--- /dev/null
+/* 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;
+}
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