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/*
* Copyright 1995-2019 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_lcl.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, prime_t *mods, BN_CTX *ctx);
static int probable_prime_dh_safe(BIGNUM *rnd, int bits,
const BIGNUM *add, const BIGNUM *rem,
BN_CTX *ctx);
#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 *bn_get0_small_factors(void)
{
return &_bignum_small_prime_factors;
}
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_prime_checks_for_size(bits);
if (bits < 2) {
/* There are no prime numbers this small. */
BNerr(BN_F_BN_GENERATE_PRIME_EX2, 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.
*/
BNerr(BN_F_BN_GENERATE_PRIME_EX2, BN_R_BITS_TOO_SMALL);
return 0;
}
mods = OPENSSL_zalloc(sizeof(*mods) * NUMPRIMES);
if (mods == NULL)
goto err;
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, mods, ctx))
goto err;
} else {
if (safe) {
if (!probable_prime_dh_safe(ret, bits, add, rem, ctx))
goto err;
} else {
if (!bn_probable_prime_dh(ret, bits, add, rem, ctx))
goto err;
}
}
if (!BN_GENCB_call(cb, 0, c1++))
/* aborted */
goto err;
if (!safe) {
i = BN_is_prime_fasttest_ex(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_fasttest_ex(ret, 1, ctx, 0, cb);
if (j == -1)
goto err;
if (j == 0)
goto loop;
j = BN_is_prime_fasttest_ex(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_MODE
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
int BN_is_prime_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
BN_GENCB *cb)
{
return BN_is_prime_fasttest_ex(a, checks, ctx_passed, 0, cb);
}
/* See FIPS 186-4 C.3.1 Miller Rabin Probabilistic Primality Test. */
int BN_is_prime_fasttest_ex(const BIGNUM *w, int checks, BN_CTX *ctx,
int do_trial_division, BN_GENCB *cb)
{
int i, status, ret = -1;
#ifndef FIPS_MODE
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) {
for (i = 1; i < NUMPRIMES; 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_MODE
if (ctx == NULL && (ctxlocal = ctx = BN_CTX_new()) == NULL)
goto err;
#endif
ret = bn_miller_rabin_is_prime(w, checks, ctx, cb, 0, &status);
if (!ret)
goto err;
ret = (status == BN_PRIMETEST_PROBABLY_PRIME);
err:
#ifndef FIPS_MODE
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 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 == BN_prime_checks)
iterations = BN_prime_checks_for_size(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, 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;
}
static int probable_prime(BIGNUM *rnd, int bits, prime_t *mods, BN_CTX *ctx)
{
int i;
BN_ULONG delta;
BN_ULONG maxdelta = BN_MASK2 - primes[NUMPRIMES - 1];
char is_single_word = bits <= BN_BITS2;
again:
/* TODO: Not all primes are private */
if (!BN_priv_rand_ex(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD, ctx))
return 0;
/* we now have a random number 'rnd' to test. */
for (i = 1; i < NUMPRIMES; i++) {
BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
if (mod == (BN_ULONG)-1)
return 0;
mods[i] = (prime_t) mod;
}
/*
* If bits is so small that it fits into a single word then we
* additionally don't want to exceed that many bits.
*/
if (is_single_word) {
BN_ULONG size_limit;
if (bits == BN_BITS2) {
/*
* Shifting by this much has undefined behaviour so we do it a
* different way
*/
size_limit = ~((BN_ULONG)0) - BN_get_word(rnd);
} else {
size_limit = (((BN_ULONG)1) << bits) - BN_get_word(rnd) - 1;
}
if (size_limit < maxdelta)
maxdelta = size_limit;
}
delta = 0;
loop:
if (is_single_word) {
BN_ULONG rnd_word = BN_get_word(rnd);
/*-
* In the case that the candidate prime is a single word then
* we check that:
* 1) It's greater than primes[i] because we shouldn't reject
* 3 as being a prime number because it's a multiple of
* three.
* 2) That it's not a multiple of a known prime. We don't
* check that rnd-1 is also coprime to all the known
* primes because there aren't many small primes where
* that's true.
*/
for (i = 1; i < NUMPRIMES && primes[i] < rnd_word; i++) {
if ((mods[i] + delta) % primes[i] == 0) {
delta += 2;
if (delta > maxdelta)
goto again;
goto loop;
}
}
} else {
for (i = 1; i < NUMPRIMES; i++) {
/*
* check that rnd is not a prime and also that gcd(rnd-1,primes)
* == 1 (except for 2)
*/
if (((mods[i] + delta) % primes[i]) <= 1) {
delta += 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;
}
int bn_probable_prime_dh(BIGNUM *rnd, int bits,
const BIGNUM *add, const BIGNUM *rem, BN_CTX *ctx)
{
int i, ret = 0;
BIGNUM *t1;
BN_CTX_start(ctx);
if ((t1 = BN_CTX_get(ctx)) == NULL)
goto err;
if (!BN_rand_ex(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD, 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, 1))
goto err;
} else {
if (!BN_add(rnd, rnd, rem))
goto err;
}
/* we now have a random number 'rand' to test. */
loop:
for (i = 1; i < NUMPRIMES; i++) {
/* check that rnd is a prime */
BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
if (mod == (BN_ULONG)-1)
goto err;
if (mod <= 1) {
if (!BN_add(rnd, rnd, add))
goto err;
goto loop;
}
}
ret = 1;
err:
BN_CTX_end(ctx);
bn_check_top(rnd);
return ret;
}
static int probable_prime_dh_safe(BIGNUM *p, int bits, const BIGNUM *padd,
const BIGNUM *rem, BN_CTX *ctx)
{
int i, ret = 0;
BIGNUM *t1, *qadd, *q;
bits--;
BN_CTX_start(ctx);
t1 = BN_CTX_get(ctx);
q = BN_CTX_get(ctx);
qadd = BN_CTX_get(ctx);
if (qadd == NULL)
goto err;
if (!BN_rshift1(qadd, padd))
goto err;
if (!BN_rand_ex(q, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD, ctx))
goto err;
/* we need ((rnd-rem) % add) == 0 */
if (!BN_mod(t1, q, qadd, ctx))
goto err;
if (!BN_sub(q, q, t1))
goto err;
if (rem == NULL) {
if (!BN_add_word(q, 1))
goto err;
} else {
if (!BN_rshift1(t1, rem))
goto err;
if (!BN_add(q, q, t1))
goto err;
}
/* we now have a random number 'rand' to test. */
if (!BN_lshift1(p, q))
goto err;
if (!BN_add_word(p, 1))
goto err;
loop:
for (i = 1; i < NUMPRIMES; i++) {
/* check that p and q are prime */
/*
* check that for p and q gcd(p-1,primes) == 1 (except for 2)
*/
BN_ULONG pmod = BN_mod_word(p, (BN_ULONG)primes[i]);
BN_ULONG qmod = BN_mod_word(q, (BN_ULONG)primes[i]);
if (pmod == (BN_ULONG)-1 || qmod == (BN_ULONG)-1)
goto err;
if (pmod == 0 || qmod == 0) {
if (!BN_add(p, p, padd))
goto err;
if (!BN_add(q, q, qadd))
goto err;
goto loop;
}
}
ret = 1;
err:
BN_CTX_end(ctx);
bn_check_top(p);
return ret;
}
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