/* Produce table of composites from list of primes and trial value.
** trial must be odd. List of primes must not include 2.
** sieve should have dimension >= MAXPRIME/2, where MAXPRIME is largest
** prime in list of primes. After this function is finished,
** if sieve[i] is non-zero, then (trial + 2*i) is composite.
** Each prime used in the sieve costs one division of trial, and eliminates
** one or more values from the search space. (3 eliminates 1/3 of the values
** alone!) Each value left in the search space costs 1 or more modular
** exponentations. So, these divisions are a bargain!
*/
mp_err mpp_sieve(mp_int *trial, const mp_digit *primes, mp_size nPrimes,
unsigned char *sieve, mp_size nSieve)
{
mp_err res;
mp_digit rem;
mp_size ix;
unsigned long offset;
memset(sieve, 0, nSieve);
for(ix = 0; ix < nPrimes; ix++) {
mp_digit prime = primes[ix];
mp_size i;
if((res = mp_mod_d(trial, prime, &rem)) != MP_OKAY)
return res;
if (rem == 0) {
offset = 0;
} else {
offset = prime - (rem / 2);
}
for (i = offset; i < nSieve ; i += prime) {
sieve[i] = 1;
}
}
return MP_OKAY;
}
int mp_isprime(mp_int *a)
{
mp_int b;
mp_digit d;
int r, res;
/* do trial division */
for (r = 0; r < 256; r++) {
mp_mod_d(a, primes[r], &d);
if (d == 0) {
return MP_NO;
}
}
/* now do 8 miller rabins */
mp_init(&b);
for (r = 0; r < 128; r++) {
mp_set(&b, primes[r]);
mp_prime_miller_rabin(a, &b, &res);
if (res == MP_NO) {
return MP_NO;
}
}
return MP_YES;
}
int mp_isdivisible_d(mp_int *a, mp_digit d)
{
int err;
mp_digit res;
if ((err = mp_mod_d(a, d, &res)) != MP_OKAY) {
return err;
}
if (res == 0) {
return MP_YES;
}
return MP_NO;
}
/* modi */
static int modi(void *a, unsigned long b, unsigned long *c)
{
mp_digit tmp;
int err;
LTC_ARGCHK(a != NULL);
LTC_ARGCHK(c != NULL);
if ((err = mpi_to_ltc_error(mp_mod_d(a, b, &tmp))) != CRYPT_OK) {
return err;
}
*c = tmp;
return CRYPT_OK;
}
mp_err mpp_divis_d(mp_int *a, mp_digit d)
{
mp_err res;
mp_digit rem;
ARGCHK(a != NULL, MP_BADARG);
if(d == 0)
return MP_NO;
if((res = mp_mod_d(a, d, &rem)) != MP_OKAY)
return res;
if(rem == 0)
return MP_YES;
else
return MP_NO;
} /* end mpp_divis_d() */
mp_err s_mpp_divp(mp_int *a, const mp_digit *vec, int size, int *which)
{
mp_err res;
mp_digit rem;
int ix;
for(ix = 0; ix < size; ix++) {
if((res = mp_mod_d(a, vec[ix], &rem)) != MP_OKAY)
return res;
if(rem == 0) {
if(which)
*which = ix;
return MP_YES;
}
}
return MP_NO;
} /* end s_mpp_divp() */
/* determines if an integers is divisible by one
* of the first PRIME_SIZE primes or not
*
* sets result to 0 if not, 1 if yes
*/
int mp_prime_is_divisible (mp_int * a, int *result)
{
int err, ix;
mp_digit res;
/* default to not */
*result = MP_NO;
for (ix = 0; ix < PRIME_SIZE; ix++) {
/* what is a mod LBL_prime_tab[ix] */
if ((err = mp_mod_d (a, ltm_prime_tab[ix], &res)) != MP_OKAY) {
return err;
}
/* is the residue zero? */
if (res == 0) {
*result = MP_YES;
return MP_OKAY;
}
}
return MP_OKAY;
}
/* Store non-zero to ret if arg is square, and zero if not */
int mp_is_square(mp_int *arg,int *ret)
{
int res;
mp_digit c;
mp_int t;
unsigned long r;
/* Default to Non-square :) */
*ret = MP_NO;
if (arg->sign == MP_NEG) {
return MP_VAL;
}
/* digits used? (TSD) */
if (arg->used == 0) {
return MP_OKAY;
}
/* First check mod 128 (suppose that DIGIT_BIT is at least 7) */
if (rem_128[127 & DIGIT(arg,0)] == 1) {
return MP_OKAY;
}
/* Next check mod 105 (3*5*7) */
if ((res = mp_mod_d(arg,105,&c)) != MP_OKAY) {
return res;
}
if (rem_105[c] == 1) {
return MP_OKAY;
}
/* product of primes less than 2^31 */
if ((res = mp_init_set_int(&t,11L*13L*17L*19L*23L*29L*31L)) != MP_OKAY) {
return res;
}
if ((res = mp_mod(arg,&t,&t)) != MP_OKAY) {
goto ERR;
}
r = mp_get_int(&t);
/* Check for other prime modules, note it's not an ERROR but we must
* free "t" so the easiest way is to goto ERR. We know that res
* is already equal to MP_OKAY from the mp_mod call
*/
if ( (1L<<(r%11)) & 0x5C4L ) goto ERR;
if ( (1L<<(r%13)) & 0x9E4L ) goto ERR;
if ( (1L<<(r%17)) & 0x5CE8L ) goto ERR;
if ( (1L<<(r%19)) & 0x4F50CL ) goto ERR;
if ( (1L<<(r%23)) & 0x7ACCA0L ) goto ERR;
if ( (1L<<(r%29)) & 0xC2EDD0CL ) goto ERR;
if ( (1L<<(r%31)) & 0x6DE2B848L ) goto ERR;
/* Final check - is sqr(sqrt(arg)) == arg ? */
if ((res = mp_sqrt(arg,&t)) != MP_OKAY) {
goto ERR;
}
if ((res = mp_sqr(&t,&t)) != MP_OKAY) {
goto ERR;
}
*ret = (mp_cmp_mag(&t,arg) == MP_EQ) ? MP_YES : MP_NO;
ERR:mp_clear(&t);
return res;
}
/* finds the next prime after the number "a" using "t" trials
* of Miller-Rabin.
*
* bbs_style = 1 means the prime must be congruent to 3 mod 4
*/
int mp_prime_next_prime(mp_int *a, int t, int bbs_style)
{
int err, res, x, y;
mp_digit res_tab[PRIME_SIZE], step, kstep;
mp_int b;
/* ensure t is valid */
if (t <= 0 || t > PRIME_SIZE) {
return MP_VAL;
}
/* force positive */
a->sign = MP_ZPOS;
/* simple algo if a is less than the largest prime in the table */
if (mp_cmp_d(a, ltm_prime_tab[PRIME_SIZE-1]) == MP_LT) {
/* find which prime it is bigger than */
for (x = PRIME_SIZE - 2; x >= 0; x--) {
if (mp_cmp_d(a, ltm_prime_tab[x]) != MP_LT) {
if (bbs_style == 1) {
/* ok we found a prime smaller or
* equal [so the next is larger]
*
* however, the prime must be
* congruent to 3 mod 4
*/
if ((ltm_prime_tab[x + 1] & 3) != 3) {
/* scan upwards for a prime congruent to 3 mod 4 */
for (y = x + 1; y < PRIME_SIZE; y++) {
if ((ltm_prime_tab[y] & 3) == 3) {
mp_set(a, ltm_prime_tab[y]);
return MP_OKAY;
}
}
}
} else {
mp_set(a, ltm_prime_tab[x + 1]);
return MP_OKAY;
}
}
}
/* at this point a maybe 1 */
if (mp_cmp_d(a, 1) == MP_EQ) {
mp_set(a, 2);
return MP_OKAY;
}
/* fall through to the sieve */
}
/* generate a prime congruent to 3 mod 4 or 1/3 mod 4? */
if (bbs_style == 1) {
kstep = 4;
} else {
kstep = 2;
}
/* at this point we will use a combination of a sieve and Miller-Rabin */
if (bbs_style == 1) {
/* if a mod 4 != 3 subtract the correct value to make it so */
if ((a->dp[0] & 3) != 3) {
if ((err = mp_sub_d(a, (a->dp[0] & 3) + 1, a)) != MP_OKAY) { return err; };
}
} else {
if (mp_iseven(a) == 1) {
/* force odd */
if ((err = mp_sub_d(a, 1, a)) != MP_OKAY) {
return err;
}
}
}
/* generate the restable */
for (x = 1; x < PRIME_SIZE; x++) {
if ((err = mp_mod_d(a, ltm_prime_tab[x], res_tab + x)) != MP_OKAY) {
return err;
}
}
/* init temp used for Miller-Rabin Testing */
if ((err = mp_init(&b)) != MP_OKAY) {
return err;
}
for (;;) {
/* skip to the next non-trivially divisible candidate */
step = 0;
do {
/* y == 1 if any residue was zero [e.g. cannot be prime] */
y = 0;
/* increase step to next candidate */
step += kstep;
/* compute the new residue without using division */
for (x = 1; x < PRIME_SIZE; x++) {
/* add the step to each residue */
res_tab[x] += kstep;
/* subtract the modulus [instead of using division] */
if (res_tab[x] >= ltm_prime_tab[x]) {
res_tab[x] -= ltm_prime_tab[x];
}
/* set flag if zero */
if (res_tab[x] == 0) {
y = 1;
}
}
} while (y == 1 && step < ((((mp_digit)1)<<DIGIT_BIT) - kstep));
/* add the step */
if ((err = mp_add_d(a, step, a)) != MP_OKAY) {
goto LBL_ERR;
}
/* if didn't pass sieve and step == MAX then skip test */
if (y == 1 && step >= ((((mp_digit)1)<<DIGIT_BIT) - kstep)) {
continue;
}
/* is this prime? */
for (x = 0; x < t; x++) {
mp_set(&b, ltm_prime_tab[t]);
if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
goto LBL_ERR;
}
if (res == MP_NO) {
break;
}
}
if (res == MP_YES) {
break;
}
}
err = MP_OKAY;
LBL_ERR:
mp_clear(&b);
return err;
}
int main(int argc, char *argv[])
{
int ix;
mp_int a, b, c, m;
mp_digit r;
if(argc < 4) {
fprintf(stderr, "Usage: %s <a> <b> <m>\n", argv[0]);
return 1;
}
printf("Test 4: Modular arithmetic\n\n");
mp_init(&a);
mp_init(&b);
mp_init(&m);
mp_read_radix(&a, argv[1], 10);
mp_read_radix(&b, argv[2], 10);
mp_read_radix(&m, argv[3], 10);
printf("a = "); mp_print(&a, stdout); fputc('\n', stdout);
printf("b = "); mp_print(&b, stdout); fputc('\n', stdout);
printf("m = "); mp_print(&m, stdout); fputc('\n', stdout);
mp_init(&c);
printf("\nc = a (mod m)\n");
mp_mod(&a, &m, &c);
printf("c = "); mp_print(&c, stdout); fputc('\n', stdout);
printf("\nc = b (mod m)\n");
mp_mod(&b, &m, &c);
printf("c = "); mp_print(&c, stdout); fputc('\n', stdout);
printf("\nc = b (mod 1853)\n");
mp_mod_d(&b, 1853, &r);
printf("c = %04X\n", r);
printf("\nc = (a + b) mod m\n");
mp_addmod(&a, &b, &m, &c);
printf("c = "); mp_print(&c, stdout); fputc('\n', stdout);
printf("\nc = (a - b) mod m\n");
mp_submod(&a, &b, &m, &c);
printf("c = "); mp_print(&c, stdout); fputc('\n', stdout);
printf("\nc = (a * b) mod m\n");
mp_mulmod(&a, &b, &m, &c);
printf("c = "); mp_print(&c, stdout); fputc('\n', stdout);
printf("\nc = (a ** b) mod m\n");
mp_exptmod(&a, &b, &m, &c);
printf("c = "); mp_print(&c, stdout); fputc('\n', stdout);
printf("\nIn-place modular squaring test:\n");
for(ix = 0; ix < 5; ix++) {
printf("a = (a * a) mod m a = ");
mp_sqrmod(&a, &m, &a);
mp_print(&a, stdout);
fputc('\n', stdout);
}
mp_clear(&c);
mp_clear(&m);
mp_clear(&b);
mp_clear(&a);
return 0;
}
/*
Sets ret to nonzero value if arg is square, 0 if not
Sets t to the square root of arg if one is available, 0 if not
*/
static int mp_issquare(mp_int *arg, int *ret, mp_int *t)
{
int res;
mp_digit c;
mp_int tmp;
unsigned long r;
/* Default to Non-square :) */
*ret = MP_NO;
if (arg->sign == MP_NEG) {
return MP_VAL;
}
/* digits used? (TSD) */
if (arg->used == 0) {
return MP_OKAY;
}
/* First check mod 128 (suppose that DIGIT_BIT is at least 7) */
if (rem_128[127 & DIGIT(arg, 0)] == 1) {
mp_set_int(t, (mp_digit)(0));
return MP_OKAY;
}
/* Next check mod 105 (3*5*7) */
if ((res = mp_mod_d(arg, 105, &c)) != MP_OKAY) {
mp_set_int(t, (mp_digit)(0));
return res;
}
if (rem_105[c] == 1) {
mp_set_int(t, (mp_digit)(0));
return MP_OKAY;
}
if ((res =
mp_init_set_int(t,
11L * 13L * 17L * 19L * 23L * 29L * 31L)) != MP_OKAY) {
mp_set_int(t, (mp_digit)(0));
return res;
}
if ((res = mp_mod(arg, t, t)) != MP_OKAY) {
goto ERR;
}
r = mp_get_int(t);
/* Check for other prime modules. We know that res
* is already equal to MP_OKAY from the mp_mod call
*/
if ((1L << (r % 11)) & 0x5C4L)
goto ERR;
if ((1L << (r % 13)) & 0x9E4L)
goto ERR;
if ((1L << (r % 17)) & 0x5CE8L)
goto ERR;
if ((1L << (r % 19)) & 0x4F50CL)
goto ERR;
if ((1L << (r % 23)) & 0x7ACCA0L)
goto ERR;
if ((1L << (r % 29)) & 0xC2EDD0CL)
goto ERR;
if ((1L << (r % 31)) & 0x6DE2B848L)
goto ERR;
/* Final check - is sqr(sqrt(arg)) == arg ? */
if ((res = mp_sqrt(arg, t)) != MP_OKAY) {
goto ERR;
}
mp_init(&tmp);
if ((res = mp_sqr(t, &tmp)) != MP_OKAY) {
goto ERR;
}
*ret = (mp_cmp_mag(&tmp, arg) == MP_EQ) ? MP_YES : MP_NO;
mp_clear(&tmp);
return res;
ERR:
mp_set_int(t, (mp_digit)(0));
mp_clear(&tmp);
return res;
}
static int next_prime(mp_int *N, mp_digit step)
{
long x, s, j, total_dist;
int res;
mp_int n1, a, y, r;
mp_digit dist, residues[UPPER_LIMIT];
_ARGCHK(N != NULL);
/* first find the residues */
for (x = 0; x < (long)UPPER_LIMIT; x++) {
if (mp_mod_d(N, __prime_tab[x], &residues[x]) != MP_OKAY) {
return CRYPT_MEM;
}
}
/* init variables */
if (mp_init_multi(&r, &n1, &a, &y, NULL) != MP_OKAY) {
return CRYPT_MEM;
}
total_dist = 0;
loop:
/* while one of the residues is zero keep looping */
dist = step;
for (x = 0; (dist < (MP_DIGIT_MAX-step-1)) && (x < (long)UPPER_LIMIT); x++) {
j = (long)residues[x] + (long)dist + total_dist;
if (j % (long)__prime_tab[x] == 0) {
dist += step; x = -1;
}
}
/* recalc the total distance from where we started */
total_dist += dist;
/* add to N */
if (mp_add_d(N, dist, N) != MP_OKAY) { goto error; }
/* n1 = N - 1 */
if (mp_sub_d(N, 1, &n1) != MP_OKAY) { goto error; }
/* r = N - 1 */
if (mp_copy(&n1, &r) != MP_OKAY) { goto error; }
/* find s such that N-1 = (2^s)r */
s = 0;
while (mp_iseven(&r)) {
++s;
if (mp_div_2(&r, &r) != MP_OKAY) {
goto error;
}
}
for (x = 0; x < 8; x++) {
/* choose a */
mp_set(&a, __prime_tab[x]);
/* compute y = a^r mod n */
if (mp_exptmod(&a, &r, N, &y) != MP_OKAY) { goto error; }
/* (y != 1) AND (y != N-1) */
if ((mp_cmp_d(&y, 1) != 0) && (mp_cmp(&y, &n1) != 0)) {
/* while j <= s-1 and y != n-1 */
for (j = 1; (j <= (s-1)) && (mp_cmp(&y, &n1) != 0); j++) {
/* y = y^2 mod N */
if (mp_sqrmod(&y, N, &y) != MP_OKAY) { goto error; }
/* if y == 1 return false */
if (mp_cmp_d(&y, 1) == 0) { goto loop; }
}
/* if y != n-1 return false */
if (mp_cmp(&y, &n1) != 0) { goto loop; }
}
}
res = CRYPT_OK;
goto done;
error:
res = CRYPT_MEM;
done:
mp_clear_multi(&a, &y, &n1, &r, NULL);
#ifdef CLEAN_STACK
zeromem(residues, sizeof(residues));
#endif
return res;
}
