 0; i < tinybits; i++) { 340 if (BIT_TEST(TinySieve, i)) 341 continue; /* 2*i+3 is composite */ 342 343 /* The next tiny prime */ 344 t = 2 * i + 3; 345 346 /* Mark all multiples of t */ 347 for (j = i + t; j < tinybits; j += t) 348 BIT_SET(TinySieve, j); 349 350 sieve_large(t); 351 } 352 353 /* 354 * Start the small block search at the next possible prime. To avoid 355 * fencepost errors, the last pass is skipped. 356 */ 357 for (smallbase = TINY_NUMBER + 3; 358 smallbase < (SMALL_MAXIMUM - TINY_NUMBER); 359 smallbase += TINY_NUMBER) { 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 r = smallbase % t; 367 368 if (r == 0) { 369 s = 0; /* t divides into smallbase exactly */ 370 } else { 371 /* smallbase+s is first entry divisible by t */ 372 s = t - r; 373 } 374 375 /* 376 * The sieve omits even numbers, so ensure that 377 * smallbase+s is odd. Then, step through the sieve 378 * in increments of 2*t 379 */ 380 if (s & 1) 381 s += t; /* Make smallbase+s odd, and s even */ 382 383 /* Mark all multiples of 2*t */ 384 for (s /= 2; s < smallbits; s += t) 385 BIT_SET(SmallSieve, s); 386 } 387 388 /* 389 * SmallSieve 390 */ 391 for (i = 0; i < smallbits; i++) { 392 if (BIT_TEST(SmallSieve, i)) 393 continue; /* 2*i+smallbase is composite */ 394 395 /* The next small prime */ 396 sieve_large((2 * i) + smallbase); 397 } 398 399 memset(SmallSieve, 0, smallwords << SHIFT_BYTE); 400 } 401 402 time(&time_stop); 403 404 logit("%.24s Sieved with %u small primes in %ld seconds", 405 ctime(&time_stop), largetries, (long) (time_stop - time_start)); 406 407 for (j = r = 0; j < largebits; j++) { 408 if (BIT_TEST(LargeSieve, j)) 409 continue; /* Definitely composite, skip */ 410 411 debug2("test q = largebase+%u", 2 * j); 412 BN_set_word(q, 2 * j); 413 BN_add(q, q, largebase); 414 if (qfileout(out, QTYPE_SOPHIE_GERMAINE, QTEST_SIEVE, 415 largetries, (power - 1) /* MSB */, (0), q) == -1) { 416 ret = -1; 417 break; 418 } 419 420 r++; /* count q */ 421 } 422 423 time(&time_stop); 424 425 xfree(LargeSieve); 426 xfree(SmallSieve); 427 xfree(TinySieve); 428 429 logit("%.24s Found %u candidates", ctime(&time_stop), r); 430 431 return (ret); 432 } 433 434 /* 435 * perform a Miller-Rabin primality test 436 * on the list of candidates 437 * (checking both q and p) 438 * The result is a list of so-call "safe" primes 439 */ 440 int 441 prime_test(FILE *in, FILE *out, u_int32_t trials, 442 u_int32_t generator_wanted) 443 { 444 BIGNUM *q, *p, *a; 445 BN_CTX *ctx; 446 char *cp, *lp; 447 u_int32_t count_in = 0, count_out = 0, count_possible = 0; 448 u_int32_t generator_known, in_tests, in_tries, in_type, in_size; 449 time_t time_start, time_stop; 450 int res; 451 452 time(&time_start); 453 454 p = BN_new(); 455 q = BN_new(); 456 ctx = BN_CTX_new(); 457 458 debug2("%.24s Final %u Miller-Rabin trials (%x generator)", 459 ctime(&time_start), trials, generator_wanted); 460 461 res = 0; 462 lp = xmalloc(QLINESIZE + 1); 463 while (fgets(lp, QLINESIZE, in) != NULL) { 464 int ll = strlen(lp); 465 466 count_in++; 467 if (ll < 14 || *lp == '!' || *lp == '#') { 468 debug2("%10u: comment or short line", count_in); 469 continue; 470 } 471 472 /* XXX - fragile parser */ 473 /* time */ 474 cp = &lp[14]; /* (skip) */ 475 476 /* type */ 477 in_type = strtoul(cp, &cp, 10); 478 479 /* tests */ 480 in_tests = strtoul(cp, &cp, 10); 481 482 if (in_tests & QTEST_COMPOSITE) { 483 debug2("%10u: known composite", count_in); 484 continue; 485 } 486 /* tries */ 487 in_tries = strtoul(cp, &cp, 10); 488 489 /* size (most significant bit) */ 490 in_size = strtoul(cp, &cp, 10); 491 492 /* generator (hex) */ 493 generator_known = strtoul(cp, &cp, 16); 494 495 /* Skip white space */ 496 cp += strspn(cp, " "); 497 498 /* modulus (hex) */ 499 switch (in_type) { 500 case QTYPE_SOPHIE_GERMAINE: 501 debug2("%10u: (%u) Sophie-Germaine", count_in, in_type); 502 a = q; 503 BN_hex2bn(&a, cp); 504 /* p = 2*q + 1 */ 505 BN_lshift(p, q, 1); 506 BN_add_word(p, 1); 507 in_size += 1; 508 generator_known = 0; 509 break; 510 default: 511 debug2("%10u: (%u)", count_in, in_type); 512 a = p; 513 BN_hex2bn(&a, cp); 514 /* q = (p-1) / 2 */ 515 BN_rshift(q, p, 1); 516 break; 517 } 518 519 /* 520 * due to earlier inconsistencies in interpretation, check 521 * the proposed bit size. 522 */ 523 if (BN_num_bits(p) != (in_size + 1)) { 524 debug2("%10u: bit size %u mismatch", count_in, in_size); 525 continue; 526 } 527 if (in_size < QSIZE_MINIMUM) { 528 debug2("%10u: bit size %u too short", count_in, in_size); 529 continue; 530 } 531 532 if (in_tests & QTEST_MILLER_RABIN) 533 in_tries += trials; 534 else 535 in_tries = trials; 536 /* 537 * guess unknown generator 538 */ 539 if (generator_known == 0) { 540 if (BN_mod_word(p, 24) == 11) 541 generator_known = 2; 542 else if (BN_mod_word(p, 12) == 5) 543 generator_known = 3; 544 else { 545 u_int32_t r = BN_mod_word(p, 10); 546 547 if (r == 3 || r == 7) { 548 generator_known = 5; 549 } 550 } 551 } 552 /* 553 * skip tests when desired generator doesn't match 554 */ 555 if (generator_wanted > 0 && 556 generator_wanted != generator_known) { 557 debug2("%10u: generator %d != %d", 558 count_in, generator_known, generator_wanted); 559 continue; 560 } 561 562 count_possible++; 563 564 /* 565 * The (1/4)^N performance bound on Miller-Rabin is 566 * extremely pessimistic, so don't spend a lot of time 567 * really verifying that q is prime until after we know 568 * that p is also prime. A single pass will weed out the 569 * vast majority of composite q's. 570 */ 571 if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) { 572 debug2("%10u: q failed first possible prime test", 573 count_in); 574 continue; 575 } 576 577 /* 578 * q is possibly prime, so go ahead and really make sure 579 * that p is prime. If it is, then we can go back and do 580 * the same for q. If p is composite, chances are that 581 * will show up on the first Rabin-Miller iteration so it 582 * doesn't hurt to specify a high iteration count. 583 */ 584 if (!BN_is_prime(p, trials, NULL, ctx, NULL)) { 585 debug2("%10u: p is not prime", count_in); 586 continue; 587 } 588 debug("%10u: p is almost certainly prime", count_in); 589 590 /* recheck q more rigorously */ 591 if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) { 592 debug("%10u: q is not prime", count_in); 593 continue; 594 } 595 debug("%10u: q is almost certainly prime", count_in); 596 597 if (qfileout(out, QTYPE_SAFE, (in_tests | QTEST_MILLER_RABIN), 598 in_tries, in_size, generator_known, p)) { 599 res = -1; 600 break; 601 } 602 603 count_out++; 604 } 605 606 time(&time_stop); 607 xfree(lp); 608 BN_free(p); 609 BN_free(q); 610 BN_CTX_free(ctx); 611 612 logit("%.24s Found %u safe primes of %u candidates in %ld seconds", 613 ctime(&time_stop), count_out, count_possible, 614 (long) (time_stop - time_start)); 615 616 return (res); 617 } 618