503 lines
12 KiB
C
503 lines
12 KiB
C
/* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
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* and Bodo Moeller for the OpenSSL project. */
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/* ====================================================================
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* Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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*
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* 1. Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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*
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in
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* the documentation and/or other materials provided with the
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* distribution.
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*
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* 3. All advertising materials mentioning features or use of this
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* software must display the following acknowledgment:
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* "This product includes software developed by the OpenSSL Project
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* for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
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*
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* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
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* endorse or promote products derived from this software without
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* prior written permission. For written permission, please contact
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* openssl-core@openssl.org.
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*
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* 5. Products derived from this software may not be called "OpenSSL"
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* nor may "OpenSSL" appear in their names without prior written
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* permission of the OpenSSL Project.
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*
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* 6. Redistributions of any form whatsoever must retain the following
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* acknowledgment:
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* "This product includes software developed by the OpenSSL Project
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* for use in the OpenSSL Toolkit (http://www.openssl.org/)"
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*
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* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
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* EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
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* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
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* ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
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* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
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* STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
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* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
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* OF THE POSSIBILITY OF SUCH DAMAGE.
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* ====================================================================
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*
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* This product includes cryptographic software written by Eric Young
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* (eay@cryptsoft.com). This product includes software written by Tim
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* Hudson (tjh@cryptsoft.com). */
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#include <openssl/bn.h>
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#include <openssl/err.h>
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#include "internal.h"
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BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) {
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// Compute a square root of |a| mod |p| using the Tonelli/Shanks algorithm
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// (cf. Henri Cohen, "A Course in Algebraic Computational Number Theory",
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// algorithm 1.5.1). |p| is assumed to be a prime.
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BIGNUM *ret = in;
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int err = 1;
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int r;
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BIGNUM *A, *b, *q, *t, *x, *y;
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int e, i, j;
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if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
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if (BN_abs_is_word(p, 2)) {
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if (ret == NULL) {
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ret = BN_new();
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}
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if (ret == NULL) {
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goto end;
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}
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if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
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if (ret != in) {
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BN_free(ret);
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}
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return NULL;
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}
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return ret;
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}
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OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
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return (NULL);
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}
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if (BN_is_zero(a) || BN_is_one(a)) {
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if (ret == NULL) {
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ret = BN_new();
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}
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if (ret == NULL) {
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goto end;
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}
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if (!BN_set_word(ret, BN_is_one(a))) {
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if (ret != in) {
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BN_free(ret);
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}
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return NULL;
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}
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return ret;
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}
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BN_CTX_start(ctx);
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A = BN_CTX_get(ctx);
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b = BN_CTX_get(ctx);
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q = BN_CTX_get(ctx);
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t = BN_CTX_get(ctx);
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x = BN_CTX_get(ctx);
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y = BN_CTX_get(ctx);
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if (y == NULL) {
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goto end;
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}
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if (ret == NULL) {
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ret = BN_new();
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}
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if (ret == NULL) {
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goto end;
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}
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// A = a mod p
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if (!BN_nnmod(A, a, p, ctx)) {
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goto end;
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}
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// now write |p| - 1 as 2^e*q where q is odd
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e = 1;
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while (!BN_is_bit_set(p, e)) {
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e++;
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}
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// we'll set q later (if needed)
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if (e == 1) {
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// The easy case: (|p|-1)/2 is odd, so 2 has an inverse
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// modulo (|p|-1)/2, and square roots can be computed
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// directly by modular exponentiation.
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// We have
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// 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2),
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// so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1.
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if (!BN_rshift(q, p, 2)) {
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goto end;
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}
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q->neg = 0;
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if (!BN_add_word(q, 1) ||
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!BN_mod_exp_mont(ret, A, q, p, ctx, NULL)) {
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goto end;
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}
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err = 0;
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goto vrfy;
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}
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if (e == 2) {
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// |p| == 5 (mod 8)
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//
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// In this case 2 is always a non-square since
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// Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
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// So if a really is a square, then 2*a is a non-square.
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// Thus for
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// b := (2*a)^((|p|-5)/8),
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// i := (2*a)*b^2
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// we have
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// i^2 = (2*a)^((1 + (|p|-5)/4)*2)
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// = (2*a)^((p-1)/2)
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// = -1;
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// so if we set
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// x := a*b*(i-1),
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// then
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// x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
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// = a^2 * b^2 * (-2*i)
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// = a*(-i)*(2*a*b^2)
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// = a*(-i)*i
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// = a.
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//
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// (This is due to A.O.L. Atkin,
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// <URL:
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//http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
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// November 1992.)
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// t := 2*a
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if (!BN_mod_lshift1_quick(t, A, p)) {
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goto end;
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}
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// b := (2*a)^((|p|-5)/8)
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if (!BN_rshift(q, p, 3)) {
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goto end;
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}
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q->neg = 0;
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if (!BN_mod_exp_mont(b, t, q, p, ctx, NULL)) {
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goto end;
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}
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// y := b^2
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if (!BN_mod_sqr(y, b, p, ctx)) {
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goto end;
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}
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// t := (2*a)*b^2 - 1
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if (!BN_mod_mul(t, t, y, p, ctx) ||
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!BN_sub_word(t, 1)) {
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goto end;
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}
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// x = a*b*t
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if (!BN_mod_mul(x, A, b, p, ctx) ||
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!BN_mod_mul(x, x, t, p, ctx)) {
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goto end;
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}
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if (!BN_copy(ret, x)) {
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goto end;
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}
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err = 0;
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goto vrfy;
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}
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// e > 2, so we really have to use the Tonelli/Shanks algorithm.
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// First, find some y that is not a square.
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if (!BN_copy(q, p)) {
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goto end; // use 'q' as temp
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}
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q->neg = 0;
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i = 2;
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do {
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// For efficiency, try small numbers first;
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// if this fails, try random numbers.
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if (i < 22) {
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if (!BN_set_word(y, i)) {
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goto end;
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}
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} else {
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if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) {
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goto end;
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}
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if (BN_ucmp(y, p) >= 0) {
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if (!(p->neg ? BN_add : BN_sub)(y, y, p)) {
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goto end;
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}
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}
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// now 0 <= y < |p|
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if (BN_is_zero(y)) {
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if (!BN_set_word(y, i)) {
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goto end;
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}
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}
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}
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r = bn_jacobi(y, q, ctx); // here 'q' is |p|
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if (r < -1) {
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goto end;
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}
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if (r == 0) {
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// m divides p
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OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
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goto end;
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}
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} while (r == 1 && ++i < 82);
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if (r != -1) {
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// Many rounds and still no non-square -- this is more likely
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// a bug than just bad luck.
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// Even if p is not prime, we should have found some y
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// such that r == -1.
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OPENSSL_PUT_ERROR(BN, BN_R_TOO_MANY_ITERATIONS);
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goto end;
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}
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// Here's our actual 'q':
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if (!BN_rshift(q, q, e)) {
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goto end;
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}
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// Now that we have some non-square, we can find an element
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// of order 2^e by computing its q'th power.
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if (!BN_mod_exp_mont(y, y, q, p, ctx, NULL)) {
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goto end;
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}
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if (BN_is_one(y)) {
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OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
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goto end;
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}
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// Now we know that (if p is indeed prime) there is an integer
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// k, 0 <= k < 2^e, such that
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//
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// a^q * y^k == 1 (mod p).
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//
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// As a^q is a square and y is not, k must be even.
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// q+1 is even, too, so there is an element
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//
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// X := a^((q+1)/2) * y^(k/2),
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//
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// and it satisfies
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//
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// X^2 = a^q * a * y^k
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// = a,
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//
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// so it is the square root that we are looking for.
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// t := (q-1)/2 (note that q is odd)
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if (!BN_rshift1(t, q)) {
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goto end;
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}
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// x := a^((q-1)/2)
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if (BN_is_zero(t)) // special case: p = 2^e + 1
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{
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if (!BN_nnmod(t, A, p, ctx)) {
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goto end;
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}
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if (BN_is_zero(t)) {
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// special case: a == 0 (mod p)
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BN_zero(ret);
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err = 0;
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goto end;
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} else if (!BN_one(x)) {
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goto end;
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}
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} else {
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if (!BN_mod_exp_mont(x, A, t, p, ctx, NULL)) {
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goto end;
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}
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if (BN_is_zero(x)) {
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// special case: a == 0 (mod p)
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BN_zero(ret);
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err = 0;
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goto end;
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}
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}
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// b := a*x^2 (= a^q)
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if (!BN_mod_sqr(b, x, p, ctx) ||
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!BN_mod_mul(b, b, A, p, ctx)) {
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goto end;
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}
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// x := a*x (= a^((q+1)/2))
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if (!BN_mod_mul(x, x, A, p, ctx)) {
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goto end;
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}
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while (1) {
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// Now b is a^q * y^k for some even k (0 <= k < 2^E
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// where E refers to the original value of e, which we
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// don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
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//
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// We have a*b = x^2,
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// y^2^(e-1) = -1,
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// b^2^(e-1) = 1.
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if (BN_is_one(b)) {
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if (!BN_copy(ret, x)) {
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goto end;
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}
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err = 0;
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goto vrfy;
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}
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// find smallest i such that b^(2^i) = 1
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i = 1;
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if (!BN_mod_sqr(t, b, p, ctx)) {
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goto end;
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}
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while (!BN_is_one(t)) {
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i++;
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if (i == e) {
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OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
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goto end;
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}
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if (!BN_mod_mul(t, t, t, p, ctx)) {
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goto end;
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}
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}
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// t := y^2^(e - i - 1)
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if (!BN_copy(t, y)) {
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goto end;
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}
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for (j = e - i - 1; j > 0; j--) {
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if (!BN_mod_sqr(t, t, p, ctx)) {
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goto end;
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}
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}
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if (!BN_mod_mul(y, t, t, p, ctx) ||
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!BN_mod_mul(x, x, t, p, ctx) ||
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!BN_mod_mul(b, b, y, p, ctx)) {
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goto end;
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}
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e = i;
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}
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vrfy:
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if (!err) {
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// verify the result -- the input might have been not a square
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// (test added in 0.9.8)
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if (!BN_mod_sqr(x, ret, p, ctx)) {
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err = 1;
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}
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if (!err && 0 != BN_cmp(x, A)) {
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OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
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err = 1;
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}
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}
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end:
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if (err) {
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if (ret != in) {
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BN_clear_free(ret);
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}
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ret = NULL;
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}
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BN_CTX_end(ctx);
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return ret;
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}
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int BN_sqrt(BIGNUM *out_sqrt, const BIGNUM *in, BN_CTX *ctx) {
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BIGNUM *estimate, *tmp, *delta, *last_delta, *tmp2;
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int ok = 0, last_delta_valid = 0;
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if (in->neg) {
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OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER);
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return 0;
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}
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if (BN_is_zero(in)) {
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BN_zero(out_sqrt);
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return 1;
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}
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BN_CTX_start(ctx);
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if (out_sqrt == in) {
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estimate = BN_CTX_get(ctx);
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} else {
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estimate = out_sqrt;
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}
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tmp = BN_CTX_get(ctx);
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last_delta = BN_CTX_get(ctx);
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delta = BN_CTX_get(ctx);
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if (estimate == NULL || tmp == NULL || last_delta == NULL || delta == NULL) {
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OPENSSL_PUT_ERROR(BN, ERR_R_MALLOC_FAILURE);
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goto err;
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}
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// We estimate that the square root of an n-bit number is 2^{n/2}.
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if (!BN_lshift(estimate, BN_value_one(), BN_num_bits(in)/2)) {
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goto err;
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}
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// This is Newton's method for finding a root of the equation |estimate|^2 -
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// |in| = 0.
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for (;;) {
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// |estimate| = 1/2 * (|estimate| + |in|/|estimate|)
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if (!BN_div(tmp, NULL, in, estimate, ctx) ||
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!BN_add(tmp, tmp, estimate) ||
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!BN_rshift1(estimate, tmp) ||
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// |tmp| = |estimate|^2
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!BN_sqr(tmp, estimate, ctx) ||
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// |delta| = |in| - |tmp|
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!BN_sub(delta, in, tmp)) {
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OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB);
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goto err;
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}
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delta->neg = 0;
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// The difference between |in| and |estimate| squared is required to always
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// decrease. This ensures that the loop always terminates, but I don't have
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// a proof that it always finds the square root for a given square.
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if (last_delta_valid && BN_cmp(delta, last_delta) >= 0) {
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break;
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}
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last_delta_valid = 1;
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tmp2 = last_delta;
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last_delta = delta;
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delta = tmp2;
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}
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if (BN_cmp(tmp, in) != 0) {
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OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
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goto err;
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}
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ok = 1;
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err:
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if (ok && out_sqrt == in && !BN_copy(out_sqrt, estimate)) {
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ok = 0;
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}
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BN_CTX_end(ctx);
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return ok;
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}
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