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/*
* 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) {
ERR_raise(ERR_LIB_BN, ERR_R_MALLOC_FAILURE);
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);
}
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
ret = ossl_bn_miller_rabin_is_prime(w, checks, ctx, cb, 0, &status);
if (!ret)
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;
}