1 /* LibTomMath, multiple-precision integer library -- Tom St Denis
3 * LibTomMath is a library that provides multiple-precision
4 * integer arithmetic as well as number theoretic functionality.
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.
10 * The library is free for all purposes without any express
13 * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org
24 #include <tma_class.h>
26 /* Assure these -Pekka */
32 #define MIN(x,y) ((x)<(y)?(x):(y))
34 #define MAX(x,y) ((x)>(y)?(x):(y))
39 /* C++ compilers don't like assigning void * to mp_digit * */
40 #define OPT_CAST(x) (x *)
44 /* C on the other hand doesn't care */
50 /* detect 64-bit mode if possible */
51 #if defined(__x86_64__)
52 #if !(defined(MP_64BIT) && defined(MP_16BIT) && defined(MP_8BIT))
57 /* some default configurations.
59 * A "mp_digit" must be able to hold DIGIT_BIT + 1 bits
60 * A "mp_word" must be able to hold 2*DIGIT_BIT + 1 bits
62 * At the very least a mp_digit must be able to hold 7 bits
63 * [any size beyond that is ok provided it doesn't overflow the data type]
66 typedef unsigned char mp_digit;
67 typedef unsigned short mp_word;
68 #elif defined(MP_16BIT)
69 typedef unsigned short mp_digit;
70 typedef unsigned long mp_word;
71 #elif defined(MP_64BIT)
72 /* for GCC only on supported platforms */
74 typedef unsigned long long ulong64;
75 typedef signed long long long64;
78 typedef unsigned long mp_digit;
79 typedef unsigned long mp_word __attribute__ ((mode(TI)));
83 /* this is the default case, 28-bit digits */
85 /* this is to make porting into LibTomCrypt easier :-) */
87 #if defined(_MSC_VER) || defined(__BORLANDC__)
88 typedef unsigned __int64 ulong64;
89 typedef signed __int64 long64;
91 typedef unsigned long long ulong64;
92 typedef signed long long long64;
96 typedef unsigned long mp_digit;
97 typedef ulong64 mp_word;
100 /* this is an extension that uses 31-bit digits */
103 /* default case is 28-bit digits, defines MP_28BIT as a handy macro to test */
109 /* define heap macros */
111 /* default to libc stuff */
113 #define XMALLOC malloc
115 #define XREALLOC realloc
116 #define XCALLOC calloc
118 /* prototypes for our heap functions */
119 extern void *XMALLOC(size_t n);
120 extern void *REALLOC(void *p, size_t n);
121 extern void *XCALLOC(size_t n, size_t s);
122 extern void XFREE(void *p);
127 /* otherwise the bits per digit is calculated automatically from the size of a mp_digit */
129 #define DIGIT_BIT ((int)((CHAR_BIT * sizeof(mp_digit) - 1))) /* bits per digit */
132 #define MP_DIGIT_BIT DIGIT_BIT
133 #define MP_MASK ((((mp_digit)1)<<((mp_digit)DIGIT_BIT))-((mp_digit)1))
134 #define MP_DIGIT_MAX MP_MASK
137 #define MP_LT -1 /* less than */
138 #define MP_EQ 0 /* equal to */
139 #define MP_GT 1 /* greater than */
141 #define MP_ZPOS 0 /* positive integer */
142 #define MP_NEG 1 /* negative */
144 #define MP_OKAY 0 /* ok result */
145 #define MP_MEM -2 /* out of mem */
146 #define MP_VAL -3 /* invalid input */
147 #define MP_RANGE MP_VAL
149 #define MP_YES 1 /* yes response */
150 #define MP_NO 0 /* no response */
152 /* Primality generation flags */
153 #define LTM_PRIME_BBS 0x0001 /* BBS style prime */
154 #define LTM_PRIME_SAFE 0x0002 /* Safe prime (p-1)/2 == prime */
155 #define LTM_PRIME_2MSB_OFF 0x0004 /* force 2nd MSB to 0 */
156 #define LTM_PRIME_2MSB_ON 0x0008 /* force 2nd MSB to 1 */
160 /* you'll have to tune these... */
161 extern int KARATSUBA_MUL_CUTOFF,
162 KARATSUBA_SQR_CUTOFF,
166 /* define this to use lower memory usage routines (exptmods mostly) */
167 /* #define MP_LOW_MEM */
169 /* default precision */
172 #define MP_PREC 64 /* default digits of precision */
174 #define MP_PREC 8 /* default digits of precision */
178 /* size of comba arrays, should be at least 2 * 2**(BITS_PER_WORD - BITS_PER_DIGIT*2) */
179 #define MP_WARRAY (1 << (sizeof(mp_word) * CHAR_BIT - 2 * DIGIT_BIT + 1))
181 /* the infamous mp_int structure */
183 int used, alloc, sign;
187 /* callback for mp_prime_random, should fill dst with random bytes and return how many read [upto len] */
188 typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat);
191 #define USED(m) ((m)->used)
192 #define DIGIT(m,k) ((m)->dp[(k)])
193 #define SIGN(m) ((m)->sign)
195 /* error code to char* string */
196 char *mp_error_to_string(int code);
198 /* ---> init and deinit bignum functions <--- */
200 int mp_init(mp_int *a);
203 void mp_clear(mp_int *a);
205 /* init a null terminated series of arguments */
206 int mp_init_multi(mp_int *mp, ...);
208 /* clear a null terminated series of arguments */
209 void mp_clear_multi(mp_int *mp, ...);
211 /* exchange two ints */
212 void mp_exch(mp_int *a, mp_int *b);
214 /* shrink ram required for a bignum */
215 int mp_shrink(mp_int *a);
217 /* grow an int to a given size */
218 int mp_grow(mp_int *a, int size);
220 /* init to a given number of digits */
221 int mp_init_size(mp_int *a, int size);
223 /* ---> Basic Manipulations <--- */
224 #define mp_iszero(a) (((a)->used == 0) ? MP_YES : MP_NO)
225 #define mp_iseven(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 0)) ? MP_YES : MP_NO)
226 #define mp_isodd(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 1)) ? MP_YES : MP_NO)
229 void mp_zero(mp_int *a);
232 void mp_set(mp_int *a, mp_digit b);
234 /* set a 32-bit const */
235 int mp_set_int(mp_int *a, unsigned long b);
237 /* get a 32-bit value */
238 unsigned long mp_get_int(mp_int * a);
240 /* initialize and set a digit */
241 int mp_init_set (mp_int * a, mp_digit b);
243 /* initialize and set 32-bit value */
244 int mp_init_set_int (mp_int * a, unsigned long b);
247 int mp_copy(mp_int *a, mp_int *b);
249 /* inits and copies, a = b */
250 int mp_init_copy(mp_int *a, mp_int *b);
252 /* trim unused digits */
253 void mp_clamp(mp_int *a);
255 /* ---> digit manipulation <--- */
257 /* right shift by "b" digits */
258 void mp_rshd(mp_int *a, int b);
260 /* left shift by "b" digits */
261 int mp_lshd(mp_int *a, int b);
264 int mp_div_2d(mp_int *a, int b, mp_int *c, mp_int *d);
267 int mp_div_2(mp_int *a, mp_int *b);
270 int mp_mul_2d(mp_int *a, int b, mp_int *c);
273 int mp_mul_2(mp_int *a, mp_int *b);
276 int mp_mod_2d(mp_int *a, int b, mp_int *c);
278 /* computes a = 2**b */
279 int mp_2expt(mp_int *a, int b);
281 /* Counts the number of lsbs which are zero before the first zero bit */
282 int mp_cnt_lsb(mp_int *a);
286 /* makes a pseudo-random int of a given size */
287 int mp_rand(mp_int *a, int digits);
289 /* ---> binary operations <--- */
291 int mp_xor(mp_int *a, mp_int *b, mp_int *c);
294 int mp_or(mp_int *a, mp_int *b, mp_int *c);
297 int mp_and(mp_int *a, mp_int *b, mp_int *c);
299 /* ---> Basic arithmetic <--- */
302 int mp_neg(mp_int *a, mp_int *b);
305 int mp_abs(mp_int *a, mp_int *b);
308 int mp_cmp(mp_int *a, mp_int *b);
310 /* compare |a| to |b| */
311 int mp_cmp_mag(mp_int *a, mp_int *b);
314 int mp_add(mp_int *a, mp_int *b, mp_int *c);
317 int mp_sub(mp_int *a, mp_int *b, mp_int *c);
320 int mp_mul(mp_int *a, mp_int *b, mp_int *c);
323 int mp_sqr(mp_int *a, mp_int *b);
325 /* a/b => cb + d == a */
326 int mp_div(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
328 /* c = a mod b, 0 <= c < b */
329 int mp_mod(mp_int *a, mp_int *b, mp_int *c);
331 /* ---> single digit functions <--- */
333 /* compare against a single digit */
334 int mp_cmp_d(mp_int *a, mp_digit b);
337 int mp_add_d(mp_int *a, mp_digit b, mp_int *c);
340 int mp_sub_d(mp_int *a, mp_digit b, mp_int *c);
343 int mp_mul_d(mp_int *a, mp_digit b, mp_int *c);
345 /* a/b => cb + d == a */
346 int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d);
348 /* a/3 => 3c + d == a */
349 int mp_div_3(mp_int *a, mp_int *c, mp_digit *d);
352 int mp_expt_d(mp_int *a, mp_digit b, mp_int *c);
354 /* c = a mod b, 0 <= c < b */
355 int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c);
357 /* ---> number theory <--- */
359 /* d = a + b (mod c) */
360 int mp_addmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
362 /* d = a - b (mod c) */
363 int mp_submod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
365 /* d = a * b (mod c) */
366 int mp_mulmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
368 /* c = a * a (mod b) */
369 int mp_sqrmod(mp_int *a, mp_int *b, mp_int *c);
371 /* c = 1/a (mod b) */
372 int mp_invmod(mp_int *a, mp_int *b, mp_int *c);
375 int mp_gcd(mp_int *a, mp_int *b, mp_int *c);
377 /* produces value such that U1*a + U2*b = U3 */
378 int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3);
380 /* c = [a, b] or (a*b)/(a, b) */
381 int mp_lcm(mp_int *a, mp_int *b, mp_int *c);
383 /* finds one of the b'th root of a, such that |c|**b <= |a|
385 * returns error if a < 0 and b is even
387 int mp_n_root(mp_int *a, mp_digit b, mp_int *c);
389 /* special sqrt algo */
390 int mp_sqrt(mp_int *arg, mp_int *ret);
392 /* is number a square? */
393 int mp_is_square(mp_int *arg, int *ret);
395 /* computes the jacobi c = (a | n) (or Legendre if b is prime) */
396 int mp_jacobi(mp_int *a, mp_int *n, int *c);
398 /* used to setup the Barrett reduction for a given modulus b */
399 int mp_reduce_setup(mp_int *a, mp_int *b);
401 /* Barrett Reduction, computes a (mod b) with a precomputed value c
403 * Assumes that 0 < a <= b*b, note if 0 > a > -(b*b) then you can merely
404 * compute the reduction as -1 * mp_reduce(mp_abs(a)) [pseudo code].
406 int mp_reduce(mp_int *a, mp_int *b, mp_int *c);
408 /* setups the montgomery reduction */
409 int mp_montgomery_setup(mp_int *a, mp_digit *mp);
411 /* computes a = B**n mod b without division or multiplication useful for
412 * normalizing numbers in a Montgomery system.
414 int mp_montgomery_calc_normalization(mp_int *a, mp_int *b);
416 /* computes x/R == x (mod N) via Montgomery Reduction */
417 int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
419 /* returns 1 if a is a valid DR modulus */
420 int mp_dr_is_modulus(mp_int *a);
422 /* sets the value of "d" required for mp_dr_reduce */
423 void mp_dr_setup(mp_int *a, mp_digit *d);
425 /* reduces a modulo b using the Diminished Radix method */
426 int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp);
428 /* returns true if a can be reduced with mp_reduce_2k */
429 int mp_reduce_is_2k(mp_int *a);
431 /* determines k value for 2k reduction */
432 int mp_reduce_2k_setup(mp_int *a, mp_digit *d);
434 /* reduces a modulo b where b is of the form 2**p - k [0 <= a] */
435 int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d);
437 /* returns true if a can be reduced with mp_reduce_2k_l */
438 int mp_reduce_is_2k_l(mp_int *a);
440 /* determines k value for 2k reduction */
441 int mp_reduce_2k_setup_l(mp_int *a, mp_int *d);
443 /* reduces a modulo b where b is of the form 2**p - k [0 <= a] */
444 int mp_reduce_2k_l(mp_int *a, mp_int *n, mp_int *d);
446 /* d = a**b (mod c) */
447 int mp_exptmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
449 /* ---> Primes <--- */
451 /* number of primes */
453 #define PRIME_SIZE 31
455 #define PRIME_SIZE 256
458 /* table of first PRIME_SIZE primes */
459 extern const mp_digit ltm_prime_tab[];
461 /* result=1 if a is divisible by one of the first PRIME_SIZE primes */
462 int mp_prime_is_divisible(mp_int *a, int *result);
464 /* performs one Fermat test of "a" using base "b".
465 * Sets result to 0 if composite or 1 if probable prime
467 int mp_prime_fermat(mp_int *a, mp_int *b, int *result);
469 /* performs one Miller-Rabin test of "a" using base "b".
470 * Sets result to 0 if composite or 1 if probable prime
472 int mp_prime_miller_rabin(mp_int *a, mp_int *b, int *result);
474 /* This gives [for a given bit size] the number of trials required
475 * such that Miller-Rabin gives a prob of failure lower than 2^-96
477 int mp_prime_rabin_miller_trials(int size);
479 /* performs t rounds of Miller-Rabin on "a" using the first
480 * t prime bases. Also performs an initial sieve of trial
481 * division. Determines if "a" is prime with probability
482 * of error no more than (1/4)**t.
484 * Sets result to 1 if probably prime, 0 otherwise
486 int mp_prime_is_prime(mp_int *a, int t, int *result);
488 /* finds the next prime after the number "a" using "t" trials
491 * bbs_style = 1 means the prime must be congruent to 3 mod 4
493 int mp_prime_next_prime(mp_int *a, int t, int bbs_style);
495 /* makes a truly random prime of a given size (bytes),
496 * call with bbs = 1 if you want it to be congruent to 3 mod 4
498 * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can
499 * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself
502 * The prime generated will be larger than 2^(8*size).
504 #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)
506 /* makes a truly random prime of a given size (bits),
508 * Flags are as follows:
510 * LTM_PRIME_BBS - make prime congruent to 3 mod 4
511 * LTM_PRIME_SAFE - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS)
512 * LTM_PRIME_2MSB_OFF - make the 2nd highest bit zero
513 * LTM_PRIME_2MSB_ON - make the 2nd highest bit one
515 * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can
516 * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself
520 int mp_prime_random_ex(mp_int *a, int t, int size, int flags, ltm_prime_callback cb, void *dat);
522 /* ---> radix conversion <--- */
523 int mp_count_bits(mp_int *a);
525 int mp_unsigned_bin_size(mp_int *a);
526 int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c);
527 int mp_to_unsigned_bin(mp_int *a, unsigned char *b);
528 int mp_to_unsigned_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen);
530 int mp_signed_bin_size(mp_int *a);
531 int mp_read_signed_bin(mp_int *a, unsigned char *b, int c);
532 int mp_to_signed_bin(mp_int *a, unsigned char *b);
533 int mp_to_signed_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen);
535 int mp_read_radix(mp_int *a, const char *str, int radix);
536 int mp_toradix(mp_int *a, char *str, int radix);
537 int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen);
538 int mp_radix_size(mp_int *a, int radix, int *size);
540 int mp_fread(mp_int *a, int radix, FILE *stream);
541 int mp_fwrite(mp_int *a, int radix, FILE *stream);
543 #define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len))
544 #define mp_raw_size(mp) mp_signed_bin_size(mp)
545 #define mp_toraw(mp, str) mp_to_signed_bin((mp), (str))
546 #define mp_read_mag(mp, str, len) mp_read_unsigned_bin((mp), (str), (len))
547 #define mp_mag_size(mp) mp_unsigned_bin_size(mp)
548 #define mp_tomag(mp, str) mp_to_unsigned_bin((mp), (str))
550 #define mp_tobinary(M, S) mp_toradix((M), (S), 2)
551 #define mp_tooctal(M, S) mp_toradix((M), (S), 8)
552 #define mp_todecimal(M, S) mp_toradix((M), (S), 10)
553 #define mp_tohex(M, S) mp_toradix((M), (S), 16)
555 /* lowlevel functions, do not call! */
556 int s_mp_add(mp_int *a, mp_int *b, mp_int *c);
557 int s_mp_sub(mp_int *a, mp_int *b, mp_int *c);
558 #define s_mp_mul(a, b, c) s_mp_mul_digs(a, b, c, (a)->used + (b)->used + 1)
559 int fast_s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
560 int s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
561 int fast_s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
562 int s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
563 int fast_s_mp_sqr(mp_int *a, mp_int *b);
564 int s_mp_sqr(mp_int *a, mp_int *b);
565 int mp_karatsuba_mul(mp_int *a, mp_int *b, mp_int *c);
566 int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c);
567 int mp_karatsuba_sqr(mp_int *a, mp_int *b);
568 int mp_toom_sqr(mp_int *a, mp_int *b);
569 int fast_mp_invmod(mp_int *a, mp_int *b, mp_int *c);
570 int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c);
571 int fast_mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
572 int mp_exptmod_fast(mp_int *G, mp_int *X, mp_int *P, mp_int *Y, int mode);
573 int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int mode);
574 void bn_reverse(unsigned char *s, int len);
576 extern const char *mp_s_rmap;