Head

GCC Code Coverage Report

Directory:	./		Exec	Total	Coverage
File:	lib/libcrypto/bn/bn_sqrt.c	Lines:	87	121	71.9 %
Date:	2017-11-13	Branches:	103	194	53.1 %


/* $OpenBSD: bn_sqrt.c,v 1.9 2017/01/29 17:49:22 beck Exp $ */
/* 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/err.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;
		}

		BNerror(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);
	if ((A = BN_CTX_get(ctx)) == NULL)
		goto end;
	if ((b = BN_CTX_get(ctx)) == NULL)
		goto end;
	if ((q = BN_CTX_get(ctx)) == NULL)
		goto end;
	if ((t = BN_CTX_get(ctx)) == NULL)
		goto end;
	if ((x = BN_CTX_get(ctx)) == NULL)
		goto end;
	if ((y = BN_CTX_get(ctx)) == 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_ct(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_ct(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) {
					if (!BN_add(y, y, p))
						goto end;
				} else {
					if (!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 */
			BNerror(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.
		 */
		BNerror(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_ct(y, y, q, p, ctx))
		goto end;
	if (BN_is_one(y)) {
		BNerror(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_ct(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) {
				BNerror(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)) {
			BNerror(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;
}


Generated by: GCOVR (Version 3.3)

Line	Branch	Exec	Source
1			/* $OpenBSD: bn_sqrt.c,v 1.9 2017/01/29 17:49:22 beck Exp $ */
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 <openssl/err.h>
59
60			#include "bn_lcl.h"
61
62			BIGNUM *
63			BN_mod_sqrt(BIGNUM in, const BIGNUM a, const BIGNUM p, BN_CTX ctx)
64			/* Returns 'ret' such that
65			* ret^2 == a (mod p),
66			* using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
67			* in Algebraic Computational Number Theory", algorithm 1.5.1).
68			* 'p' must be prime!
69			*/
70			{
71			BIGNUM *ret = in;
72			int err = 1;
73			int r;
74			BIGNUM A, b, q, t, x, y;
75			int e, i, j;
76
77	✓✗✓✓ ✓✓✗✓	1524	if (!BN_is_odd(p) \|\| BN_abs_is_word(p, 1)) {
78	✓✗✓✗	30	if (BN_abs_is_word(p, 2)) {
79	✗✓	15	if (ret == NULL)
80			ret = BN_new();
81	✓✗	15	if (ret == NULL)
82			goto end;
83	✗✓	15	if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
84			if (ret != in)
85			BN_free(ret);
86			return NULL;
87			}
88			bn_check_top(ret);
89		15	return ret;
90			}
91
92			BNerror(BN_R_P_IS_NOT_PRIME);
93			return (NULL);
94			}
95
96	✓✓✓✓ ✓✓✓✗	800	if (BN_is_zero(a) \|\| BN_is_one(a)) {
97	✗✓	20	if (ret == NULL)
98			ret = BN_new();
99	✓✗	20	if (ret == NULL)
100			goto end;
101	✓✓✓✗ ✗✓	62	if (!BN_set_word(ret, BN_is_one(a))) {
102			if (ret != in)
103			BN_free(ret);
104			return NULL;
105			}
106			bn_check_top(ret);
107		20	return ret;
108			}
109
110		322	BN_CTX_start(ctx);
111	✓✗	322	if ((A = BN_CTX_get(ctx)) == NULL)
112			goto end;
113	✓✗	322	if ((b = BN_CTX_get(ctx)) == NULL)
114			goto end;
115	✓✗	322	if ((q = BN_CTX_get(ctx)) == NULL)
116			goto end;
117	✓✗	322	if ((t = BN_CTX_get(ctx)) == NULL)
118			goto end;
119	✓✗	322	if ((x = BN_CTX_get(ctx)) == NULL)
120			goto end;
121	✓✗	322	if ((y = BN_CTX_get(ctx)) == NULL)
122			goto end;
123
124	✗✓	322	if (ret == NULL)
125			ret = BN_new();
126	✓✗	322	if (ret == NULL)
127			goto end;
128
129			/* A = a mod p */
130	✓✗	322	if (!BN_nnmod(A, a, p, ctx))
131			goto end;
132
133			/* now write \|p\| - 1 as 2^eq where q is odd /
134			e = 1;
135	✓✓	2338	while (!BN_is_bit_set(p, e))
136		847	e++;
137			/* we'll set q later (if needed) */
138
139	✓✓	322	if (e == 1) {
140			/* The easy case: (\|p\|-1)/2 is odd, so 2 has an inverse
141			* modulo (\|p\|-1)/2, and square roots can be computed
142			* directly by modular exponentiation.
143			* We have
144			* 2 * (\|p\|+1)/4 == 1 (mod (\|p\|-1)/2),
145			* so we can use exponent (\|p\|+1)/4, i.e. (\|p\|-3)/4 + 1.
146			*/
147	✓✗	210	if (!BN_rshift(q, p, 2))
148			goto end;
149		210	q->neg = 0;
150	✓✗	210	if (!BN_add_word(q, 1))
151			goto end;
152	✓✗	210	if (!BN_mod_exp_ct(ret, A, q, p, ctx))
153			goto end;
154			err = 0;
155		210	goto vrfy;
156			}
157
158	✓✓	112	if (e == 2) {
159			/* \|p\| == 5 (mod 8)
160			*
161			* In this case 2 is always a non-square since
162			* Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
163			* So if a really is a square, then 2*a is a non-square.
164			* Thus for
165			* b := (2*a)^((\|p\|-5)/8),
166			* i := (2a)b^2
167			* we have
168			* i^2 = (2a)^((1 + (\|p\|-5)/4)2)
169			* = (2*a)^((p-1)/2)
170			* = -1;
171			* so if we set
172			* x := ab(i-1),
173			* then
174			* x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
175			* = a^2 * b^2 * (-2*i)
176			* = a(-i)(2ab^2)
177			* = a(-i)i
178			* = a.
179			*
180			* (This is due to A.O.L. Atkin,
181			* <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
182			* November 1992.)
183			*/
184
185			/* t := 2a /
186	✓✗	58	if (!BN_mod_lshift1_quick(t, A, p))
187			goto end;
188
189			/* b := (2a)^((\|p\|-5)/8) /
190	✓✗	58	if (!BN_rshift(q, p, 3))
191			goto end;
192		58	q->neg = 0;
193	✓✗	58	if (!BN_mod_exp_ct(b, t, q, p, ctx))
194			goto end;
195
196			/* y := b^2 */
197	✓✗	58	if (!BN_mod_sqr(y, b, p, ctx))
198			goto end;
199
200			/* t := (2a)b^2 - 1*/
201	✓✗	58	if (!BN_mod_mul(t, t, y, p, ctx))
202			goto end;
203	✓✗	58	if (!BN_sub_word(t, 1))
204			goto end;
205
206			/* x = abt */
207	✓✗	58	if (!BN_mod_mul(x, A, b, p, ctx))
208			goto end;
209	✓✗	58	if (!BN_mod_mul(x, x, t, p, ctx))
210			goto end;
211
212	✓✗	58	if (!BN_copy(ret, x))
213			goto end;
214			err = 0;
215		58	goto vrfy;
216			}
217
218			/* e > 2, so we really have to use the Tonelli/Shanks algorithm.
219			* First, find some y that is not a square. */
220	✓✗	54	if (!BN_copy(q, p)) goto end; /* use 'q' as temp */
221		54	q->neg = 0;
222			i = 2;
223		54	do {
224			/* For efficiency, try small numbers first;
225			* if this fails, try random numbers.
226			*/
227	✓✗	172	if (i < 22) {
228	✓✗	172	if (!BN_set_word(y, i))
229			goto end;
230			} else {
231			if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0))
232			goto end;
233			if (BN_ucmp(y, p) >= 0) {
234			if (p->neg) {
235			if (!BN_add(y, y, p))
236			goto end;
237			} else {
238			if (!BN_sub(y, y, p))
239			goto end;
240			}
241			}
242			/* now 0 <= y < \|p\| */
243			if (BN_is_zero(y))
244			if (!BN_set_word(y, i))
245			goto end;
246			}
247
248		172	r = BN_kronecker(y, q, ctx); /* here 'q' is \|p\| */
249	✓✗	172	if (r < -1)
250			goto end;
251	✗✓	172	if (r == 0) {
252			/* m divides p */
253			BNerror(BN_R_P_IS_NOT_PRIME);
254			goto end;
255			}
256	✓✗	118	}
257	✓✓	290	while (r == 1 && ++i < 82);
258
259	✗✓	54	if (r != -1) {
260			/* Many rounds and still no non-square -- this is more likely
261			* a bug than just bad luck.
262			* Even if p is not prime, we should have found some y
263			* such that r == -1.
264			*/
265			BNerror(BN_R_TOO_MANY_ITERATIONS);
266			goto end;
267			}
268
269			/* Here's our actual 'q': */
270	✓✗	54	if (!BN_rshift(q, q, e))
271			goto end;
272
273			/* Now that we have some non-square, we can find an element
274			* of order 2^e by computing its q'th power. */
275	✓✗	54	if (!BN_mod_exp_ct(y, y, q, p, ctx))
276			goto end;
277	✓✓✗✓ ✗✗	67	if (BN_is_one(y)) {
278			BNerror(BN_R_P_IS_NOT_PRIME);
279			goto end;
280			}
281
282			/* Now we know that (if p is indeed prime) there is an integer
283			* k, 0 <= k < 2^e, such that
284			*
285			* a^q * y^k == 1 (mod p).
286			*
287			* As a^q is a square and y is not, k must be even.
288			* q+1 is even, too, so there is an element
289			*
290			* X := a^((q+1)/2) * y^(k/2),
291			*
292			* and it satisfies
293			*
294			* X^2 = a^q * a * y^k
295			* = a,
296			*
297			* so it is the square root that we are looking for.
298			*/
299
300			/* t := (q-1)/2 (note that q is odd) */
301	✓✗	54	if (!BN_rshift1(t, q))
302			goto end;
303
304			/* x := a^((q-1)/2) */
305	✓✓	54	if (BN_is_zero(t)) /* special case: p = 2^e + 1 */
306			{
307	✓✗	13	if (!BN_nnmod(t, A, p, ctx))
308			goto end;
309	✗✓	13	if (BN_is_zero(t)) {
310			/* special case: a == 0 (mod p) */
311			BN_zero(ret);
312			err = 0;
313			goto end;
314	✓✗	13	} else if (!BN_one(x))
315			goto end;
316			} else {
317	✓✗	41	if (!BN_mod_exp_ct(x, A, t, p, ctx))
318			goto end;
319	✗✓	41	if (BN_is_zero(x)) {
320			/* special case: a == 0 (mod p) */
321			BN_zero(ret);
322			err = 0;
323			goto end;
324			}
325			}
326
327			/* b := ax^2 (= a^q) /
328	✓✗	54	if (!BN_mod_sqr(b, x, p, ctx))
329			goto end;
330	✓✗	54	if (!BN_mod_mul(b, b, A, p, ctx))
331			goto end;
332
333			/* x := ax (= a^((q+1)/2)) /
334	✓✗	54	if (!BN_mod_mul(x, x, A, p, ctx))
335			goto end;
336
337		388	while (1) {
338			/* Now b is a^q * y^k for some even k (0 <= k < 2^E
339			* where E refers to the original value of e, which we
340			* don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
341			*
342			* We have a*b = x^2,
343			* y^2^(e-1) = -1,
344			* b^2^(e-1) = 1.
345			*/
346
347	✓✓✓✓ ✓✗	521	if (BN_is_one(b)) {
348	✓✗	54	if (!BN_copy(ret, x))
349			goto end;
350			err = 0;
351		54	goto vrfy;
352			}
353
354
355			/* find smallest i such that b^(2^i) = 1 */
356			i = 1;
357	✓✗	334	if (!BN_mod_sqr(t, b, p, ctx))
358			goto end;
359	✓✓✓✓ ✓✓	39026	while (!BN_is_one(t)) {
360		12444	i++;
361	✗✓	12444	if (i == e) {
362			BNerror(BN_R_NOT_A_SQUARE);
363			goto end;
364			}
365	✓✗	12444	if (!BN_mod_mul(t, t, t, p, ctx))
366			goto end;
367			}
368
369
370			/* t := y^2^(e - i - 1) */
371	✓✗	334	if (!BN_copy(t, y))
372			goto end;
373	✓✓	1520	for (j = e - i - 1; j > 0; j--) {
374	✓✗	426	if (!BN_mod_sqr(t, t, p, ctx))
375			goto end;
376			}
377	✓✗	334	if (!BN_mod_mul(y, t, t, p, ctx))
378			goto end;
379	✓✗	334	if (!BN_mod_mul(x, x, t, p, ctx))
380			goto end;
381	✓✗	334	if (!BN_mod_mul(b, b, y, p, ctx))
382			goto end;
383			e = i;
384			}
385
386			vrfy:
387	✓✗	322	if (!err) {
388			/* verify the result -- the input might have been not a square
389			* (test added in 0.9.8) */
390
391	✗✓	322	if (!BN_mod_sqr(x, ret, p, ctx))
392			err = 1;
393
394	✓✗✗✓	644	if (!err && 0 != BN_cmp(x, A)) {
395			BNerror(BN_R_NOT_A_SQUARE);
396			err = 1;
397			}
398			}
399
400			end:
401	✗✓	322	if (err) {
402			if (ret != NULL && ret != in) {
403			BN_clear_free(ret);
404			}
405			ret = NULL;
406			}
407		322	BN_CTX_end(ctx);
408			bn_check_top(ret);
409		322	return ret;
410		357	}