-XXX
+/* crypto/bn/bn_mod.c */
+/* 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 "cryptlib.h"
+#include "bn_lcl.h"
+
+
+BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
+/* Returns 'ret' such that
+ * ret^2 == a (mod p),
+ * using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
+ * in Algebraic Computational Number Theory", algorithm 1.5.1).
+ * 'p' must be prime!
+ */
+ {
+ BIGNUM *ret = in;
+ int err = 1;
+ int r;
+ BIGNUM *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)))
+ {
+ BN_free(ret);
+ return NULL;
+ }
+ return ret;
+ }
+
+ BNerr(BN_F_BN_MOD_SQRT, 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)))
+ {
+ BN_free(ret);
+ return NULL;
+ }
+ return ret;
+ }
+
+#if 0 /* if BN_mod_sqrt is used with correct input, this just wastes time */
+ r = BN_kronecker(a, p, ctx);
+ if (r < -1) return NULL;
+ if (r == -1)
+ {
+ BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
+ return(NULL);
+ }
+#endif
+
+ BN_CTX_start(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;
+
+ /* now write |p| - 1 as 2^e*q where q is odd */
+ e = 1;
+ while (!BN_is_bit_set(p, e))
+ e++;
+ if (e > 2)
+ /* we don't need this q if e = 1 or 2 */
+ if (!BN_rshift(q, p, e)) goto end;
+ q->neg = 0;
+
+ 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;
+ if (!BN_add_word(q, 1)) goto end;
+ if (!BN_mod_exp(ret, a, q, p, ctx)) goto end;
+ err = 0;
+ goto end;
+ }
+
+ 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.)
+ */
+
+ /* make sure that a is reduced modulo p */
+ if (a->neg || BN_ucmp(a, p) >= 0)
+ {
+ if (!BN_nnmod(x, a, p, ctx)) goto end;
+ a = x; /* use x as temporary variable */
+ }
+
+ /* t := 2*a */
+ if (!BN_mod_lshift1_quick(t, a, p)) goto end;
+
+ /* b := (2*a)^((p-5)/8) */
+ if (!BN_rshift(q, p, 3)) goto end;
+ if (!BN_mod_exp(b, t, q, p, ctx)) 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)) goto end;
+ if (!BN_sub_word(t, 1)) goto end;
+
+ /* x = a*b*t */
+ if (!BN_mod_mul(x, a, b, p, ctx)) goto end;
+ if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
+
+ if (!BN_copy(ret, x)) goto end;
+ err = 0;
+ goto end;
+ }
+
+ /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
+ * First, find some y that is not a square. */
+ 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_kronecker(y, p, ctx);
+ if (r < -1) goto end;
+ if (r == 0)
+ {
+ /* m divides p */
+ BNerr(BN_F_BN_MOD_SQRT, 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.
+ */
+ BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
+ 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(y, y, q, p, ctx)) goto end;
+ if (BN_is_one(y))
+ {
+ BNerr(BN_F_BN_MOD_SQRT, 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) */
+ if (!BN_zero(ret)) goto end;
+ err = 0;
+ goto end;
+ }
+ else
+ if (!BN_one(x)) goto end;
+ }
+ else
+ {
+ if (!BN_mod_exp(x, a, t, p, ctx)) goto end;
+ if (BN_is_zero(x))
+ {
+ /* special case: a == 0 (mod p) */
+ if (!BN_zero(ret)) goto end;
+ err = 0;
+ goto end;
+ }
+ }
+
+ /* b := a*x^2 (= a^q) */
+ if (!BN_mod_sqr(b, x, p, ctx)) goto end;
+ if (!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 end;
+ }
+
+
+ /* 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)
+ {
+ BNerr(BN_F_BN_MOD_SQRT, 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)) goto end;
+ if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
+ if (!BN_mod_mul(b, b, y, p, ctx)) goto end;
+ e = i;
+ }
+
+ end:
+ if (err)
+ {
+ if (ret != NULL && ret != in)
+ {
+ BN_clear_free(ret);
+ }
+ ret = NULL;
+ }
+ BN_CTX_end(ctx);
+ return ret;
+ }
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