[thirdparty/openssl.git] / crypto / bn / bn_sqrt.c

/* 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"


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;
	}
Commit	Line	Data
fea4280a	1	/* crypto/bn/bn_sqrt.c */
cd2eebfd BM	2	/* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
	3	* and Bodo Moeller for the OpenSSL project. */
	4	/* ====================================================================
	5	* Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved.
	6	*
	7	* Redistribution and use in source and binary forms, with or without
	8	* modification, are permitted provided that the following conditions
	9	* are met:
	10	*
	11	* 1. Redistributions of source code must retain the above copyright
	12	* notice, this list of conditions and the following disclaimer.
	13	*
	14	* 2. Redistributions in binary form must reproduce the above copyright
	15	* notice, this list of conditions and the following disclaimer in
	16	* the documentation and/or other materials provided with the
	17	* distribution.
	18	*
	19	* 3. All advertising materials mentioning features or use of this
	20	* software must display the following acknowledgment:
	21	* "This product includes software developed by the OpenSSL Project
	22	* for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
	23	*
	24	* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
	25	* endorse or promote products derived from this software without
	26	* prior written permission. For written permission, please contact
	27	* openssl-core@openssl.org.
	28	*
	29	* 5. Products derived from this software may not be called "OpenSSL"
	30	* nor may "OpenSSL" appear in their names without prior written
	31	* permission of the OpenSSL Project.
	32	*
	33	* 6. Redistributions of any form whatsoever must retain the following
	34	* acknowledgment:
	35	* "This product includes software developed by the OpenSSL Project
	36	* for use in the OpenSSL Toolkit (http://www.openssl.org/)"
	37	*
	38	* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
	39	* EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
	40	* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
	41	* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
	42	* ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
	43	* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
	44	* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
	45	* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
	46	* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
	47	* STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
	48	* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
	49	* OF THE POSSIBILITY OF SUCH DAMAGE.
	50	* ====================================================================
	51	*
	52	* This product includes cryptographic software written by Eric Young
	53	* (eay@cryptsoft.com). This product includes software written by Tim
	54	* Hudson (tjh@cryptsoft.com).
	55	*
	56	*/
	57
	58	#include "cryptlib.h"
	59	#include "bn_lcl.h"
	60
	61
	62	BIGNUM BN_mod_sqrt(BIGNUM in, const BIGNUM a, const BIGNUM p, BN_CTX *ctx)
	63	/* Returns 'ret' such that
	64	* ret^2 == a (mod p),
	65	* using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
66	* in Algebraic Computational Number Theory", algorithm 1.5.1).
67	* 'p' must be prime!
68	*/
69	{
70	BIGNUM *ret = in;
71	int err = 1;
72	int r;
6fb60a84	73	BIGNUM A, b, q, t, x, y;
cd2eebfd BM	74	int e, i, j;
	75
	76	if (!BN_is_odd(p) \|\| BN_abs_is_word(p, 1))
	77	{
	78	if (BN_abs_is_word(p, 2))
	79	{
	80	if (ret == NULL)
	81	ret = BN_new();
	82	if (ret == NULL)
	83	goto end;
	84	if (!BN_set_word(ret, BN_is_bit_set(a, 0)))
	85	{
7534d131 BM	86	if (ret != in)
7534d131 BM	87	BN_free(ret);
cd2eebfd BM	88	return NULL;
cd2eebfd BM	89	}
d870740c	90	bn_check_top(ret);
cd2eebfd BM	91	return ret;
	92	}
	93
	94	BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
	95	return(NULL);
	96	}
	97
bac68541 BM	98	if (BN_is_zero(a) \|\| BN_is_one(a))
	99	{
	100	if (ret == NULL)
	101	ret = BN_new();
	102	if (ret == NULL)
	103	goto end;
	104	if (!BN_set_word(ret, BN_is_one(a)))
	105	{
7534d131 BM	106	if (ret != in)
7534d131 BM	107	BN_free(ret);
bac68541 BM	108	return NULL;
bac68541 BM	109	}
d870740c	110	bn_check_top(ret);
bac68541 BM	111	return ret;
	112	}
	113
cd2eebfd	114	BN_CTX_start(ctx);
6fb60a84	115	A = BN_CTX_get(ctx);
cd2eebfd BM	116	b = BN_CTX_get(ctx);
	117	q = BN_CTX_get(ctx);
	118	t = BN_CTX_get(ctx);
	119	x = BN_CTX_get(ctx);
	120	y = BN_CTX_get(ctx);
	121	if (y == NULL) goto end;
	122
	123	if (ret == NULL)
	124	ret = BN_new();
	125	if (ret == NULL) goto end;
	126
6fb60a84 BM	127	/* A = a mod p */
	128	if (!BN_nnmod(A, a, p, ctx)) goto end;
	129
cd2eebfd BM	130	/* now write \|p\| - 1 as 2^eq where q is odd /
	131	e = 1;
	132	while (!BN_is_bit_set(p, e))
	133	e++;
80d89e6a	134	/* we'll set q later (if needed) */
cd2eebfd BM	135
	136	if (e == 1)
	137	{
3e9a08ec TH	138	/*-
3e9a08ec TH	139	* The easy case: (\|p\|-1)/2 is odd, so 2 has an inverse
80d89e6a	140	* modulo (\|p\|-1)/2, and square roots can be computed
cd2eebfd BM	141	* directly by modular exponentiation.
cd2eebfd BM	142	* We have
80d89e6a BM	143	* 2 * (\|p\|+1)/4 == 1 (mod (\|p\|-1)/2),
80d89e6a BM	144	* so we can use exponent (\|p\|+1)/4, i.e. (\|p\|-3)/4 + 1.
cd2eebfd	145	*/
bac68541	146	if (!BN_rshift(q, p, 2)) goto end;
bc5f2740	147	q->neg = 0;
bac68541	148	if (!BN_add_word(q, 1)) goto end;
6fb60a84	149	if (!BN_mod_exp(ret, A, q, p, ctx)) goto end;
cd2eebfd	150	err = 0;
6fb60a84	151	goto vrfy;
cd2eebfd BM	152	}
cd2eebfd BM	153
bac68541 BM	154	if (e == 2)
bac68541 BM	155	{
3e9a08ec TH	156	/*-
3e9a08ec TH	157	* \|p\| == 5 (mod 8)
bac68541 BM	158	*
	159	* In this case 2 is always a non-square since
	160	* Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
	161	* So if a really is a square, then 2*a is a non-square.
	162	* Thus for
80d89e6a	163	* b := (2*a)^((\|p\|-5)/8),
bac68541 BM	164	* i := (2a)b^2
bac68541 BM	165	* we have
80d89e6a	166	* i^2 = (2a)^((1 + (\|p\|-5)/4)2)
bac68541 BM	167	* = (2*a)^((p-1)/2)
	168	* = -1;
	169	* so if we set
	170	* x := ab(i-1),
	171	* then
	172	* x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
	173	* = a^2 * b^2 * (-2*i)
	174	* = a(-i)(2ab^2)
	175	* = a(-i)i
	176	* = a.
	177	*
	178	* (This is due to A.O.L. Atkin,
	179	* <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
	180	* November 1992.)
	181	*/
	182
bac68541	183	/* t := 2a /
6fb60a84	184	if (!BN_mod_lshift1_quick(t, A, p)) goto end;
bac68541	185
80d89e6a	186	/* b := (2a)^((\|p\|-5)/8) /
bac68541	187	if (!BN_rshift(q, p, 3)) goto end;
bc5f2740	188	q->neg = 0;
bac68541 BM	189	if (!BN_mod_exp(b, t, q, p, ctx)) goto end;
	190
	191	/* y := b^2 */
	192	if (!BN_mod_sqr(y, b, p, ctx)) goto end;
	193
	194	/* t := (2a)b^2 - 1*/
	195	if (!BN_mod_mul(t, t, y, p, ctx)) goto end;
aa66eba7	196	if (!BN_sub_word(t, 1)) goto end;
bac68541 BM	197
bac68541 BM	198	/* x = abt */
6fb60a84	199	if (!BN_mod_mul(x, A, b, p, ctx)) goto end;
bac68541 BM	200	if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
	201
	202	if (!BN_copy(ret, x)) goto end;
	203	err = 0;
6fb60a84	204	goto vrfy;
bac68541 BM	205	}
	206
	207	/* e > 2, so we really have to use the Tonelli/Shanks algorithm.
cd2eebfd	208	* First, find some y that is not a square. */
80d89e6a BM	209	if (!BN_copy(q, p)) goto end; /* use 'q' as temp */
80d89e6a BM	210	q->neg = 0;
25439b76	211	i = 2;
cd2eebfd BM	212	do
	213	{
	214	/* For efficiency, try small numbers first;
	215	* if this fails, try random numbers.
	216	*/
25439b76	217	if (i < 22)
cd2eebfd BM	218	{
	219	if (!BN_set_word(y, i)) goto end;
	220	}
	221	else
	222	{
	223	if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) goto end;
	224	if (BN_ucmp(y, p) >= 0)
	225	{
	226	if (!(p->neg ? BN_add : BN_sub)(y, y, p)) goto end;
	227	}
	228	/* now 0 <= y < \|p\| */
	229	if (BN_is_zero(y))
	230	if (!BN_set_word(y, i)) goto end;
	231	}
	232
80d89e6a	233	r = BN_kronecker(y, q, ctx); /* here 'q' is \|p\| */
cd2eebfd BM	234	if (r < -1) goto end;
	235	if (r == 0)
	236	{
	237	/* m divides p */
	238	BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
	239	goto end;
	240	}
	241	}
25439b76	242	while (r == 1 && ++i < 82);
cd2eebfd BM	243
	244	if (r != -1)
	245	{
	246	/* Many rounds and still no non-square -- this is more likely
	247	* a bug than just bad luck.
	248	* Even if p is not prime, we should have found some y
	249	* such that r == -1.
	250	*/
	251	BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
	252	goto end;
	253	}
	254
80d89e6a BM	255	/* Here's our actual 'q': */
80d89e6a BM	256	if (!BN_rshift(q, q, e)) goto end;
cd2eebfd BM	257
	258	/* Now that we have some non-square, we can find an element
	259	* of order 2^e by computing its q'th power. */
	260	if (!BN_mod_exp(y, y, q, p, ctx)) goto end;
	261	if (BN_is_one(y))
	262	{
	263	BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
	264	goto end;
	265	}
	266
3e9a08ec TH	267	/*-
3e9a08ec TH	268	* Now we know that (if p is indeed prime) there is an integer
cd2eebfd BM	269	* k, 0 <= k < 2^e, such that
	270	*
	271	* a^q * y^k == 1 (mod p).
	272	*
	273	* As a^q is a square and y is not, k must be even.
	274	* q+1 is even, too, so there is an element
	275	*
	276	* X := a^((q+1)/2) * y^(k/2),
	277	*
	278	* and it satisfies
	279	*
	280	* X^2 = a^q * a * y^k
	281	* = a,
	282	*
	283	* so it is the square root that we are looking for.
	284	*/
	285
	286	/* t := (q-1)/2 (note that q is odd) */
	287	if (!BN_rshift1(t, q)) goto end;
	288
	289	/* x := a^((q-1)/2) */
	290	if (BN_is_zero(t)) /* special case: p = 2^e + 1 */
	291	{
6fb60a84	292	if (!BN_nnmod(t, A, p, ctx)) goto end;
cd2eebfd BM	293	if (BN_is_zero(t))
	294	{
	295	/* special case: a == 0 (mod p) */
b6358c89	296	BN_zero(ret);
cd2eebfd BM	297	err = 0;
	298	goto end;
	299	}
	300	else
	301	if (!BN_one(x)) goto end;
	302	}
	303	else
	304	{
6fb60a84	305	if (!BN_mod_exp(x, A, t, p, ctx)) goto end;
cd2eebfd BM	306	if (BN_is_zero(x))
	307	{
	308	/* special case: a == 0 (mod p) */
b6358c89	309	BN_zero(ret);
cd2eebfd BM	310	err = 0;
	311	goto end;
	312	}
	313	}
	314
	315	/* b := ax^2 (= a^q) /
	316	if (!BN_mod_sqr(b, x, p, ctx)) goto end;
6fb60a84	317	if (!BN_mod_mul(b, b, A, p, ctx)) goto end;
cd2eebfd BM	318
cd2eebfd BM	319	/* x := ax (= a^((q+1)/2)) /
6fb60a84	320	if (!BN_mod_mul(x, x, A, p, ctx)) goto end;
cd2eebfd BM	321
	322	while (1)
	323	{
3e9a08ec TH	324	/*-
3e9a08ec TH	325	* Now b is a^q * y^k for some even k (0 <= k < 2^E
cd2eebfd BM	326	* where E refers to the original value of e, which we
	327	* don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
	328	*
	329	* We have a*b = x^2,
	330	* y^2^(e-1) = -1,
	331	* b^2^(e-1) = 1.
	332	*/
	333
	334	if (BN_is_one(b))
	335	{
	336	if (!BN_copy(ret, x)) goto end;
	337	err = 0;
6fb60a84	338	goto vrfy;
cd2eebfd BM	339	}
	340
	341
	342	/* find smallest i such that b^(2^i) = 1 */
	343	i = 1;
	344	if (!BN_mod_sqr(t, b, p, ctx)) goto end;
	345	while (!BN_is_one(t))
	346	{
	347	i++;
	348	if (i == e)
	349	{
	350	BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
	351	goto end;
	352	}
	353	if (!BN_mod_mul(t, t, t, p, ctx)) goto end;
	354	}
	355
	356
	357	/* t := y^2^(e - i - 1) */
	358	if (!BN_copy(t, y)) goto end;
	359	for (j = e - i - 1; j > 0; j--)
	360	{
	361	if (!BN_mod_sqr(t, t, p, ctx)) goto end;
	362	}
	363	if (!BN_mod_mul(y, t, t, p, ctx)) goto end;
	364	if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
	365	if (!BN_mod_mul(b, b, y, p, ctx)) goto end;
	366	e = i;
	367	}
	368
6fb60a84 BM	369	vrfy:
	370	if (!err)
	371	{
	372	/* verify the result -- the input might have been not a square
	373	* (test added in 0.9.8) */
	374
	375	if (!BN_mod_sqr(x, ret, p, ctx))
	376	err = 1;
	377
	378	if (!err && 0 != BN_cmp(x, A))
	379	{
	380	BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
	381	err = 1;
	382	}
	383	}
	384
cd2eebfd BM	385	end:
	386	if (err)
	387	{
	388	if (ret != NULL && ret != in)
	389	{
	390	BN_clear_free(ret);
	391	}
	392	ret = NULL;
	393	}
	394	BN_CTX_end(ctx);
d870740c	395	bn_check_top(ret);
cd2eebfd BM	396	return ret;
cd2eebfd BM	397	}