364 lines
6.6 KiB
C
364 lines
6.6 KiB
C
/* 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 "tommath.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, int li, mp_int * p, mp_int * q)
|
|
{
|
|
mp_int a, b, c, n, x, y, z, v;
|
|
int res, ii;
|
|
static const mp_digit bases[] = { 2, 3, 5, 7, 11, 13, 17, 19 };
|
|
|
|
/* 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=bases[ii] */
|
|
for (ii = 0; ii < li; ii++) {
|
|
mp_set (&x, bases[ii]);
|
|
|
|
/* 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)
|
|
continue;
|
|
|
|
/* 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)
|
|
continue;
|
|
|
|
/* 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)
|
|
continue;
|
|
|
|
/* 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)
|
|
continue;
|
|
|
|
/* 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)
|
|
continue;
|
|
|
|
/* 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)
|
|
continue;
|
|
break;
|
|
}
|
|
|
|
/* no bases worked? */
|
|
if (ii == li)
|
|
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);
|
|
}
|
|
|
|
/* get q to be the order of the large prime subgroup */
|
|
mp_sub_d (&n, 1, q);
|
|
mp_div_2 (q, q);
|
|
mp_div (q, &b, q, NULL);
|
|
|
|
mp_exch (&n, p);
|
|
|
|
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, li;
|
|
clock_t t1;
|
|
|
|
srand (time (NULL));
|
|
|
|
printf ("Enter # of bits: \n");
|
|
fgets (buf, sizeof (buf), stdin);
|
|
sscanf (buf, "%d", &k);
|
|
|
|
printf ("Enter number of bases to try (1 to 8):\n");
|
|
fgets (buf, sizeof (buf), stdin);
|
|
sscanf (buf, "%d", &li);
|
|
|
|
|
|
mp_init (&p);
|
|
mp_init (&q);
|
|
|
|
t1 = clock ();
|
|
pprime (k, li, &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;
|
|
}
|