1 /* LibTomMath, multiple-precision integer library -- Tom St Denis 2 * 3 * LibTomMath is a library that provides multiple-precision 4 * integer arithmetic as well as number theoretic functionality. 5 * 6 * The library was designed directly after the MPI library by 7 * Michael Fromberger but has been written from scratch with 8 * additional optimizations in place. 9 * 10 * The library is free for all purposes without any express 11 * guarantee it works. 12 * 13 * Tom St Denis, tomstdenis (at) gmail.com, http://math.libtomcrypt.com 14 */ 15 #ifndef BN_H_ 16 #define BN_H_ 17 18 #include <stdio.h> 19 #include <string.h> 20 #include <stdlib.h> 21 #include <ctype.h> 22 #include <limits.h> 23 24 #include "tommath_class.h" 25 26 #ifndef MIN 27 #define MIN(x,y) ((x)<(y)?(x):(y)) 28 #endif 29 30 #ifndef MAX 31 #define MAX(x,y) ((x)>(y)?(x):(y)) 32 #endif 33 34 #ifdef __cplusplus 35 extern "C" { 36 37 /* C++ compilers don't like assigning void * to mp_digit * */ 38 #define OPT_CAST(x) (x *) 39 40 #else 41 42 /* C on the other hand doesn't care */ 43 #define OPT_CAST(x) 44 45 #endif 46 47 48 /* detect 64-bit mode if possible */ 49 #if defined(__x86_64__) 50 #if !(defined(MP_64BIT) && defined(MP_16BIT) && defined(MP_8BIT)) 51 #define MP_64BIT 52 #endif 53 #endif 54 55 /* some default configurations. 56 * 57 * A "mp_digit" must be able to hold DIGIT_BIT + 1 bits 58 * A "mp_word" must be able to hold 2*DIGIT_BIT + 1 bits 59 * 60 * At the very least a mp_digit must be able to hold 7 bits 61 * [any size beyond that is ok provided it doesn't overflow the data type] 62 */ 63 #ifdef MP_8BIT 64 typedef unsigned char mp_digit; 65 typedef unsigned short mp_word; 66 #elif defined(MP_16BIT) 67 typedef unsigned short mp_digit; 68 typedef unsigned long mp_word; 69 #elif defined(MP_64BIT) 70 /* for GCC only on supported platforms */ 71 #ifndef CRYPT 72 typedef unsigned long long ulong64; 73 typedef signed long long long64; 74 #endif 75 76 typedef unsigned long mp_digit; 77 typedef unsigned long mp_word __attribute__ ((mode(TI))); 78 79 #define DIGIT_BIT 60 80 #else 81 /* this is the default case, 28-bit digits */ 82 83 /* this is to make porting into LibTomCrypt easier :-) */ 84 #ifndef CRYPT 85 #if defined(_MSC_VER) || defined(__BORLANDC__) 86 typedef unsigned __int64 ulong64; 87 typedef signed __int64 long64; 88 #else 89 typedef unsigned long long ulong64; 90 typedef signed long long long64; 91 #endif 92 #endif 93 94 typedef unsigned long mp_digit; 95 typedef ulong64 mp_word; 96 97 #ifdef MP_31BIT 98 /* this is an extension that uses 31-bit digits */ 99 #define DIGIT_BIT 31 100 #else 101 /* default case is 28-bit digits, defines MP_28BIT as a handy macro to test */ 102 #define DIGIT_BIT 28 103 #define MP_28BIT 104 #endif 105 #endif 106 107 /* define heap macros */ 108 #ifndef CRYPT 109 /* default to libc stuff */ 110 #ifndef XMALLOC 111 #define XMALLOC malloc 112 #define XFREE free 113 #define XREALLOC realloc 114 #define XCALLOC calloc 115 #else 116 /* prototypes for our heap functions */ 117 extern void *XMALLOC(size_t n); 118 extern void *XREALLOC(void *p, size_t n); 119 extern void *XCALLOC(size_t n, size_t s); 120 extern void XFREE(void *p); 121 #endif 122 #endif 123 124 125 /* otherwise the bits per digit is calculated automatically from the size of a mp_digit */ 126 #ifndef DIGIT_BIT 127 #define DIGIT_BIT ((int)((CHAR_BIT * sizeof(mp_digit) - 1))) /* bits per digit */ 128 #endif 129 130 #define MP_DIGIT_BIT DIGIT_BIT 131 #define MP_MASK ((((mp_digit)1)<<((mp_digit)DIGIT_BIT))-((mp_digit)1)) 132 #define MP_DIGIT_MAX MP_MASK 133 134 /* equalities */ 135 #define MP_LT -1 /* less than */ 136 #define MP_EQ 0 /* equal to */ 137 #define MP_GT 1 /* greater than */ 138 139 #define MP_ZPOS 0 /* positive integer */ 140 #define MP_NEG 1 /* negative */ 141 142 #define MP_OKAY 0 /* ok result */ 143 #define MP_MEM -2 /* out of mem */ 144 #define MP_VAL -3 /* invalid input */ 145 #define MP_RANGE MP_VAL 146 147 #define MP_YES 1 /* yes response */ 148 #define MP_NO 0 /* no response */ 149 150 /* Primality generation flags */ 151 #define LTM_PRIME_BBS 0x0001 /* BBS style prime */ 152 #define LTM_PRIME_SAFE 0x0002 /* Safe prime (p-1)/2 == prime */ 153 #define LTM_PRIME_2MSB_ON 0x0008 /* force 2nd MSB to 1 */ 154 155 typedef int mp_err; 156 157 /* you'll have to tune these... */ 158 extern int KARATSUBA_MUL_CUTOFF, 159 KARATSUBA_SQR_CUTOFF, 160 TOOM_MUL_CUTOFF, 161 TOOM_SQR_CUTOFF; 162 163 /* define this to use lower memory usage routines (exptmods mostly) */ 164 /* #define MP_LOW_MEM */ 165 166 /* default precision */ 167 #ifndef MP_PREC 168 #ifndef MP_LOW_MEM 169 #define MP_PREC 32 /* default digits of precision */ 170 #else 171 #define MP_PREC 8 /* default digits of precision */ 172 #endif 173 #endif 174 175 /* size of comba arrays, should be at least 2 * 2**(BITS_PER_WORD - BITS_PER_DIGIT*2) */ 176 #define MP_WARRAY (1 << (sizeof(mp_word) * CHAR_BIT - 2 * DIGIT_BIT + 1)) 177 178 /* the infamous mp_int structure */ 179 typedef struct { 180 int used, alloc, sign; 181 mp_digit *dp; 182 } mp_int; 183 184 /* callback for mp_prime_random, should fill dst with random bytes and return how many read [upto len] */ 185 typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat); 186 187 188 #define USED(m) ((m)->used) 189 #define DIGIT(m,k) ((m)->dp[(k)]) 190 #define SIGN(m) ((m)->sign) 191 192 /* error code to char* string */ 193 char *mp_error_to_string(int code); 194 195 /* ---> init and deinit bignum functions <--- */ 196 /* init a bignum */ 197 int mp_init(mp_int *a); 198 199 /* free a bignum */ 200 void mp_clear(mp_int *a); 201 202 /* init a null terminated series of arguments */ 203 int mp_init_multi(mp_int *mp, ...); 204 205 /* clear a null terminated series of arguments */ 206 void mp_clear_multi(mp_int *mp, ...); 207 208 /* exchange two ints */ 209 void mp_exch(mp_int *a, mp_int *b); 210 211 /* shrink ram required for a bignum */ 212 int mp_shrink(mp_int *a); 213 214 /* grow an int to a given size */ 215 int mp_grow(mp_int *a, int size); 216 217 /* init to a given number of digits */ 218 int mp_init_size(mp_int *a, int size); 219 220 /* ---> Basic Manipulations <--- */ 221 #define mp_iszero(a) (((a)->used == 0) ? MP_YES : MP_NO) 222 #define mp_iseven(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 0)) ? MP_YES : MP_NO) 223 #define mp_isodd(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 1)) ? MP_YES : MP_NO) 224 225 /* set to zero */ 226 void mp_zero(mp_int *a); 227 228 /* set to a digit */ 229 void mp_set(mp_int *a, mp_digit b); 230 231 /* set a 32-bit const */ 232 int mp_set_int(mp_int *a, unsigned long b); 233 234 /* get a 32-bit value */ 235 unsigned long mp_get_int(mp_int * a); 236 237 /* initialize and set a digit */ 238 int mp_init_set (mp_int * a, mp_digit b); 239 240 /* initialize and set 32-bit value */ 241 int mp_init_set_int (mp_int * a, unsigned long b); 242 243 /* copy, b = a */ 244 int mp_copy(mp_int *a, mp_int *b); 245 246 /* inits and copies, a = b */ 247 int mp_init_copy(mp_int *a, mp_int *b); 248 249 /* trim unused digits */ 250 void mp_clamp(mp_int *a); 251 252 /* ---> digit manipulation <--- */ 253 254 /* right shift by "b" digits */ 255 void mp_rshd(mp_int *a, int b); 256 257 /* left shift by "b" digits */ 258 int mp_lshd(mp_int *a, int b); 259 260 /* c = a / 2**b */ 261 int mp_div_2d(mp_int *a, int b, mp_int *c, mp_int *d); 262 263 /* b = a/2 */ 264 int mp_div_2(mp_int *a, mp_int *b); 265 266 /* c = a * 2**b */ 267 int mp_mul_2d(mp_int *a, int b, mp_int *c); 268 269 /* b = a*2 */ 270 int mp_mul_2(mp_int *a, mp_int *b); 271 272 /* c = a mod 2**d */ 273 int mp_mod_2d(mp_int *a, int b, mp_int *c); 274 275 /* computes a = 2**b */ 276 int mp_2expt(mp_int *a, int b); 277 278 /* Counts the number of lsbs which are zero before the first zero bit */ 279 int mp_cnt_lsb(mp_int *a); 280 281 /* I Love Earth! */ 282 283 /* makes a pseudo-random int of a given size */ 284 int mp_rand(mp_int *a, int digits); 285 286 /* ---> binary operations <--- */ 287 /* c = a XOR b */ 288 int mp_xor(mp_int *a, mp_int *b, mp_int *c); 289 290 /* c = a OR b */ 291 int mp_or(mp_int *a, mp_int *b, mp_int *c); 292 293 /* c = a AND b */ 294 int mp_and(mp_int *a, mp_int *b, mp_int *c); 295 296 /* ---> Basic arithmetic <--- */ 297 298 /* b = -a */ 299 int mp_neg(mp_int *a, mp_int *b); 300 301 /* b = |a| */ 302 int mp_abs(mp_int *a, mp_int *b); 303 304 /* compare a to b */ 305 int mp_cmp(mp_int *a, mp_int *b); 306 307 /* compare |a| to |b| */ 308 int mp_cmp_mag(mp_int *a, mp_int *b); 309 310 /* c = a + b */ 311 int mp_add(mp_int *a, mp_int *b, mp_int *c); 312 313 /* c = a - b */ 314 int mp_sub(mp_int *a, mp_int *b, mp_int *c); 315 316 /* c = a * b */ 317 int mp_mul(mp_int *a, mp_int *b, mp_int *c); 318 319 /* b = a*a */ 320 int mp_sqr(mp_int *a, mp_int *b); 321 322 /* a/b => cb + d == a */ 323 int mp_div(mp_int *a, mp_int *b, mp_int *c, mp_int *d); 324 325 /* c = a mod b, 0 <= c < b */ 326 int mp_mod(mp_int *a, mp_int *b, mp_int *c); 327 328 /* ---> single digit functions <--- */ 329 330 /* compare against a single digit */ 331 int mp_cmp_d(mp_int *a, mp_digit b); 332 333 /* c = a + b */ 334 int mp_add_d(mp_int *a, mp_digit b, mp_int *c); 335 336 /* c = a - b */ 337 int mp_sub_d(mp_int *a, mp_digit b, mp_int *c); 338 339 /* c = a * b */ 340 int mp_mul_d(mp_int *a, mp_digit b, mp_int *c); 341 342 /* a/b => cb + d == a */ 343 int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d); 344 345 /* a/3 => 3c + d == a */ 346 int mp_div_3(mp_int *a, mp_int *c, mp_digit *d); 347 348 /* c = a**b */ 349 int mp_expt_d(mp_int *a, mp_digit b, mp_int *c); 350 351 /* c = a mod b, 0 <= c < b */ 352 int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c); 353 354 /* ---> number theory <--- */ 355 356 /* d = a + b (mod c) */ 357 int mp_addmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); 358 359 /* d = a - b (mod c) */ 360 int mp_submod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); 361 362 /* d = a * b (mod c) */ 363 int mp_mulmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); 364 365 /* c = a * a (mod b) */ 366 int mp_sqrmod(mp_int *a, mp_int *b, mp_int *c); 367 368 /* c = 1/a (mod b) */ 369 int mp_invmod(mp_int *a, mp_int *b, mp_int *c); 370 371 /* c = (a, b) */ 372 int mp_gcd(mp_int *a, mp_int *b, mp_int *c); 373 374 /* produces value such that U1*a + U2*b = U3 */ 375 int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3); 376 377 /* c = [a, b] or (a*b)/(a, b) */ 378 int mp_lcm(mp_int *a, mp_int *b, mp_int *c); 379 380 /* finds one of the b'th root of a, such that |c|**b <= |a| 381 * 382 * returns error if a < 0 and b is even 383 */ 384 int mp_n_root(mp_int *a, mp_digit b, mp_int *c); 385 386 /* special sqrt algo */ 387 int mp_sqrt(mp_int *arg, mp_int *ret); 388 389 /* is number a square? */ 390 int mp_is_square(mp_int *arg, int *ret); 391 392 /* computes the jacobi c = (a | n) (or Legendre if b is prime) */ 393 int mp_jacobi(mp_int *a, mp_int *n, int *c); 394 395 /* used to setup the Barrett reduction for a given modulus b */ 396 int mp_reduce_setup(mp_int *a, mp_int *b); 397 398 /* Barrett Reduction, computes a (mod b) with a precomputed value c 399 * 400 * Assumes that 0 < a <= b*b, note if 0 > a > -(b*b) then you can merely 401 * compute the reduction as -1 * mp_reduce(mp_abs(a)) [pseudo code]. 402 */ 403 int mp_reduce(mp_int *a, mp_int *b, mp_int *c); 404 405 /* setups the montgomery reduction */ 406 int mp_montgomery_setup(mp_int *a, mp_digit *mp); 407 408 /* computes a = B**n mod b without division or multiplication useful for 409 * normalizing numbers in a Montgomery system. 410 */ 411 int mp_montgomery_calc_normalization(mp_int *a, mp_int *b); 412 413 /* computes x/R == x (mod N) via Montgomery Reduction */ 414 int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp); 415 416 /* returns 1 if a is a valid DR modulus */ 417 int mp_dr_is_modulus(mp_int *a); 418 419 /* sets the value of "d" required for mp_dr_reduce */ 420 void mp_dr_setup(mp_int *a, mp_digit *d); 421 422 /* reduces a modulo b using the Diminished Radix method */ 423 int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp); 424 425 /* returns true if a can be reduced with mp_reduce_2k */ 426 int mp_reduce_is_2k(mp_int *a); 427 428 /* determines k value for 2k reduction */ 429 int mp_reduce_2k_setup(mp_int *a, mp_digit *d); 430 431 /* reduces a modulo b where b is of the form 2**p - k [0 <= a] */ 432 int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d); 433 434 /* returns true if a can be reduced with mp_reduce_2k_l */ 435 int mp_reduce_is_2k_l(mp_int *a); 436 437 /* determines k value for 2k reduction */ 438 int mp_reduce_2k_setup_l(mp_int *a, mp_int *d); 439 440 /* reduces a modulo b where b is of the form 2**p - k [0 <= a] */ 441 int mp_reduce_2k_l(mp_int *a, mp_int *n, mp_int *d); 442 443 /* d = a**b (mod c) */ 444 int mp_exptmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); 445 446 /* ---> Primes <--- */ 447 448 /* number of primes */ 449 #ifdef MP_8BIT 450 #define PRIME_SIZE 31 451 #else 452 #define PRIME_SIZE 256 453 #endif 454 455 /* table of first PRIME_SIZE primes */ 456 extern const mp_digit ltm_prime_tab[]; 457 458 /* result=1 if a is divisible by one of the first PRIME_SIZE primes */ 459 int mp_prime_is_divisible(mp_int *a, int *result); 460 461 /* performs one Fermat test of "a" using base "b". 462 * Sets result to 0 if composite or 1 if probable prime 463 */ 464 int mp_prime_fermat(mp_int *a, mp_int *b, int *result); 465 466 /* performs one Miller-Rabin test of "a" using base "b". 467 * Sets result to 0 if composite or 1 if probable prime 468 */ 469 int mp_prime_miller_rabin(mp_int *a, mp_int *b, int *result); 470 471 /* This gives [for a given bit size] the number of trials required 472 * such that Miller-Rabin gives a prob of failure lower than 2^-96 473 */ 474 int mp_prime_rabin_miller_trials(int size); 475 476 /* performs t rounds of Miller-Rabin on "a" using the first 477 * t prime bases. Also performs an initial sieve of trial 478 * division. Determines if "a" is prime with probability 479 * of error no more than (1/4)**t. 480 * 481 * Sets result to 1 if probably prime, 0 otherwise 482 */ 483 int mp_prime_is_prime(mp_int *a, int t, int *result); 484 485 /* finds the next prime after the number "a" using "t" trials 486 * of Miller-Rabin. 487 * 488 * bbs_style = 1 means the prime must be congruent to 3 mod 4 489 */ 490 int mp_prime_next_prime(mp_int *a, int t, int bbs_style); 491 492 /* makes a truly random prime of a given size (bytes), 493 * call with bbs = 1 if you want it to be congruent to 3 mod 4 494 * 495 * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can 496 * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself 497 * so it can be NULL 498 * 499 * The prime generated will be larger than 2^(8*size). 500 */ 501 #define mp_prime_random(a, t, size, bbs, cb, dat) mp_prime_random_ex(a, t, ((size) * 8) + 1, (bbs==1)?LTM_PRIME_BBS:0, cb, dat) 502 503 /* makes a truly random prime of a given size (bits), 504 * 505 * Flags are as follows: 506 * 507 * LTM_PRIME_BBS - make prime congruent to 3 mod 4 508 * LTM_PRIME_SAFE - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS) 509 * LTM_PRIME_2MSB_OFF - make the 2nd highest bit zero 510 * LTM_PRIME_2MSB_ON - make the 2nd highest bit one 511 * 512 * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can 513 * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself 514 * so it can be NULL 515 * 516 */ 517 int mp_prime_random_ex(mp_int *a, int t, int size, int flags, ltm_prime_callback cb, void *dat); 518 519 /* ---> radix conversion <--- */ 520 int mp_count_bits(mp_int *a); 521 522 int mp_unsigned_bin_size(mp_int *a); 523 int mp_read_unsigned_bin(mp_int *a, const unsigned char *b, int c); 524 int mp_to_unsigned_bin(mp_int *a, unsigned char *b); 525 int mp_to_unsigned_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen); 526 527 int mp_signed_bin_size(mp_int *a); 528 int mp_read_signed_bin(mp_int *a, const unsigned char *b, int c); 529 int mp_to_signed_bin(mp_int *a, unsigned char *b); 530 int mp_to_signed_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen); 531 532 int mp_read_radix(mp_int *a, const char *str, int radix); 533 int mp_toradix(mp_int *a, char *str, int radix); 534 int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen); 535 int mp_radix_size(mp_int *a, int radix, int *size); 536 537 int mp_fread(mp_int *a, int radix, FILE *stream); 538 int mp_fwrite(mp_int *a, int radix, FILE *stream); 539 540 #define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len)) 541 #define mp_raw_size(mp) mp_signed_bin_size(mp) 542 #define mp_toraw(mp, str) mp_to_signed_bin((mp), (str)) 543 #define mp_read_mag(mp, str, len) mp_read_unsigned_bin((mp), (str), (len)) 544 #define mp_mag_size(mp) mp_unsigned_bin_size(mp) 545 #define mp_tomag(mp, str) mp_to_unsigned_bin((mp), (str)) 546 547 #define mp_tobinary(M, S) mp_toradix((M), (S), 2) 548 #define mp_tooctal(M, S) mp_toradix((M), (S), 8) 549 #define mp_todecimal(M, S) mp_toradix((M), (S), 10) 550 #define mp_tohex(M, S) mp_toradix((M), (S), 16) 551 552 /* lowlevel functions, do not call! */ 553 int s_mp_add(mp_int *a, mp_int *b, mp_int *c); 554 int s_mp_sub(mp_int *a, mp_int *b, mp_int *c); 555 #define s_mp_mul(a, b, c) s_mp_mul_digs(a, b, c, (a)->used + (b)->used + 1) 556 int fast_s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs); 557 int s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs); 558 int fast_s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs); 559 int s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs); 560 int fast_s_mp_sqr(mp_int *a, mp_int *b); 561 int s_mp_sqr(mp_int *a, mp_int *b); 562 int mp_karatsuba_mul(mp_int *a, mp_int *b, mp_int *c); 563 int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c); 564 int mp_karatsuba_sqr(mp_int *a, mp_int *b); 565 int mp_toom_sqr(mp_int *a, mp_int *b); 566 int fast_mp_invmod(mp_int *a, mp_int *b, mp_int *c); 567 int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c); 568 int fast_mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp); 569 int mp_exptmod_fast(mp_int *G, mp_int *X, mp_int *P, mp_int *Y, int mode); 570 int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int mode); 571 void bn_reverse(unsigned char *s, int len); 572 573 extern const char *mp_s_rmap; 574 575 #ifdef __cplusplus 576 } 577 #endif 578 579 #endif 580 581 582 /* $Source: /cvs/libtom/libtommath/tommath.h,v $ */ 583 /* $Revision: 1.8 $ */ 584 /* $Date: 2006/03/31 14:18:44 $ */ 585
