Mirror of BoringSSL (grpc依赖)
https://boringssl.googlesource.com/boringssl
You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
502 lines
12 KiB
502 lines
12 KiB
/* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de> |
|
* and Bodo Moeller for the OpenSSL project. */ |
|
/* ==================================================================== |
|
* Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved. |
|
* |
|
* Redistribution and use in source and binary forms, with or without |
|
* modification, are permitted provided that the following conditions |
|
* are met: |
|
* |
|
* 1. Redistributions of source code must retain the above copyright |
|
* notice, this list of conditions and the following disclaimer. |
|
* |
|
* 2. Redistributions in binary form must reproduce the above copyright |
|
* notice, this list of conditions and the following disclaimer in |
|
* the documentation and/or other materials provided with the |
|
* distribution. |
|
* |
|
* 3. All advertising materials mentioning features or use of this |
|
* software must display the following acknowledgment: |
|
* "This product includes software developed by the OpenSSL Project |
|
* for use in the OpenSSL Toolkit. (http://www.openssl.org/)" |
|
* |
|
* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to |
|
* endorse or promote products derived from this software without |
|
* prior written permission. For written permission, please contact |
|
* openssl-core@openssl.org. |
|
* |
|
* 5. Products derived from this software may not be called "OpenSSL" |
|
* nor may "OpenSSL" appear in their names without prior written |
|
* permission of the OpenSSL Project. |
|
* |
|
* 6. Redistributions of any form whatsoever must retain the following |
|
* acknowledgment: |
|
* "This product includes software developed by the OpenSSL Project |
|
* for use in the OpenSSL Toolkit (http://www.openssl.org/)" |
|
* |
|
* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY |
|
* EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE |
|
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR |
|
* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR |
|
* ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, |
|
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT |
|
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; |
|
* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) |
|
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, |
|
* STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) |
|
* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED |
|
* OF THE POSSIBILITY OF SUCH DAMAGE. |
|
* ==================================================================== |
|
* |
|
* This product includes cryptographic software written by Eric Young |
|
* (eay@cryptsoft.com). This product includes software written by Tim |
|
* Hudson (tjh@cryptsoft.com). */ |
|
|
|
#include <openssl/bn.h> |
|
|
|
#include <openssl/err.h> |
|
|
|
#include "internal.h" |
|
|
|
|
|
BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) { |
|
// Compute a square root of |a| mod |p| using the Tonelli/Shanks algorithm |
|
// (cf. Henri Cohen, "A Course in Algebraic Computational Number Theory", |
|
// algorithm 1.5.1). |p| is assumed to be a prime. |
|
|
|
BIGNUM *ret = in; |
|
int err = 1; |
|
int r; |
|
BIGNUM *A, *b, *q, *t, *x, *y; |
|
int e, i, j; |
|
|
|
if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) { |
|
if (BN_abs_is_word(p, 2)) { |
|
if (ret == NULL) { |
|
ret = BN_new(); |
|
} |
|
if (ret == NULL) { |
|
goto end; |
|
} |
|
if (!BN_set_word(ret, BN_is_bit_set(a, 0))) { |
|
if (ret != in) { |
|
BN_free(ret); |
|
} |
|
return NULL; |
|
} |
|
return ret; |
|
} |
|
|
|
OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME); |
|
return (NULL); |
|
} |
|
|
|
if (BN_is_zero(a) || BN_is_one(a)) { |
|
if (ret == NULL) { |
|
ret = BN_new(); |
|
} |
|
if (ret == NULL) { |
|
goto end; |
|
} |
|
if (!BN_set_word(ret, BN_is_one(a))) { |
|
if (ret != in) { |
|
BN_free(ret); |
|
} |
|
return NULL; |
|
} |
|
return ret; |
|
} |
|
|
|
BN_CTX_start(ctx); |
|
A = BN_CTX_get(ctx); |
|
b = BN_CTX_get(ctx); |
|
q = BN_CTX_get(ctx); |
|
t = BN_CTX_get(ctx); |
|
x = BN_CTX_get(ctx); |
|
y = BN_CTX_get(ctx); |
|
if (y == NULL) { |
|
goto end; |
|
} |
|
|
|
if (ret == NULL) { |
|
ret = BN_new(); |
|
} |
|
if (ret == NULL) { |
|
goto end; |
|
} |
|
|
|
// A = a mod p |
|
if (!BN_nnmod(A, a, p, ctx)) { |
|
goto end; |
|
} |
|
|
|
// now write |p| - 1 as 2^e*q where q is odd |
|
e = 1; |
|
while (!BN_is_bit_set(p, e)) { |
|
e++; |
|
} |
|
// we'll set q later (if needed) |
|
|
|
if (e == 1) { |
|
// The easy case: (|p|-1)/2 is odd, so 2 has an inverse |
|
// modulo (|p|-1)/2, and square roots can be computed |
|
// directly by modular exponentiation. |
|
// We have |
|
// 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2), |
|
// so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1. |
|
if (!BN_rshift(q, p, 2)) { |
|
goto end; |
|
} |
|
q->neg = 0; |
|
if (!BN_add_word(q, 1) || |
|
!BN_mod_exp_mont(ret, A, q, p, ctx, NULL)) { |
|
goto end; |
|
} |
|
err = 0; |
|
goto vrfy; |
|
} |
|
|
|
if (e == 2) { |
|
// |p| == 5 (mod 8) |
|
// |
|
// In this case 2 is always a non-square since |
|
// Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime. |
|
// So if a really is a square, then 2*a is a non-square. |
|
// Thus for |
|
// b := (2*a)^((|p|-5)/8), |
|
// i := (2*a)*b^2 |
|
// we have |
|
// i^2 = (2*a)^((1 + (|p|-5)/4)*2) |
|
// = (2*a)^((p-1)/2) |
|
// = -1; |
|
// so if we set |
|
// x := a*b*(i-1), |
|
// then |
|
// x^2 = a^2 * b^2 * (i^2 - 2*i + 1) |
|
// = a^2 * b^2 * (-2*i) |
|
// = a*(-i)*(2*a*b^2) |
|
// = a*(-i)*i |
|
// = a. |
|
// |
|
// (This is due to A.O.L. Atkin, |
|
// <URL: |
|
//http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>, |
|
// November 1992.) |
|
|
|
// t := 2*a |
|
if (!bn_mod_lshift1_consttime(t, A, p, ctx)) { |
|
goto end; |
|
} |
|
|
|
// b := (2*a)^((|p|-5)/8) |
|
if (!BN_rshift(q, p, 3)) { |
|
goto end; |
|
} |
|
q->neg = 0; |
|
if (!BN_mod_exp_mont(b, t, q, p, ctx, NULL)) { |
|
goto end; |
|
} |
|
|
|
// y := b^2 |
|
if (!BN_mod_sqr(y, b, p, ctx)) { |
|
goto end; |
|
} |
|
|
|
// t := (2*a)*b^2 - 1 |
|
if (!BN_mod_mul(t, t, y, p, ctx) || |
|
!BN_sub_word(t, 1)) { |
|
goto end; |
|
} |
|
|
|
// x = a*b*t |
|
if (!BN_mod_mul(x, A, b, p, ctx) || |
|
!BN_mod_mul(x, x, t, p, ctx)) { |
|
goto end; |
|
} |
|
|
|
if (!BN_copy(ret, x)) { |
|
goto end; |
|
} |
|
err = 0; |
|
goto vrfy; |
|
} |
|
|
|
// e > 2, so we really have to use the Tonelli/Shanks algorithm. |
|
// First, find some y that is not a square. |
|
if (!BN_copy(q, p)) { |
|
goto end; // use 'q' as temp |
|
} |
|
q->neg = 0; |
|
i = 2; |
|
do { |
|
// For efficiency, try small numbers first; |
|
// if this fails, try random numbers. |
|
if (i < 22) { |
|
if (!BN_set_word(y, i)) { |
|
goto end; |
|
} |
|
} else { |
|
if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) { |
|
goto end; |
|
} |
|
if (BN_ucmp(y, p) >= 0) { |
|
if (!(p->neg ? BN_add : BN_sub)(y, y, p)) { |
|
goto end; |
|
} |
|
} |
|
// now 0 <= y < |p| |
|
if (BN_is_zero(y)) { |
|
if (!BN_set_word(y, i)) { |
|
goto end; |
|
} |
|
} |
|
} |
|
|
|
r = bn_jacobi(y, q, ctx); // here 'q' is |p| |
|
if (r < -1) { |
|
goto end; |
|
} |
|
if (r == 0) { |
|
// m divides p |
|
OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME); |
|
goto end; |
|
} |
|
} while (r == 1 && ++i < 82); |
|
|
|
if (r != -1) { |
|
// Many rounds and still no non-square -- this is more likely |
|
// a bug than just bad luck. |
|
// Even if p is not prime, we should have found some y |
|
// such that r == -1. |
|
OPENSSL_PUT_ERROR(BN, BN_R_TOO_MANY_ITERATIONS); |
|
goto end; |
|
} |
|
|
|
// Here's our actual 'q': |
|
if (!BN_rshift(q, q, e)) { |
|
goto end; |
|
} |
|
|
|
// Now that we have some non-square, we can find an element |
|
// of order 2^e by computing its q'th power. |
|
if (!BN_mod_exp_mont(y, y, q, p, ctx, NULL)) { |
|
goto end; |
|
} |
|
if (BN_is_one(y)) { |
|
OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME); |
|
goto end; |
|
} |
|
|
|
// Now we know that (if p is indeed prime) there is an integer |
|
// k, 0 <= k < 2^e, such that |
|
// |
|
// a^q * y^k == 1 (mod p). |
|
// |
|
// As a^q is a square and y is not, k must be even. |
|
// q+1 is even, too, so there is an element |
|
// |
|
// X := a^((q+1)/2) * y^(k/2), |
|
// |
|
// and it satisfies |
|
// |
|
// X^2 = a^q * a * y^k |
|
// = a, |
|
// |
|
// so it is the square root that we are looking for. |
|
|
|
// t := (q-1)/2 (note that q is odd) |
|
if (!BN_rshift1(t, q)) { |
|
goto end; |
|
} |
|
|
|
// x := a^((q-1)/2) |
|
if (BN_is_zero(t)) // special case: p = 2^e + 1 |
|
{ |
|
if (!BN_nnmod(t, A, p, ctx)) { |
|
goto end; |
|
} |
|
if (BN_is_zero(t)) { |
|
// special case: a == 0 (mod p) |
|
BN_zero(ret); |
|
err = 0; |
|
goto end; |
|
} else if (!BN_one(x)) { |
|
goto end; |
|
} |
|
} else { |
|
if (!BN_mod_exp_mont(x, A, t, p, ctx, NULL)) { |
|
goto end; |
|
} |
|
if (BN_is_zero(x)) { |
|
// special case: a == 0 (mod p) |
|
BN_zero(ret); |
|
err = 0; |
|
goto end; |
|
} |
|
} |
|
|
|
// b := a*x^2 (= a^q) |
|
if (!BN_mod_sqr(b, x, p, ctx) || |
|
!BN_mod_mul(b, b, A, p, ctx)) { |
|
goto end; |
|
} |
|
|
|
// x := a*x (= a^((q+1)/2)) |
|
if (!BN_mod_mul(x, x, A, p, ctx)) { |
|
goto end; |
|
} |
|
|
|
while (1) { |
|
// Now b is a^q * y^k for some even k (0 <= k < 2^E |
|
// where E refers to the original value of e, which we |
|
// don't keep in a variable), and x is a^((q+1)/2) * y^(k/2). |
|
// |
|
// We have a*b = x^2, |
|
// y^2^(e-1) = -1, |
|
// b^2^(e-1) = 1. |
|
|
|
if (BN_is_one(b)) { |
|
if (!BN_copy(ret, x)) { |
|
goto end; |
|
} |
|
err = 0; |
|
goto vrfy; |
|
} |
|
|
|
|
|
// find smallest i such that b^(2^i) = 1 |
|
i = 1; |
|
if (!BN_mod_sqr(t, b, p, ctx)) { |
|
goto end; |
|
} |
|
while (!BN_is_one(t)) { |
|
i++; |
|
if (i == e) { |
|
OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE); |
|
goto end; |
|
} |
|
if (!BN_mod_mul(t, t, t, p, ctx)) { |
|
goto end; |
|
} |
|
} |
|
|
|
|
|
// t := y^2^(e - i - 1) |
|
if (!BN_copy(t, y)) { |
|
goto end; |
|
} |
|
for (j = e - i - 1; j > 0; j--) { |
|
if (!BN_mod_sqr(t, t, p, ctx)) { |
|
goto end; |
|
} |
|
} |
|
if (!BN_mod_mul(y, t, t, p, ctx) || |
|
!BN_mod_mul(x, x, t, p, ctx) || |
|
!BN_mod_mul(b, b, y, p, ctx)) { |
|
goto end; |
|
} |
|
e = i; |
|
} |
|
|
|
vrfy: |
|
if (!err) { |
|
// verify the result -- the input might have been not a square |
|
// (test added in 0.9.8) |
|
|
|
if (!BN_mod_sqr(x, ret, p, ctx)) { |
|
err = 1; |
|
} |
|
|
|
if (!err && 0 != BN_cmp(x, A)) { |
|
OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE); |
|
err = 1; |
|
} |
|
} |
|
|
|
end: |
|
if (err) { |
|
if (ret != in) { |
|
BN_clear_free(ret); |
|
} |
|
ret = NULL; |
|
} |
|
BN_CTX_end(ctx); |
|
return ret; |
|
} |
|
|
|
int BN_sqrt(BIGNUM *out_sqrt, const BIGNUM *in, BN_CTX *ctx) { |
|
BIGNUM *estimate, *tmp, *delta, *last_delta, *tmp2; |
|
int ok = 0, last_delta_valid = 0; |
|
|
|
if (in->neg) { |
|
OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER); |
|
return 0; |
|
} |
|
if (BN_is_zero(in)) { |
|
BN_zero(out_sqrt); |
|
return 1; |
|
} |
|
|
|
BN_CTX_start(ctx); |
|
if (out_sqrt == in) { |
|
estimate = BN_CTX_get(ctx); |
|
} else { |
|
estimate = out_sqrt; |
|
} |
|
tmp = BN_CTX_get(ctx); |
|
last_delta = BN_CTX_get(ctx); |
|
delta = BN_CTX_get(ctx); |
|
if (estimate == NULL || tmp == NULL || last_delta == NULL || delta == NULL) { |
|
OPENSSL_PUT_ERROR(BN, ERR_R_MALLOC_FAILURE); |
|
goto err; |
|
} |
|
|
|
// We estimate that the square root of an n-bit number is 2^{n/2}. |
|
if (!BN_lshift(estimate, BN_value_one(), BN_num_bits(in)/2)) { |
|
goto err; |
|
} |
|
|
|
// This is Newton's method for finding a root of the equation |estimate|^2 - |
|
// |in| = 0. |
|
for (;;) { |
|
// |estimate| = 1/2 * (|estimate| + |in|/|estimate|) |
|
if (!BN_div(tmp, NULL, in, estimate, ctx) || |
|
!BN_add(tmp, tmp, estimate) || |
|
!BN_rshift1(estimate, tmp) || |
|
// |tmp| = |estimate|^2 |
|
!BN_sqr(tmp, estimate, ctx) || |
|
// |delta| = |in| - |tmp| |
|
!BN_sub(delta, in, tmp)) { |
|
OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB); |
|
goto err; |
|
} |
|
|
|
delta->neg = 0; |
|
// The difference between |in| and |estimate| squared is required to always |
|
// decrease. This ensures that the loop always terminates, but I don't have |
|
// a proof that it always finds the square root for a given square. |
|
if (last_delta_valid && BN_cmp(delta, last_delta) >= 0) { |
|
break; |
|
} |
|
|
|
last_delta_valid = 1; |
|
|
|
tmp2 = last_delta; |
|
last_delta = delta; |
|
delta = tmp2; |
|
} |
|
|
|
if (BN_cmp(tmp, in) != 0) { |
|
OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE); |
|
goto err; |
|
} |
|
|
|
ok = 1; |
|
|
|
err: |
|
if (ok && out_sqrt == in && !BN_copy(out_sqrt, estimate)) { |
|
ok = 0; |
|
} |
|
BN_CTX_end(ctx); |
|
return ok; |
|
}
|
|
|