freebl/mpi/mpprime.c

/*
 *  mpprime.c
 *
 *  Utilities for finding and working with prime and pseudo-prime
 *  integers
 *
 * This Source Code Form is subject to the terms of the Mozilla Public
 * License, v. 2.0. If a copy of the MPL was not distributed with this
 * file, You can obtain one at http://mozilla.org/MPL/2.0/. */

#include "mpi-priv.h"
#include "mpprime.h"
#include "mplogic.h"
#include <stdlib.h>
#include <string.h>

#define SMALL_TABLE 0 /* determines size of hard-wired prime table */

#define RANDOM() rand()

#include "primes.c" /* pull in the prime digit table */

/*
   Test if any of a given vector of digits divides a.  If not, MP_NO
   is returned; otherwise, MP_YES is returned and 'which' is set to
   the index of the integer in the vector which divided a.
 */
mp_err s_mpp_divp(mp_int *a, const mp_digit *vec, int size, int *which);

/* {{{ mpp_divis(a, b) */

/*
  mpp_divis(a, b)

  Returns MP_YES if a is divisible by b, or MP_NO if it is not.
 */

mp_err
mpp_divis(mp_int *a, mp_int *b)
{
    mp_err res;
    mp_int rem;

    if ((res = mp_init(&rem)) != MP_OKAY)
        return res;

    if ((res = mp_mod(a, b, &rem)) != MP_OKAY)
        goto CLEANUP;

    if (mp_cmp_z(&rem) == 0)
        res = MP_YES;
    else
        res = MP_NO;

CLEANUP:
    mp_clear(&rem);
    return res;

} /* end mpp_divis() */

/* }}} */

/* {{{ mpp_divis_d(a, d) */

/*
  mpp_divis_d(a, d)

  Return MP_YES if a is divisible by d, or MP_NO if it is not.
 */

mp_err
mpp_divis_d(mp_int *a, mp_digit d)
{
    mp_err res;
    mp_digit rem;

    ARGCHK(a != NULL, MP_BADARG);

    if (d == 0)
        return MP_NO;

    if ((res = mp_mod_d(a, d, &rem)) != MP_OKAY)
        return res;

    if (rem == 0)
        return MP_YES;
    else
        return MP_NO;

} /* end mpp_divis_d() */

/* }}} */

/* {{{ mpp_random(a) */

/*
  mpp_random(a)

  Assigns a random value to a.  This value is generated using the
  standard C library's rand() function, so it should not be used for
  cryptographic purposes, but it should be fine for primality testing,
  since all we really care about there is good statistical properties.

  As many digits as a currently has are filled with random digits.
 */

mp_err
mpp_random(mp_int *a)

{
    mp_digit next = 0;
    unsigned int ix, jx;

    ARGCHK(a != NULL, MP_BADARG);

    for (ix = 0; ix < USED(a); ix++) {
        for (jx = 0; jx < sizeof(mp_digit); jx++) {
            next = (next << CHAR_BIT) | (RANDOM() & UCHAR_MAX);
        }
        DIGIT(a, ix) = next;
    }

    return MP_OKAY;

} /* end mpp_random() */

/* }}} */

/* {{{ mpp_random_size(a, prec) */

mp_err
mpp_random_size(mp_int *a, mp_size prec)
{
    mp_err res;

    ARGCHK(a != NULL && prec > 0, MP_BADARG);

    if ((res = s_mp_pad(a, prec)) != MP_OKAY)
        return res;

    return mpp_random(a);

} /* end mpp_random_size() */

/* }}} */

/* {{{ mpp_divis_vector(a, vec, size, which) */

/*
  mpp_divis_vector(a, vec, size, which)

  Determines if a is divisible by any of the 'size' digits in vec.
  Returns MP_YES and sets 'which' to the index of the offending digit,
  if it is; returns MP_NO if it is not.
 */

mp_err
mpp_divis_vector(mp_int *a, const mp_digit *vec, int size, int *which)
{
    ARGCHK(a != NULL && vec != NULL && size > 0, MP_BADARG);

    return s_mpp_divp(a, vec, size, which);

} /* end mpp_divis_vector() */

/* }}} */

/* {{{ mpp_divis_primes(a, np) */

/*
  mpp_divis_primes(a, np)

  Test whether a is divisible by any of the first 'np' primes.  If it
  is, returns MP_YES and sets *np to the value of the digit that did
  it.  If not, returns MP_NO.
 */
mp_err
mpp_divis_primes(mp_int *a, mp_digit *np)
{
    int size, which;
    mp_err res;

    ARGCHK(a != NULL && np != NULL, MP_BADARG);

    size = (int)*np;
    if (size > prime_tab_size)
        size = prime_tab_size;

    res = mpp_divis_vector(a, prime_tab, size, &which);
    if (res == MP_YES)
        *np = prime_tab[which];

    return res;

} /* end mpp_divis_primes() */

/* }}} */

/* {{{ mpp_fermat(a, w) */

/*
  Using w as a witness, try pseudo-primality testing based on Fermat's
  little theorem.  If a is prime, and (w, a) = 1, then w^a == w (mod
  a).  So, we compute z = w^a (mod a) and compare z to w; if they are
  equal, the test passes and we return MP_YES.  Otherwise, we return
  MP_NO.
 */
mp_err
mpp_fermat(mp_int *a, mp_digit w)
{
    mp_int base, test;
    mp_err res;

    if ((res = mp_init(&base)) != MP_OKAY)
        return res;

    mp_set(&base, w);

    if ((res = mp_init(&test)) != MP_OKAY)
        goto TEST;

    /* Compute test = base^a (mod a) */
    if ((res = mp_exptmod(&base, a, a, &test)) != MP_OKAY)
        goto CLEANUP;

    if (mp_cmp(&base, &test) == 0)
        res = MP_YES;
    else
        res = MP_NO;

CLEANUP:
    mp_clear(&test);
TEST:
    mp_clear(&base);

    return res;

} /* end mpp_fermat() */

/* }}} */

/*
  Perform the fermat test on each of the primes in a list until
  a) one of them shows a is not prime, or
  b) the list is exhausted.
  Returns:  MP_YES if it passes tests.
        MP_NO  if fermat test reveals it is composite
        Some MP error code if some other error occurs.
 */
mp_err
mpp_fermat_list(mp_int *a, const mp_digit *primes, mp_size nPrimes)
{
    mp_err rv = MP_YES;

    while (nPrimes-- > 0 && rv == MP_YES) {
        rv = mpp_fermat(a, *primes++);
    }
    return rv;
}

/* {{{ mpp_pprime(a, nt) */

/*
  mpp_pprime(a, nt)

  Performs nt iteration of the Miller-Rabin probabilistic primality
  test on a.  Returns MP_YES if the tests pass, MP_NO if one fails.
  If MP_NO is returned, the number is definitely composite.  If MP_YES
  is returned, it is probably prime (but that is not guaranteed).
 */

mp_err
mpp_pprime(mp_int *a, int nt)
{
    mp_err res;
    mp_int x, amo, m, z; /* "amo" = "a minus one" */
    int iter;
    unsigned int jx;
    mp_size b;

    ARGCHK(a != NULL, MP_BADARG);

    MP_DIGITS(&x) = 0;
    MP_DIGITS(&amo) = 0;
    MP_DIGITS(&m) = 0;
    MP_DIGITS(&z) = 0;

    /* Initialize temporaries... */
    MP_CHECKOK(mp_init(&amo));
    /* Compute amo = a - 1 for what follows...    */
    MP_CHECKOK(mp_sub_d(a, 1, &amo));

    b = mp_trailing_zeros(&amo);
    if (!b) { /* a was even ? */
        res = MP_NO;
        goto CLEANUP;
    }

    MP_CHECKOK(mp_init_size(&x, MP_USED(a)));
    MP_CHECKOK(mp_init(&z));
    MP_CHECKOK(mp_init(&m));
    MP_CHECKOK(mp_div_2d(&amo, b, &m, 0));

    /* Do the test nt times... */
    for (iter = 0; iter < nt; iter++) {

        /* Choose a random value for 1 < x < a      */
        MP_CHECKOK(s_mp_pad(&x, USED(a)));
        mpp_random(&x);
        MP_CHECKOK(mp_mod(&x, a, &x));
        if (mp_cmp_d(&x, 1) <= 0) {
            iter--;   /* don't count this iteration */
            continue; /* choose a new x */
        }

        /* Compute z = (x ** m) mod a               */
        MP_CHECKOK(mp_exptmod(&x, &m, a, &z));

        if (mp_cmp_d(&z, 1) == 0 || mp_cmp(&z, &amo) == 0) {
            res = MP_YES;
            continue;
        }

        res = MP_NO; /* just in case the following for loop never executes. */
        for (jx = 1; jx < b; jx++) {
            /* z = z^2 (mod a) */
            MP_CHECKOK(mp_sqrmod(&z, a, &z));
            res = MP_NO; /* previous line set res to MP_YES */

            if (mp_cmp_d(&z, 1) == 0) {
                break;
            }
            if (mp_cmp(&z, &amo) == 0) {
                res = MP_YES;
                break;
            }
        } /* end testing loop */

        /* If the test passes, we will continue iterating, but a failed
           test means the candidate is definitely NOT prime, so we will
           immediately break out of this loop
         */
        if (res == MP_NO)
            break;

    } /* end iterations loop */

CLEANUP:
    mp_clear(&m);
    mp_clear(&z);
    mp_clear(&x);
    mp_clear(&amo);
    return res;

} /* end mpp_pprime() */

/* }}} */

/* Produce table of composites from list of primes and trial value.
** trial must be odd. List of primes must not include 2.
** sieve should have dimension >= MAXPRIME/2, where MAXPRIME is largest
** prime in list of primes.  After this function is finished,
** if sieve[i] is non-zero, then (trial + 2*i) is composite.
** Each prime used in the sieve costs one division of trial, and eliminates
** one or more values from the search space. (3 eliminates 1/3 of the values
** alone!)  Each value left in the search space costs 1 or more modular
** exponentations.  So, these divisions are a bargain!
*/
mp_err
mpp_sieve(mp_int *trial, const mp_digit *primes, mp_size nPrimes,
          unsigned char *sieve, mp_size nSieve)
{
    mp_err res;
    mp_digit rem;
    mp_size ix;
    unsigned long offset;

    memset(sieve, 0, nSieve);

    for (ix = 0; ix < nPrimes; ix++) {
        mp_digit prime = primes[ix];
        mp_size i;
        if ((res = mp_mod_d(trial, prime, &rem)) != MP_OKAY)
            return res;

        if (rem == 0) {
            offset = 0;
        } else {
            offset = prime - rem;
        }

        for (i = offset; i < nSieve * 2; i += prime) {
            if (i % 2 == 0) {
                sieve[i / 2] = 1;
            }
        }
    }

    return MP_OKAY;
}

#define SIEVE_SIZE 32 * 1024

mp_err
mpp_make_prime(mp_int *start, mp_size nBits, mp_size strong,
               unsigned long *nTries)
{
    mp_digit np;
    mp_err res;
    unsigned int i = 0;
    mp_int trial;
    mp_int q;
    mp_size num_tests;
    unsigned char *sieve;

    ARGCHK(start != 0, MP_BADARG);
    ARGCHK(nBits > 16, MP_RANGE);

    sieve = malloc(SIEVE_SIZE);
    ARGCHK(sieve != NULL, MP_MEM);

    MP_DIGITS(&trial) = 0;
    MP_DIGITS(&q) = 0;
    MP_CHECKOK(mp_init(&trial));
    MP_CHECKOK(mp_init(&q));
    /* values originally taken from table 4.4,
   * HandBook of Applied Cryptography, augmented by FIPS-186
   * requirements, Table C.2 and C.3 */
    if (nBits >= 2000) {
        num_tests = 3;
    } else if (nBits >= 1536) {
        num_tests = 4;
    } else if (nBits >= 1024) {
        num_tests = 5;
    } else if (nBits >= 550) {
        num_tests = 6;
    } else if (nBits >= 450) {
        num_tests = 7;
    } else if (nBits >= 400) {
        num_tests = 8;
    } else if (nBits >= 350) {
        num_tests = 9;
    } else if (nBits >= 300) {
        num_tests = 10;
    } else if (nBits >= 250) {
        num_tests = 20;
    } else if (nBits >= 200) {
        num_tests = 41;
    } else if (nBits >= 100) {
        num_tests = 38; /* funny anomaly in the FIPS tables, for aux primes, the
                         * required more iterations for larger aux primes */
    } else
        num_tests = 50;

    if (strong)
        --nBits;
    MP_CHECKOK(mpl_set_bit(start, nBits - 1, 1));
    MP_CHECKOK(mpl_set_bit(start, 0, 1));
    for (i = mpl_significant_bits(start) - 1; i >= nBits; --i) {
        MP_CHECKOK(mpl_set_bit(start, i, 0));
    }
    /* start sieveing with prime value of 3. */
    MP_CHECKOK(mpp_sieve(start, prime_tab + 1, prime_tab_size - 1,
                         sieve, SIEVE_SIZE));

#ifdef DEBUG_SIEVE
    res = 0;
    for (i = 0; i < SIEVE_SIZE; ++i) {
        if (!sieve[i])
            ++res;
    }
    fprintf(stderr, "sieve found %d potential primes.\n", res);
#define FPUTC(x, y) fputc(x, y)
#else
#define FPUTC(x, y)
#endif

    res = MP_NO;
    for (i = 0; i < SIEVE_SIZE; ++i) {
        if (sieve[i]) /* this number is composite */
            continue;
        MP_CHECKOK(mp_add_d(start, 2 * i, &trial));
        FPUTC('.', stderr);
        /* run a Fermat test */
        res = mpp_fermat(&trial, 2);
        if (res != MP_OKAY) {
            if (res == MP_NO)
                continue; /* was composite */
            goto CLEANUP;
        }

        FPUTC('+', stderr);
        /* If that passed, run some Miller-Rabin tests  */
        res = mpp_pprime(&trial, num_tests);
        if (res != MP_OKAY) {
            if (res == MP_NO)
                continue; /* was composite */
            goto CLEANUP;
        }
        FPUTC('!', stderr);

        if (!strong)
            break; /* success !! */

        /* At this point, we have strong evidence that our candidate
           is itself prime.  If we want a strong prime, we need now
           to test q = 2p + 1 for primality...
        */
        MP_CHECKOK(mp_mul_2(&trial, &q));
        MP_CHECKOK(mp_add_d(&q, 1, &q));

        /* Test q for small prime divisors ... */
        np = prime_tab_size;
        res = mpp_divis_primes(&q, &np);
        if (res == MP_YES) { /* is composite */
            mp_clear(&q);
            continue;
        }
        if (res != MP_NO)
            goto CLEANUP;

        /* And test with Fermat, as with its parent ... */
        res = mpp_fermat(&q, 2);
        if (res != MP_YES) {
            mp_clear(&q);
            if (res == MP_NO)
                continue; /* was composite */
            goto CLEANUP;
        }

        /* And test with Miller-Rabin, as with its parent ... */
        res = mpp_pprime(&q, num_tests);
        if (res != MP_YES) {
            mp_clear(&q);
            if (res == MP_NO)
                continue; /* was composite */
            goto CLEANUP;
        }

        /* If it passed, we've got a winner */
        mp_exch(&q, &trial);
        mp_clear(&q);
        break;

    } /* end of loop through sieved values */
    if (res == MP_YES)
        mp_exch(&trial, start);
CLEANUP:
    mp_clear(&trial);
    mp_clear(&q);
    if (nTries)
        *nTries += i;
    if (sieve != NULL) {
        memset(sieve, 0, SIEVE_SIZE);
        free(sieve);
    }
    return res;
}

/*========================================================================*/
/*------------------------------------------------------------------------*/
/* Static functions visible only to the library internally                */

/* {{{ s_mpp_divp(a, vec, size, which) */

/*
   Test for divisibility by members of a vector of digits.  Returns
   MP_NO if a is not divisible by any of them; returns MP_YES and sets
   'which' to the index of the offender, if it is.  Will stop on the
   first digit against which a is divisible.
 */

mp_err
s_mpp_divp(mp_int *a, const mp_digit *vec, int size, int *which)
{
    mp_err res;
    mp_digit rem;

    int ix;

    for (ix = 0; ix < size; ix++) {
        if ((res = mp_mod_d(a, vec[ix], &rem)) != MP_OKAY)
            return res;

        if (rem == 0) {
            if (which)
                *which = ix;
            return MP_YES;
        }
    }

    return MP_NO;

} /* end s_mpp_divp() */

/* }}} */

/*------------------------------------------------------------------------*/
/* HERE THERE BE DRAGONS                                                  */