/* 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;

	}