kc3-lang/libtommath/bn_mp_prime_is_prime.c

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• Hash : b19f529c
  Author :
  Date : 2018-05-27T22:05:52
  

  Corrected 128 bit entry in bn_mp_prime_miller_rabin_rials.c and extended it slightly

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• 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 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] == 0 || a->dp[0] == 1) {
           *result = 0;
           return MP_OKAY;
        }
        if (a->dp[0] == 2) {
           *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,2)) != 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".
         */
                                     /* 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 {                       /* 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 << 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;
              mp_div_2d(&b, len, &b, NULL);
           }
  
           /* Although the chance for b <= 3 is miniscule, try again. */
           if (mp_cmp_d(&b,3) != 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
  
  /* ref:         $Format:%D$ */
  /* git commit:  $Format:%H$ */
  /* commit time: $Format:%ai$ */
  

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