libtommath/etc/pprime.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 "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, 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;
}