/* crypto/bn/bn_sqrt.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"

 EXPORT_C 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 *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;

 				}

 			bn_check_top(ret);

 			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)))

 			{

 			if (ret != in)

 				BN_free(ret);

 			return NULL;

 			}

 		bn_check_top(ret);

 		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)) goto end;

 		if (!BN_mod_exp(ret, A, q, p, ctx)) 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_quick(t, A, p)) goto end;

 		/* b := (2*a)^((|p|-5)/8) */

 		if (!BN_rshift(q, p, 3)) goto end;

 		q->neg = 0;

 		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 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_kronecker(y, q, ctx); /* here 'q' is |p| */

 		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;

 		}

 	/* 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(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) */

 			BN_zero(ret);

 			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) */

 			BN_zero(ret);

 			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 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)

 				{

 				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;

 		}

  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))

 			{

 			BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);

 			err = 1;

 			}

 		}

  end:

 	if (err)

 		{

 		if (ret != NULL && ret != in)

 			{

 			BN_clear_free(ret);

 			}

 		ret = NULL;

 		}

 	BN_CTX_end(ctx);

 	bn_check_top(ret);

 	return ret;

 	}