/*

 * Copyright 1995-2021 The OpenSSL Project Authors. All Rights Reserved.

 *

 * Licensed under the Apache License 2.0 (the "License").  You may not use

 * this file except in compliance with the License.  You can obtain a copy

 * in the file LICENSE in the source distribution or at

 * https://www.openssl.org/source/license.html

 */

#include <stdio.h>

#include <time.h>

#include "internal/cryptlib.h"

#include "bn_local.h"

/*

 * The quick sieve algorithm approach to weeding out primes is Philip

 * Zimmermann's, as implemented in PGP.  I have had a read of his comments

 * and implemented my own version.

 */

#include "bn_prime.h"

static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods,

                          BN_CTX *ctx);

static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods,

                             const BIGNUM *add, const BIGNUM *rem,

                             BN_CTX *ctx);

static int bn_is_prime_int(const BIGNUM *w, int checks, BN_CTX *ctx,

                           int do_trial_division, BN_GENCB *cb);

#define square(x) ((BN_ULONG)(x) * (BN_ULONG)(x))

#if BN_BITS2 == 64

# define BN_DEF(lo, hi) (BN_ULONG)hi<<32|lo

#else

# define BN_DEF(lo, hi) lo, hi

#endif

/*

 * See SP800 89 5.3.3 (Step f)

 * The product of the set of primes ranging from 3 to 751

 * Generated using process in test/bn_internal_test.c test_bn_small_factors().

 * This includes 751 (which is not currently included in SP 800-89).

 */

static const BN_ULONG small_prime_factors[] = {

    BN_DEF(0x3ef4e3e1, 0xc4309333), BN_DEF(0xcd2d655f, 0x71161eb6),

    BN_DEF(0x0bf94862, 0x95e2238c), BN_DEF(0x24f7912b, 0x3eb233d3),

    BN_DEF(0xbf26c483, 0x6b55514b), BN_DEF(0x5a144871, 0x0a84d817),

    BN_DEF(0x9b82210a, 0x77d12fee), BN_DEF(0x97f050b3, 0xdb5b93c2),

    BN_DEF(0x4d6c026b, 0x4acad6b9), BN_DEF(0x54aec893, 0xeb7751f3),

    BN_DEF(0x36bc85c4, 0xdba53368), BN_DEF(0x7f5ec78e, 0xd85a1b28),

    BN_DEF(0x6b322244, 0x2eb072d8), BN_DEF(0x5e2b3aea, 0xbba51112),

    BN_DEF(0x0e2486bf, 0x36ed1a6c), BN_DEF(0xec0c5727, 0x5f270460),

    (BN_ULONG)0x000017b1

};

#define BN_SMALL_PRIME_FACTORS_TOP OSSL_NELEM(small_prime_factors)

static const BIGNUM _bignum_small_prime_factors = {

    (BN_ULONG *)small_prime_factors,

    BN_SMALL_PRIME_FACTORS_TOP,

    BN_SMALL_PRIME_FACTORS_TOP,

    0,

    BN_FLG_STATIC_DATA

};

const BIGNUM *ossl_bn_get0_small_factors(void)

{

    return &_bignum_small_prime_factors;

}

/*

 * Calculate the number of trial divisions that gives the best speed in

 * combination with Miller-Rabin prime test, based on the sized of the prime.

 */

static int calc_trial_divisions(int bits)

{

    if (bits <= 512)

        return 64;

    else if (bits <= 1024)

        return 128;

    else if (bits <= 2048)

        return 384;

    else if (bits <= 4096)

        return 1024;

    return NUMPRIMES;

}

/*

 * Use a minimum of 64 rounds of Miller-Rabin, which should give a false

 * positive rate of 2^-128. If the size of the prime is larger than 2048

 * the user probably wants a higher security level than 128, so switch

 * to 128 rounds giving a false positive rate of 2^-256.

 * Returns the number of rounds.

 */

static int bn_mr_min_checks(int bits)

{

    if (bits > 2048)

        return 128;

    return 64;

}

int BN_GENCB_call(BN_GENCB *cb, int a, int b)

{

    /* No callback means continue */

    if (!cb)

        return 1;

    switch (cb->ver) {

    case 1:

        /* Deprecated-style callbacks */

        if (!cb->cb.cb_1)

            return 1;

        cb->cb.cb_1(a, b, cb->arg);

        return 1;

    case 2:

        /* New-style callbacks */

        return cb->cb.cb_2(a, b, cb);

    default:

        break;

    }

    /* Unrecognised callback type */

    return 0;

}

int BN_generate_prime_ex2(BIGNUM *ret, int bits, int safe,

                          const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb,

                          BN_CTX *ctx)

{

    BIGNUM *t;

    int found = 0;

    int i, j, c1 = 0;

    prime_t *mods = NULL;

    int checks = bn_mr_min_checks(bits);

    if (bits < 2) {

        /* There are no prime numbers this small. */

        ERR_raise(ERR_LIB_BN, BN_R_BITS_TOO_SMALL);

        return 0;

    } else if (add == NULL && safe && bits < 6 && bits != 3) {

        /*

         * The smallest safe prime (7) is three bits.

         * But the following two safe primes with less than 6 bits (11, 23)

         * are unreachable for BN_rand with BN_RAND_TOP_TWO.

         */

        ERR_raise(ERR_LIB_BN, BN_R_BITS_TOO_SMALL);

        return 0;

    }

    mods = OPENSSL_zalloc(sizeof(*mods) * NUMPRIMES);

    if (mods == NULL)

        return 0;

    BN_CTX_start(ctx);

    t = BN_CTX_get(ctx);

    if (t == NULL)

        goto err;

 loop:

    /* make a random number and set the top and bottom bits */

    if (add == NULL) {

        if (!probable_prime(ret, bits, safe, mods, ctx))

            goto err;

    } else {

        if (!probable_prime_dh(ret, bits, safe, mods, add, rem, ctx))

            goto err;

    }

    if (!BN_GENCB_call(cb, 0, c1++))

        /* aborted */

        goto err;

    if (!safe) {

        i = bn_is_prime_int(ret, checks, ctx, 0, cb);

        if (i == -1)

            goto err;

        if (i == 0)

            goto loop;

    } else {

        /*

         * for "safe prime" generation, check that (p-1)/2 is prime. Since a

         * prime is odd, We just need to divide by 2

         */

        if (!BN_rshift1(t, ret))

            goto err;

        for (i = 0; i < checks; i++) {

            j = bn_is_prime_int(ret, 1, ctx, 0, cb);

            if (j == -1)

                goto err;

            if (j == 0)

                goto loop;

            j = bn_is_prime_int(t, 1, ctx, 0, cb);

            if (j == -1)

                goto err;

            if (j == 0)

                goto loop;

            if (!BN_GENCB_call(cb, 2, c1 - 1))

                goto err;

            /* We have a safe prime test pass */

        }

    }

    /* we have a prime :-) */

    found = 1;

 err:

    OPENSSL_free(mods);

    BN_CTX_end(ctx);

    bn_check_top(ret);

    return found;

}

#ifndef FIPS_MODULE

int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe,

                         const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb)

{

    BN_CTX *ctx = BN_CTX_new();

    int retval;

    if (ctx == NULL)

        return 0;

    retval = BN_generate_prime_ex2(ret, bits, safe, add, rem, cb, ctx);

    BN_CTX_free(ctx);

    return retval;

}

#endif

#ifndef OPENSSL_NO_DEPRECATED_3_0

int BN_is_prime_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,

                   BN_GENCB *cb)

{

    return ossl_bn_check_prime(a, checks, ctx_passed, 0, cb);

}

int BN_is_prime_fasttest_ex(const BIGNUM *w, int checks, BN_CTX *ctx,

                            int do_trial_division, BN_GENCB *cb)

{

    return ossl_bn_check_prime(w, checks, ctx, do_trial_division, cb);

}

#endif

/* Wrapper around bn_is_prime_int that sets the minimum number of checks */

int ossl_bn_check_prime(const BIGNUM *w, int checks, BN_CTX *ctx,

                        int do_trial_division, BN_GENCB *cb)

{

    int min_checks = bn_mr_min_checks(BN_num_bits(w));

    if (checks < min_checks)

        checks = min_checks;

    return bn_is_prime_int(w, checks, ctx, do_trial_division, cb);

}

/*

 * Use this only for key generation.

 * It always uses trial division. The number of checks

 * (MR rounds) passed in is used without being clamped to a minimum value.

 */

int ossl_bn_check_generated_prime(const BIGNUM *w, int checks, BN_CTX *ctx,

                                  BN_GENCB *cb)

{

    return bn_is_prime_int(w, checks, ctx, 1, cb);

}

int BN_check_prime(const BIGNUM *p, BN_CTX *ctx, BN_GENCB *cb)

{

    return ossl_bn_check_prime(p, 0, ctx, 1, cb);

}

/*

 * Tests that |w| is probably prime

 * See FIPS 186-4 C.3.1 Miller Rabin Probabilistic Primality Test.

 *

 * Returns 0 when composite, 1 when probable prime, -1 on error.

 */

static int bn_is_prime_int(const BIGNUM *w, int checks, BN_CTX *ctx,

                           int do_trial_division, BN_GENCB *cb)

{

    int i, status, ret = -1;

#ifndef FIPS_MODULE

    BN_CTX *ctxlocal = NULL;

#else

    if (ctx == NULL)

        return -1;

#endif

    /* w must be bigger than 1 */

    if (BN_cmp(w, BN_value_one()) <= 0)

        return 0;

    /* w must be odd */

    if (BN_is_odd(w)) {

        /* Take care of the really small prime 3 */

        if (BN_is_word(w, 3))

            return 1;

    } else {

        /* 2 is the only even prime */

        return BN_is_word(w, 2);

    }

    /* first look for small factors */

    if (do_trial_division) {

        int trial_divisions = calc_trial_divisions(BN_num_bits(w));

        for (i = 1; i < trial_divisions; i++) {

            BN_ULONG mod = BN_mod_word(w, primes[i]);

            if (mod == (BN_ULONG)-1)

                return -1;

            if (mod == 0)

                return BN_is_word(w, primes[i]);

        }

        if (!BN_GENCB_call(cb, 1, -1))

            return -1;

    }

#ifndef FIPS_MODULE

    if (ctx == NULL && (ctxlocal = ctx = BN_CTX_new()) == NULL)

        goto err;

#endif

    if (!ossl_bn_miller_rabin_is_prime(w, checks, ctx, cb, 0, &status)) {

        ret = -1;

        goto err;

    }

    ret = (status == BN_PRIMETEST_PROBABLY_PRIME);

err:

#ifndef FIPS_MODULE

    BN_CTX_free(ctxlocal);

#endif

    return ret;

}

/*

 * Refer to FIPS 186-4 C.3.2 Enhanced Miller-Rabin Probabilistic Primality Test.

 * OR C.3.1 Miller-Rabin Probabilistic Primality Test (if enhanced is zero).

 * The Step numbers listed in the code refer to the enhanced case.

 *

 * if enhanced is set, then status returns one of the following:

 *     BN_PRIMETEST_PROBABLY_PRIME

 *     BN_PRIMETEST_COMPOSITE_WITH_FACTOR

 *     BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME

 * if enhanced is zero, then status returns either

 *     BN_PRIMETEST_PROBABLY_PRIME or

 *     BN_PRIMETEST_COMPOSITE

 *

 * returns 0 if there was an error, otherwise it returns 1.

 */

int ossl_bn_miller_rabin_is_prime(const BIGNUM *w, int iterations, BN_CTX *ctx,

                                  BN_GENCB *cb, int enhanced, int *status)

{

    int i, j, a, ret = 0;

    BIGNUM *g, *w1, *w3, *x, *m, *z, *b;

    BN_MONT_CTX *mont = NULL;

    /* w must be odd */

    if (!BN_is_odd(w))

        return 0;

    BN_CTX_start(ctx);

    g = BN_CTX_get(ctx);

    w1 = BN_CTX_get(ctx);

    w3 = BN_CTX_get(ctx);

    x = BN_CTX_get(ctx);

    m = BN_CTX_get(ctx);

    z = BN_CTX_get(ctx);

    b = BN_CTX_get(ctx);

    if (!(b != NULL

            /* w1 := w - 1 */

            && BN_copy(w1, w)

            && BN_sub_word(w1, 1)

            /* w3 := w - 3 */

            && BN_copy(w3, w)

            && BN_sub_word(w3, 3)))

        goto err;

    /* check w is larger than 3, otherwise the random b will be too small */

    if (BN_is_zero(w3) || BN_is_negative(w3))

        goto err;

    /* (Step 1) Calculate largest integer 'a' such that 2^a divides w-1 */

    a = 1;

    while (!BN_is_bit_set(w1, a))

        a++;

    /* (Step 2) m = (w-1) / 2^a */

    if (!BN_rshift(m, w1, a))

        goto err;

    /* Montgomery setup for computations mod a */

    mont = BN_MONT_CTX_new();

    if (mont == NULL || !BN_MONT_CTX_set(mont, w, ctx))

        goto err;

    if (iterations == 0)

        iterations = bn_mr_min_checks(BN_num_bits(w));

    /* (Step 4) */

    for (i = 0; i < iterations; ++i) {

        /* (Step 4.1) obtain a Random string of bits b where 1 < b < w-1 */

        if (!BN_priv_rand_range_ex(b, w3, 0, ctx)

                || !BN_add_word(b, 2)) /* 1 < b < w-1 */

            goto err;

        if (enhanced) {

            /* (Step 4.3) */

            if (!BN_gcd(g, b, w, ctx))

                goto err;

            /* (Step 4.4) */

            if (!BN_is_one(g)) {

                *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;

                ret = 1;

                goto err;

            }

        }

        /* (Step 4.5) z = b^m mod w */

        if (!BN_mod_exp_mont(z, b, m, w, ctx, mont))

            goto err;

        /* (Step 4.6) if (z = 1 or z = w-1) */

        if (BN_is_one(z) || BN_cmp(z, w1) == 0)

            goto outer_loop;

        /* (Step 4.7) for j = 1 to a-1 */

        for (j = 1; j < a ; ++j) {

            /* (Step 4.7.1 - 4.7.2) x = z. z = x^2 mod w */

            if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx))

                goto err;

            /* (Step 4.7.3) */

            if (BN_cmp(z, w1) == 0)

                goto outer_loop;

            /* (Step 4.7.4) */

            if (BN_is_one(z))

                goto composite;

        }

        /* At this point z = b^((w-1)/2) mod w */

        /* (Steps 4.8 - 4.9) x = z, z = x^2 mod w */

        if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx))

            goto err;

        /* (Step 4.10) */

        if (BN_is_one(z))

            goto composite;

        /* (Step 4.11) x = b^(w-1) mod w */

        if (!BN_copy(x, z))

            goto err;

composite:

        if (enhanced) {

            /* (Step 4.1.2) g = GCD(x-1, w) */

            if (!BN_sub_word(x, 1) || !BN_gcd(g, x, w, ctx))

                goto err;

            /* (Steps 4.1.3 - 4.1.4) */

            if (BN_is_one(g))

                *status = BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME;

            else

                *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;

        } else {

            *status = BN_PRIMETEST_COMPOSITE;

        }

        ret = 1;

        goto err;

outer_loop: ;

        /* (Step 4.1.5) */

        if (!BN_GENCB_call(cb, 1, i))

            goto err;

    }

    /* (Step 5) */

    *status = BN_PRIMETEST_PROBABLY_PRIME;

    ret = 1;

err:

    BN_clear(g);

    BN_clear(w1);

    BN_clear(w3);

    BN_clear(x);

    BN_clear(m);

    BN_clear(z);

    BN_clear(b);

    BN_CTX_end(ctx);

    BN_MONT_CTX_free(mont);

    return ret;

}

/*

 * Generate a random number of |bits| bits that is probably prime by sieving.

 * If |safe| != 0, it generates a safe prime.

 * |mods| is a preallocated array that gets reused when called again.

 *

 * The probably prime is saved in |rnd|.

 *

 * Returns 1 on success and 0 on error.

 */

static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods,

                          BN_CTX *ctx)

{

    int i;

    BN_ULONG delta;

    int trial_divisions = calc_trial_divisions(bits);

    BN_ULONG maxdelta = BN_MASK2 - primes[trial_divisions - 1];

 again:

    if (!BN_priv_rand_ex(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD, 0,

                         ctx))

        return 0;

    if (safe && !BN_set_bit(rnd, 1))

        return 0;

    /* we now have a random number 'rnd' to test. */

    for (i = 1; i < trial_divisions; i++) {

        BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);

        if (mod == (BN_ULONG)-1)

            return 0;

        mods[i] = (prime_t) mod;

    }

    delta = 0;

 loop:

    for (i = 1; i < trial_divisions; i++) {

        /*

         * check that rnd is a prime and also that

         * gcd(rnd-1,primes) == 1 (except for 2)

         * do the second check only if we are interested in safe primes

         * in the case that the candidate prime is a single word then

         * we check only the primes up to sqrt(rnd)

         */

        if (bits <= 31 && delta <= 0x7fffffff

                && square(primes[i]) > BN_get_word(rnd) + delta)

            break;

        if (safe ? (mods[i] + delta) % primes[i] <= 1

                 : (mods[i] + delta) % primes[i] == 0) {

            delta += safe ? 4 : 2;

            if (delta > maxdelta)

                goto again;

            goto loop;

        }

    }

    if (!BN_add_word(rnd, delta))

        return 0;

    if (BN_num_bits(rnd) != bits)

        goto again;

    bn_check_top(rnd);

    return 1;

}

/*

 * Generate a random number |rnd| of |bits| bits that is probably prime

 * and satisfies |rnd| % |add| == |rem| by sieving.

 * If |safe| != 0, it generates a safe prime.

 * |mods| is a preallocated array that gets reused when called again.

 *

 * Returns 1 on success and 0 on error.

 */

static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods,

                             const BIGNUM *add, const BIGNUM *rem,

                             BN_CTX *ctx)

{

    int i, ret = 0;

    BIGNUM *t1;

    BN_ULONG delta;

    int trial_divisions = calc_trial_divisions(bits);

    BN_ULONG maxdelta = BN_MASK2 - primes[trial_divisions - 1];

    BN_CTX_start(ctx);

    if ((t1 = BN_CTX_get(ctx)) == NULL)

        goto err;

    if (maxdelta > BN_MASK2 - BN_get_word(add))

        maxdelta = BN_MASK2 - BN_get_word(add);

 again:

    if (!BN_rand_ex(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD, 0, ctx))

        goto err;

    /* we need ((rnd-rem) % add) == 0 */

    if (!BN_mod(t1, rnd, add, ctx))

        goto err;

    if (!BN_sub(rnd, rnd, t1))

        goto err;

    if (rem == NULL) {

        if (!BN_add_word(rnd, safe ? 3u : 1u))

            goto err;

    } else {

        if (!BN_add(rnd, rnd, rem))

            goto err;

    }

    if (BN_num_bits(rnd) < bits

            || BN_get_word(rnd) < (safe ? 5u : 3u)) {

        if (!BN_add(rnd, rnd, add))

            goto err;

    }

    /* we now have a random number 'rnd' to test. */

    for (i = 1; i < trial_divisions; i++) {

        BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);

        if (mod == (BN_ULONG)-1)

            goto err;

        mods[i] = (prime_t) mod;

    }

    delta = 0;

 loop:

    for (i = 1; i < trial_divisions; i++) {

        /* check that rnd is a prime */

        if (bits <= 31 && delta <= 0x7fffffff

                && square(primes[i]) > BN_get_word(rnd) + delta)

            break;

        /* rnd mod p == 1 implies q = (rnd-1)/2 is divisible by p */

        if (safe ? (mods[i] + delta) % primes[i] <= 1

                 : (mods[i] + delta) % primes[i] == 0) {

            delta += BN_get_word(add);

            if (delta > maxdelta)

                goto again;

            goto loop;

        }

    }

    if (!BN_add_word(rnd, delta))

        goto err;

    ret = 1;

 err:

    BN_CTX_end(ctx);

    bn_check_top(rnd);

    return ret;

}