/* Generates provable primes
*
* See http://iahu.ca:8080/papers/pp.pdf for more info.
*
* Tom St Denis, tomstdenis@iahu.ca, http://tom.iahu.ca
*/
#include <time.h>
#include "bn.h"
/* fast square root */
static mp_digit i_sqrt(mp_word x)
{
mp_word x1, x2;
x2 = x;
do {
x1 = x2;
x2 = x1 - ((x1 * x1) - x)/(2*x1);
} while (x1 != x2);
if (x1*x1 > x) {
--x1;
}
return x1;
}
/* generates a prime digit */
static mp_digit prime_digit()
{
mp_digit r, x, y, next;
/* make a DIGIT_BIT-bit random number */
for (r = x = 0; x < DIGIT_BIT; x++) {
r = (r << 1) | (rand() & 1);
}
/* now force it odd */
r |= 1;
/* force it to be >30 */
if (r < 30) {
r += 30;
}
/* get square root, since if 'r' is composite its factors must be < than this */
y = i_sqrt(r);
next = (y+1)*(y+1);
do {
r += 2; /* next candidate */
/* update sqrt ? */
if (next <= r) {
++y;
next = (y+1)*(y+1);
}
/* loop if divisible by 3,5,7,11,13,17,19,23,29 */
if ((r % 3) == 0) { x = 0; continue; }
if ((r % 5) == 0) { x = 0; continue; }
if ((r % 7) == 0) { x = 0; continue; }
if ((r % 11) == 0) { x = 0; continue; }
if ((r % 13) == 0) { x = 0; continue; }
if ((r % 17) == 0) { x = 0; continue; }
if ((r % 19) == 0) { x = 0; continue; }
if ((r % 23) == 0) { x = 0; continue; }
if ((r % 29) == 0) { x = 0; continue; }
/* now check if r is divisible by x + k={1,7,11,13,17,19,23,29} */
for (x = 30; x <= y; x += 30) {
if ((r % (x+1)) == 0) { x = 0; break; }
if ((r % (x+7)) == 0) { x = 0; break; }
if ((r % (x+11)) == 0) { x = 0; break; }
if ((r % (x+13)) == 0) { x = 0; break; }
if ((r % (x+17)) == 0) { x = 0; break; }
if ((r % (x+19)) == 0) { x = 0; break; }
if ((r % (x+23)) == 0) { x = 0; break; }
if ((r % (x+29)) == 0) { x = 0; break; }
}
} while (x == 0);
return r;
}
/* makes a prime of at least k bits */
int pprime(int k, mp_int *p, mp_int *q)
{
mp_int a, b, c, n, x, y, z, v;
int res;
/* single digit ? */
if (k <= (int)DIGIT_BIT) {
mp_set(p, prime_digit());
return MP_OKAY;
}
if ((res = mp_init(&c)) != MP_OKAY) {
return res;
}
if ((res = mp_init(&v)) != MP_OKAY) {
goto __C;
}
/* product of first 50 primes */
if ((res = mp_read_radix(&v, "19078266889580195013601891820992757757219839668357012055907516904309700014933909014729740190", 10)) != MP_OKAY) {
goto __V;
}
if ((res = mp_init(&a)) != MP_OKAY) {
goto __V;
}
/* set the prime */
mp_set(&a, prime_digit());
if ((res = mp_init(&b)) != MP_OKAY) {
goto __A;
}
if ((res = mp_init(&n)) != MP_OKAY) {
goto __B;
}
if ((res = mp_init(&x)) != MP_OKAY) {
goto __N;
}
if ((res = mp_init(&y)) != MP_OKAY) {
goto __X;
}
if ((res = mp_init(&z)) != MP_OKAY) {
goto __Y;
}
/* now loop making the single digit */
while (mp_count_bits(&a) < k) {
printf("prime has %4d bits left\r", k - mp_count_bits(&a)); fflush(stdout);
top:
mp_set(&b, prime_digit());
/* now compute z = a * b * 2 */
if ((res = mp_mul(&a, &b, &z)) != MP_OKAY) { /* z = a * b */
goto __Z;
}
if ((res = mp_copy(&z, &c)) != MP_OKAY) { /* c = a * b */
goto __Z;
}
if ((res = mp_mul_2(&z, &z)) != MP_OKAY) { /* z = 2 * a * b */
goto __Z;
}
/* n = z + 1 */
if ((res = mp_add_d(&z, 1, &n)) != MP_OKAY) { /* n = z + 1 */
goto __Z;
}
/* check (n, v) == 1 */
if ((res = mp_gcd(&n, &v, &y)) != MP_OKAY) { /* y = (n, v) */
goto __Z;
}
if (mp_cmp_d(&y, 1) != MP_EQ) goto top;
/* now try base x=2 */
mp_set(&x, 2);
/* compute x^a mod n */
if ((res = mp_exptmod(&x, &a, &n, &y)) != MP_OKAY) { /* y = x^a mod n */
goto __Z;
}
/* if y == 1 loop */
if (mp_cmp_d(&y, 1) == MP_EQ) goto top;
/* now x^2a mod n */
if ((res = mp_sqrmod(&y, &n, &y)) != MP_OKAY) { /* y = x^2a mod n */
goto __Z;
}
if (mp_cmp_d(&y, 1) == MP_EQ) goto top;
/* compute x^b mod n */
if ((res = mp_exptmod(&x, &b, &n, &y)) != MP_OKAY) { /* y = x^b mod n */
goto __Z;
}
/* if y == 1 loop */
if (mp_cmp_d(&y, 1) == MP_EQ) goto top;
/* now x^2b mod n */
if ((res = mp_sqrmod(&y, &n, &y)) != MP_OKAY) { /* y = x^2b mod n */
goto __Z;
}
if (mp_cmp_d(&y, 1) == MP_EQ) goto top;
/* compute x^c mod n == x^ab mod n */
if ((res = mp_exptmod(&x, &c, &n, &y)) != MP_OKAY) { /* y = x^ab mod n */
goto __Z;
}
/* if y == 1 loop */
if (mp_cmp_d(&y, 1) == MP_EQ) goto top;
/* now compute (x^c mod n)^2 */
if ((res = mp_sqrmod(&y, &n, &y)) != MP_OKAY) { /* y = x^2ab mod n */
goto __Z;
}
/* y should be 1 */
if (mp_cmp_d(&y, 1) != MP_EQ) goto top;
/*
{
char buf[4096];
mp_toradix(&n, buf, 10);
printf("Certificate of primality for:\n%s\n\n", buf);
mp_toradix(&a, buf, 10);
printf("A == \n%s\n\n", buf);
mp_toradix(&b, buf, 10);
printf("B == \n%s\n", buf);
printf("----------------------------------------------------------------\n");
}
*/
/* a = n */
mp_copy(&n, &a);
}
mp_exch(&n, p);
mp_exch(&b, q);
res = MP_OKAY;
__Z: mp_clear(&z);
__Y: mp_clear(&y);
__X: mp_clear(&x);
__N: mp_clear(&n);
__B: mp_clear(&b);
__A: mp_clear(&a);
__V: mp_clear(&v);
__C: mp_clear(&c);
return res;
}
int main(void)
{
mp_int p, q;
char buf[4096];
int k;
clock_t t1;
srand(time(NULL));
printf("Enter # of bits: \n");
scanf("%d", &k);
mp_init(&p);
mp_init(&q);
t1 = clock();
pprime(k, &p, &q);
t1 = clock() - t1;
printf("\n\nTook %ld ticks, %d bits\n", t1, mp_count_bits(&p));
mp_toradix(&p, buf, 10);
printf("P == %s\n", buf);
mp_toradix(&q, buf, 10);
printf("Q == %s\n", buf);
return 0;
}
