tommath/bn_mp_prime_is_prime.c

#include "tommath_private.h"
#ifdef BN_MP_PRIME_IS_PRIME_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * The library is free for all purposes without any express
 * guarantee it works.
 */

/* portable integer log of two with small footprint */
static unsigned int s_floor_ilog2(int value)
{
   unsigned int r = 0;
   while ((value >>= 1) != 0) {
      r++;
   }
   return r;
}


int mp_prime_is_prime(const mp_int *a, int t, int *result)
{
   mp_int  b;
   int     ix, err, res, p_max = 0, size_a, len;
   unsigned int fips_rand, mask;

   /* default to no */
   *result = MP_NO;

   /* valid value of t? */
   if (t > PRIME_SIZE) {
      return MP_VAL;
   }

   /* Some shortcuts */
   /* N > 3 */
   if (a->used == 1) {
      if ((a->dp[0] == 0u) || (a->dp[0] == 1u)) {
         *result = 0;
         return MP_OKAY;
      }
      if (a->dp[0] == 2u) {
         *result = 1;
         return MP_OKAY;
      }
   }

   /* N must be odd */
   if (mp_iseven(a) == MP_YES) {
      return MP_OKAY;
   }
   /* N is not a perfect square: floor(sqrt(N))^2 != N */
   if ((err = mp_is_square(a, &res)) != MP_OKAY) {
      return err;
   }
   if (res != 0) {
      return MP_OKAY;
   }

   /* is the input equal to one of the primes in the table? */
   for (ix = 0; ix < PRIME_SIZE; ix++) {
      if (mp_cmp_d(a, ltm_prime_tab[ix]) == MP_EQ) {
         *result = MP_YES;
         return MP_OKAY;
      }
   }
#ifdef MP_8BIT
   /* The search in the loop above was exhaustive in this case */
   if (a->used == 1 && PRIME_SIZE >= 31) {
      return MP_OKAY;
   }
#endif

   /* first perform trial division */
   if ((err = mp_prime_is_divisible(a, &res)) != MP_OKAY) {
      return err;
   }

   /* return if it was trivially divisible */
   if (res == MP_YES) {
      return MP_OKAY;
   }

   /*
       Run the Miller-Rabin test with base 2 for the BPSW test.
    */
   if ((err = mp_init_set(&b, 2uL)) != MP_OKAY) {
      return err;
   }

   if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
      goto LBL_B;
   }
   if (res == MP_NO) {
      goto LBL_B;
   }
   /*
      Rumours have it that Mathematica does a second M-R test with base 3.
      Other rumours have it that their strong L-S test is slightly different.
      It does not hurt, though, beside a bit of extra runtime.
   */
   b.dp[0]++;
   if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
      goto LBL_B;
   }
   if (res == MP_NO) {
      goto LBL_B;
   }

   /*
    * Both, the Frobenius-Underwood test and the the Lucas-Selfridge test are quite
    * slow so if speed is an issue, define LTM_USE_FIPS_ONLY to use M-R tests with
    * bases 2, 3 and t random bases.
    */
#ifndef LTM_USE_FIPS_ONLY
   if (t >= 0) {
      /*
       * Use a Frobenius-Underwood test instead of the Lucas-Selfridge test for
       * MP_8BIT (It is unknown if the Lucas-Selfridge test works with 16-bit
       * integers but the necesssary analysis is on the todo-list).
       */
#if defined (MP_8BIT) || defined (LTM_USE_FROBENIUS_TEST)
      err = mp_prime_frobenius_underwood(a, &res);
      if (err != MP_OKAY && err != MP_ITER) {
         goto LBL_B;
      }
      if (res == MP_NO) {
         goto LBL_B;
      }
#else
      if ((err = mp_prime_strong_lucas_selfridge(a, &res)) != MP_OKAY) {
         goto LBL_B;
      }
      if (res == MP_NO) {
         goto LBL_B;
      }
#endif
   }
#endif

   /* run at least one Miller-Rabin test with a random base */
   if (t == 0) {
      t = 1;
   }

   /*
      abs(t) extra rounds of M-R to extend the range of primes it can find if t < 0.
      Only recommended if the input range is known to be < 3317044064679887385961981

      It uses the bases for a deterministic M-R test if input < 3317044064679887385961981
      The caller has to check the size.

      Not for cryptographic use because with known bases strong M-R pseudoprimes can
      be constructed. Use at least one M-R test with a random base (t >= 1).

      The 1119 bit large number

      80383745745363949125707961434194210813883768828755814583748891752229742737653\
      33652186502336163960045457915042023603208766569966760987284043965408232928738\
      79185086916685732826776177102938969773947016708230428687109997439976544144845\
      34115587245063340927902227529622941498423068816854043264575340183297861112989\
      60644845216191652872597534901

      has been constructed by F. Arnault (F. Arnault, "Rabin-Miller primality test:
      composite numbers which pass it.",  Mathematics of Computation, 1995, 64. Jg.,
      Nr. 209, S. 355-361), is a semiprime with the two factors

      40095821663949960541830645208454685300518816604113250877450620473800321707011\
      96242716223191597219733582163165085358166969145233813917169287527980445796800\
      452592031836601

      20047910831974980270915322604227342650259408302056625438725310236900160853505\
      98121358111595798609866791081582542679083484572616906958584643763990222898400\
      226296015918301

      and it is a strong pseudoprime to all forty-six prime M-R bases up to 200

      It does not fail the strong Bailley-PSP test as implemented here, it is just
      given as an example, if not the reason to use the BPSW-test instead of M-R-tests
      with a sequence of primes 2...n.

   */
   if (t < 0) {
      t = -t;
      /*
          Sorenson, Jonathan; Webster, Jonathan (2015).
           "Strong Pseudoprimes to Twelve Prime Bases".
       */
      /* 0x437ae92817f9fc85b7e5 = 318665857834031151167461 */
      if ((err =   mp_read_radix(&b, "437ae92817f9fc85b7e5", 16)) != MP_OKAY) {
         goto LBL_B;
      }

      if (mp_cmp(a, &b) == MP_LT) {
         p_max = 12;
      } else {
         /* 0x2be6951adc5b22410a5fd = 3317044064679887385961981 */
         if ((err = mp_read_radix(&b, "2be6951adc5b22410a5fd", 16)) != MP_OKAY) {
            goto LBL_B;
         }

         if (mp_cmp(a, &b) == MP_LT) {
            p_max = 13;
         } else {
            err = MP_VAL;
            goto LBL_B;
         }
      }

      /* for compatibility with the current API (well, compatible within a sign's width) */
      if (p_max < t) {
         p_max = t;
      }

      if (p_max > PRIME_SIZE) {
         err = MP_VAL;
         goto LBL_B;
      }
      /* we did bases 2 and 3  already, skip them */
      for (ix = 2; ix < p_max; ix++) {
         mp_set(&b, ltm_prime_tab[ix]);
         if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
            goto LBL_B;
         }
         if (res == MP_NO) {
            goto LBL_B;
         }
      }
   }
   /*
       Do "t" M-R tests with random bases between 3 and "a".
       See Fips 186.4 p. 126ff
   */
   else if (t > 0) {
      /*
       * The mp_digit's have a defined bit-size but the size of the
       * array a.dp is a simple 'int' and this library can not assume full
       * compliance to the current C-standard (ISO/IEC 9899:2011) because
       * it gets used for small embeded processors, too. Some of those MCUs
       * have compilers that one cannot call standard compliant by any means.
       * Hence the ugly type-fiddling in the following code.
       */
      size_a = mp_count_bits(a);
      mask = (1u << s_floor_ilog2(size_a)) - 1u;
      /*
         Assuming the General Rieman hypothesis (never thought to write that in a
         comment) the upper bound can be lowered to  2*(log a)^2.
         E. Bach, "Explicit bounds for primality testing and related problems,"
         Math. Comp. 55 (1990), 355-380.

            size_a = (size_a/10) * 7;
            len = 2 * (size_a * size_a);

         E.g.: a number of size 2^2048 would be reduced to the upper limit

            floor(2048/10)*7 = 1428
            2 * 1428^2       = 4078368

         (would have been ~4030331.9962 with floats and natural log instead)
         That number is smaller than 2^28, the default bit-size of mp_digit.
      */

      /*
        How many tests, you might ask? Dana Jacobsen of Math::Prime::Util fame
        does exactly 1. In words: one. Look at the end of _GMP_is_prime() in
        Math-Prime-Util-GMP-0.50/primality.c if you do not believe it.

        The function mp_rand() goes to some length to use a cryptographically
        good PRNG. That also means that the chance to always get the same base
        in the loop is non-zero, although very low.
        If the BPSW test and/or the addtional Frobenious test have been
        performed instead of just the Miller-Rabin test with the bases 2 and 3,
        a single extra test should suffice, so such a very unlikely event
        will not do much harm.

        To preemptivly answer the dangling question: no, a witness does not
        need to be prime.
      */
      for (ix = 0; ix < t; ix++) {
         /* mp_rand() guarantees the first digit to be non-zero */
         if ((err = mp_rand(&b, 1)) != MP_OKAY) {
            goto LBL_B;
         }
         /*
          * Reduce digit before casting because mp_digit might be bigger than
          * an unsigned int and "mask" on the other side is most probably not.
          */
         fips_rand = (unsigned int)(b.dp[0] & (mp_digit) mask);
#ifdef MP_8BIT
         /*
          * One 8-bit digit is too small, so concatenate two if the size of
          * unsigned int allows for it.
          */
         if ((sizeof(unsigned int) * CHAR_BIT)/2 >= (sizeof(mp_digit) * CHAR_BIT)) {
            if ((err = mp_rand(&b, 1)) != MP_OKAY) {
               goto LBL_B;
            }
            fips_rand <<= sizeof(mp_digit) * CHAR_BIT;
            fips_rand |= (unsigned int) b.dp[0];
            fips_rand &= mask;
         }
#endif
         /* Ceil, because small numbers have a right to live, too, */
         len = (((int)fips_rand + DIGIT_BIT) / DIGIT_BIT);
         /*  Unlikely. */
         if (len < 0) {
            ix--;
            continue;
         }
         /*
          * As mentioned above, one 8-bit digit is too small and
          * although it can only happen in the unlikely case that
          * an "unsigned int" is smaller than 16 bit a simple test
          * is cheap and the correction even cheaper.
          */
#ifdef MP_8BIT
         /* All "a" < 2^8 have been caught before */
         if (len == 1) {
            len++;
         }
#endif
         if ((err = mp_rand(&b, len)) != MP_OKAY) {
            goto LBL_B;
         }
         /*
          * That number might got too big and the witness has to be
          * smaller than or equal to "a"
          */
         len = mp_count_bits(&b);
         if (len > size_a) {
            len = len - size_a;
            if ((err = mp_div_2d(&b, len, &b, NULL)) != MP_OKAY) {
               goto LBL_B;
            }
         }

         /* Although the chance for b <= 3 is miniscule, try again. */
         if (mp_cmp_d(&b, 3uL) != MP_GT) {
            ix--;
            continue;
         }
         if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
            goto LBL_B;
         }
         if (res == MP_NO) {
            goto LBL_B;
         }
      }
   }

   /* passed the test */
   *result = MP_YES;
LBL_B:
   mp_clear(&b);
   return err;
}

#endif

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