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openssl/crypto/bn/bn_prime.c

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/*
* Copyright 1995-2020 The OpenSSL Project Authors. All Rights Reserved.
*
* Licensed under the OpenSSL license (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 witness(BIGNUM *w, const BIGNUM *a, const BIGNUM *a1,
const BIGNUM *a1_odd, int k, BN_CTX *ctx,
BN_MONT_CTX *mont);
static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods);
static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods,
const BIGNUM *add, const BIGNUM *rem,
BN_CTX *ctx);
#define square(x) ((BN_ULONG)(x) * (BN_ULONG)(x))
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_ex(BIGNUM *ret, int bits, int safe,
const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb)
{
BIGNUM *t;
int found = 0;
int i, j, c1 = 0;
BN_CTX *ctx = NULL;
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_EX, 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_EX, BN_R_BITS_TOO_SMALL);
return 0;
}
mods = OPENSSL_zalloc(sizeof(*mods) * NUMPRIMES);
if (mods == NULL)
goto err;
ctx = BN_CTX_new();
if (ctx == 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, safe, mods))
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_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_CTX_free(ctx);
bn_check_top(ret);
return found;
}
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);
}
int BN_is_prime_fasttest_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
int do_trial_division, BN_GENCB *cb)
{
int i, j, ret = -1;
int k;
BN_CTX *ctx = NULL;
BIGNUM *A1, *A1_odd, *A3, *check; /* taken from ctx */
BN_MONT_CTX *mont = NULL;
/* Take care of the really small primes 2 & 3 */
if (BN_is_word(a, 2) || BN_is_word(a, 3))
return 1;
/* Check odd and bigger than 1 */
if (!BN_is_odd(a) || BN_cmp(a, BN_value_one()) <= 0)
return 0;
if (checks == BN_prime_checks)
checks = BN_prime_checks_for_size(BN_num_bits(a));
/* first look for small factors */
if (do_trial_division) {
for (i = 1; i < NUMPRIMES; i++) {
BN_ULONG mod = BN_mod_word(a, primes[i]);
if (mod == (BN_ULONG)-1)
goto err;
if (mod == 0)
return BN_is_word(a, primes[i]);
}
if (!BN_GENCB_call(cb, 1, -1))
goto err;
}
if (ctx_passed != NULL)
ctx = ctx_passed;
else if ((ctx = BN_CTX_new()) == NULL)
goto err;
BN_CTX_start(ctx);
A1 = BN_CTX_get(ctx);
A3 = BN_CTX_get(ctx);
A1_odd = BN_CTX_get(ctx);
check = BN_CTX_get(ctx);
if (check == NULL)
goto err;
/* compute A1 := a - 1 */
if (!BN_copy(A1, a) || !BN_sub_word(A1, 1))
goto err;
/* compute A3 := a - 3 */
if (!BN_copy(A3, a) || !BN_sub_word(A3, 3))
goto err;
/* write A1 as A1_odd * 2^k */
k = 1;
while (!BN_is_bit_set(A1, k))
k++;
if (!BN_rshift(A1_odd, A1, k))
goto err;
/* Montgomery setup for computations mod a */
mont = BN_MONT_CTX_new();
if (mont == NULL)
goto err;
if (!BN_MONT_CTX_set(mont, a, ctx))
goto err;
for (i = 0; i < checks; i++) {
/* 1 < check < a-1 */
if (!BN_priv_rand_range(check, A3) || !BN_add_word(check, 2))
goto err;
j = witness(check, a, A1, A1_odd, k, ctx, mont);
if (j == -1)
goto err;
if (j) {
ret = 0;
goto err;
}
if (!BN_GENCB_call(cb, 1, i))
goto err;
}
ret = 1;
err:
if (ctx != NULL) {
BN_CTX_end(ctx);
if (ctx_passed == NULL)
BN_CTX_free(ctx);
}
BN_MONT_CTX_free(mont);
return ret;
}
static int witness(BIGNUM *w, const BIGNUM *a, const BIGNUM *a1,
const BIGNUM *a1_odd, int k, BN_CTX *ctx,
BN_MONT_CTX *mont)
{
if (!BN_mod_exp_mont(w, w, a1_odd, a, ctx, mont)) /* w := w^a1_odd mod a */
return -1;
if (BN_is_one(w))
return 0; /* probably prime */
if (BN_cmp(w, a1) == 0)
return 0; /* w == -1 (mod a), 'a' is probably prime */
while (--k) {
if (!BN_mod_mul(w, w, w, a, ctx)) /* w := w^2 mod a */
return -1;
if (BN_is_one(w))
return 1; /* 'a' is composite, otherwise a previous 'w'
* would have been == -1 (mod 'a') */
if (BN_cmp(w, a1) == 0)
return 0; /* w == -1 (mod a), 'a' is probably prime */
}
/*
* If we get here, 'w' is the (a-1)/2-th power of the original 'w', and
* it is neither -1 nor +1 -- so 'a' cannot be prime
*/
bn_check_top(w);
return 1;
}
static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods)
{
int i;
BN_ULONG delta;
BN_ULONG maxdelta = BN_MASK2 - primes[NUMPRIMES - 1];
again:
/* TODO: Not all primes are private */
if (!BN_priv_rand(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD))
return 0;
if (safe && !BN_set_bit(rnd, 1))
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;
}
delta = 0;
loop:
for (i = 1; i < NUMPRIMES; 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;
}
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;
BN_ULONG maxdelta = BN_MASK2 - primes[NUMPRIMES - 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(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD))
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 < NUMPRIMES; 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 < NUMPRIMES; 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;
}
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