/src/libressl/crypto/bn/bn_sqrt.c

Source (jump to first uncovered line)
/* $OpenBSD: bn_sqrt.c,v 1.11 2022/06/20 15:02:21 tb 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)) /* use 'q' as temp */
    goto end;
  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 the smallest i with 0 < i < e such that b^(2^i) = 1. */
    for (i = 1; i < e; i++) {
      if (i == 1) {
        if (!BN_mod_sqr(t, b, p, ctx))
          goto end;
      } else {
        if (!BN_mod_sqr(t, t, p, ctx))
          goto end;
      }
      if (BN_is_one(t))
        break;
    }
    if (i >= e) {
      BNerror(BN_R_NOT_A_SQUARE);
      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;
}

Coverage Report

Created: 2022-08-24 06:31

Line	Count	Source (jump to first uncovered line)
1		/* $OpenBSD: bn_sqrt.c,v 1.11 2022/06/20 15:02:21 tb 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	1.10k	{
71	1.10k	BIGNUM *ret = in;
72	1.10k	int err = 1;
73	1.10k	int r;
74	1.10k	BIGNUM A, b, q, t, x, y;
75	1.10k	int e, i, j;
76
77	1.10k	if (!BN_is_odd(p) \|\| BN_abs_is_word(p, 1)) {
78	0	if (BN_abs_is_word(p, 2)) {
79	0	if (ret == NULL)
80	0	ret = BN_new();
81	0	if (ret == NULL)
82	0	goto end;
83	0	if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
84	0	if (ret != in)
85	0	BN_free(ret);
86	0	return NULL;
87	0	}
88	0	bn_check_top(ret);
89	0	return ret;
90	0	}
91
92	0	BNerror(BN_R_P_IS_NOT_PRIME);
93	0	return (NULL);
94	0	}
95
96	1.10k	if (BN_is_zero(a) \|\| BN_is_one(a)) {
97	0	if (ret == NULL)
98	0	ret = BN_new();
99	0	if (ret == NULL)
100	0	goto end;
101	0	if (!BN_set_word(ret, BN_is_one(a))) {
102	0	if (ret != in)
103	0	BN_free(ret);
104	0	return NULL;
105	0	}
106	0	bn_check_top(ret);
107	0	return ret;
108	0	}
109
110	1.10k	BN_CTX_start(ctx);
111	1.10k	if ((A = BN_CTX_get(ctx)) == NULL)
112	0	goto end;
113	1.10k	if ((b = BN_CTX_get(ctx)) == NULL)
114	0	goto end;
115	1.10k	if ((q = BN_CTX_get(ctx)) == NULL)
116	0	goto end;
117	1.10k	if ((t = BN_CTX_get(ctx)) == NULL)
118	0	goto end;
119	1.10k	if ((x = BN_CTX_get(ctx)) == NULL)
120	0	goto end;
121	1.10k	if ((y = BN_CTX_get(ctx)) == NULL)
122	0	goto end;
123
124	1.10k	if (ret == NULL)
125	0	ret = BN_new();
126	1.10k	if (ret == NULL)
127	0	goto end;
128
129		/* A = a mod p */
130	1.10k	if (!BN_nnmod(A, a, p, ctx))
131	0	goto end;
132
133		/* now write \|p\| - 1 as 2^eq where q is odd /
134	1.10k	e = 1;
135	1.10k	while (!BN_is_bit_set(p, e))
136	0	e++;
137		/* we'll set q later (if needed) */
138
139	1.10k	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	1.10k	if (!BN_rshift(q, p, 2))
148	0	goto end;
149	1.10k	q->neg = 0;
150	1.10k	if (!BN_add_word(q, 1))
151	0	goto end;
152	1.10k	if (!BN_mod_exp_ct(ret, A, q, p, ctx))
153	0	goto end;
154	1.10k	err = 0;
155	1.10k	goto vrfy;
156	1.10k	}
157
158	0	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	0	if (!BN_mod_lshift1_quick(t, A, p))
187	0	goto end;
188
189		/* b := (2a)^((\|p\|-5)/8) /
190	0	if (!BN_rshift(q, p, 3))
191	0	goto end;
192	0	q->neg = 0;
193	0	if (!BN_mod_exp_ct(b, t, q, p, ctx))
194	0	goto end;
195
196		/* y := b^2 */
197	0	if (!BN_mod_sqr(y, b, p, ctx))
198	0	goto end;
199
200		/* t := (2a)b^2 - 1*/
201	0	if (!BN_mod_mul(t, t, y, p, ctx))
202	0	goto end;
203	0	if (!BN_sub_word(t, 1))
204	0	goto end;
205
206		/* x = abt */
207	0	if (!BN_mod_mul(x, A, b, p, ctx))
208	0	goto end;
209	0	if (!BN_mod_mul(x, x, t, p, ctx))
210	0	goto end;
211
212	0	if (!BN_copy(ret, x))
213	0	goto end;
214	0	err = 0;
215	0	goto vrfy;
216	0	}
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	0	if (!BN_copy(q, p)) /* use 'q' as temp */
221	0	goto end;
222	0	q->neg = 0;
223	0	i = 2;
224	0	do {
225		/* For efficiency, try small numbers first;
226		* if this fails, try random numbers.
227		*/
228	0	if (i < 22) {
229	0	if (!BN_set_word(y, i))
230	0	goto end;
231	0	} else {
232	0	if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0))
233	0	goto end;
234	0	if (BN_ucmp(y, p) >= 0) {
235	0	if (p->neg) {
236	0	if (!BN_add(y, y, p))
237	0	goto end;
238	0	} else {
239	0	if (!BN_sub(y, y, p))
240	0	goto end;
241	0	}
242	0	}
243		/* now 0 <= y < \|p\| */
244	0	if (BN_is_zero(y))
245	0	if (!BN_set_word(y, i))
246	0	goto end;
247	0	}
248
249	0	r = BN_kronecker(y, q, ctx); /* here 'q' is \|p\| */
250	0	if (r < -1)
251	0	goto end;
252	0	if (r == 0) {
253		/* m divides p */
254	0	BNerror(BN_R_P_IS_NOT_PRIME);
255	0	goto end;
256	0	}
257	0	} while (r == 1 && ++i < 82);
258
259	0	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	0	BNerror(BN_R_TOO_MANY_ITERATIONS);
266	0	goto end;
267	0	}
268
269		/* Here's our actual 'q': */
270	0	if (!BN_rshift(q, q, e))
271	0	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	0	if (!BN_mod_exp_ct(y, y, q, p, ctx))
276	0	goto end;
277	0	if (BN_is_one(y)) {
278	0	BNerror(BN_R_P_IS_NOT_PRIME);
279	0	goto end;
280	0	}
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	0	if (!BN_rshift1(t, q))
302	0	goto end;
303
304		/* x := a^((q-1)/2) */
305	0	if (BN_is_zero(t)) { /* special case: p = 2^e + 1 */
306	0	if (!BN_nnmod(t, A, p, ctx))
307	0	goto end;
308	0	if (BN_is_zero(t)) {
309		/* special case: a == 0 (mod p) */
310	0	BN_zero(ret);
311	0	err = 0;
312	0	goto end;
313	0	} else if (!BN_one(x))
314	0	goto end;
315	0	} else {
316	0	if (!BN_mod_exp_ct(x, A, t, p, ctx))
317	0	goto end;
318	0	if (BN_is_zero(x)) {
319		/* special case: a == 0 (mod p) */
320	0	BN_zero(ret);
321	0	err = 0;
322	0	goto end;
323	0	}
324	0	}
325
326		/* b := ax^2 (= a^q) /
327	0	if (!BN_mod_sqr(b, x, p, ctx))
328	0	goto end;
329	0	if (!BN_mod_mul(b, b, A, p, ctx))
330	0	goto end;
331
332		/* x := ax (= a^((q+1)/2)) /
333	0	if (!BN_mod_mul(x, x, A, p, ctx))
334	0	goto end;
335
336	0	while (1) {
337		/* Now b is a^q * y^k for some even k (0 <= k < 2^E
338		* where E refers to the original value of e, which we
339		* don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
340		*
341		* We have a*b = x^2,
342		* y^2^(e-1) = -1,
343		* b^2^(e-1) = 1.
344		*/
345
346	0	if (BN_is_one(b)) {
347	0	if (!BN_copy(ret, x))
348	0	goto end;
349	0	err = 0;
350	0	goto vrfy;
351	0	}
352
353		/* Find the smallest i with 0 < i < e such that b^(2^i) = 1. */
354	0	for (i = 1; i < e; i++) {
355	0	if (i == 1) {
356	0	if (!BN_mod_sqr(t, b, p, ctx))
357	0	goto end;
358	0	} else {
359	0	if (!BN_mod_sqr(t, t, p, ctx))
360	0	goto end;
361	0	}
362	0	if (BN_is_one(t))
363	0	break;
364	0	}
365	0	if (i >= e) {
366	0	BNerror(BN_R_NOT_A_SQUARE);
367	0	goto end;
368	0	}
369
370		/* t := y^2^(e - i - 1) */
371	0	if (!BN_copy(t, y))
372	0	goto end;
373	0	for (j = e - i - 1; j > 0; j--) {
374	0	if (!BN_mod_sqr(t, t, p, ctx))
375	0	goto end;
376	0	}
377	0	if (!BN_mod_mul(y, t, t, p, ctx))
378	0	goto end;
379	0	if (!BN_mod_mul(x, x, t, p, ctx))
380	0	goto end;
381	0	if (!BN_mod_mul(b, b, y, p, ctx))
382	0	goto end;
383	0	e = i;
384	0	}
385
386	1.10k	vrfy:
387	1.10k	if (!err) {
388		/* verify the result -- the input might have been not a square
389		* (test added in 0.9.8) */
390
391	1.10k	if (!BN_mod_sqr(x, ret, p, ctx))
392	0	err = 1;
393
394	1.10k	if (!err && 0 != BN_cmp(x, A)) {
395	441	BNerror(BN_R_NOT_A_SQUARE);
396	441	err = 1;
397	441	}
398	1.10k	}
399
400	1.10k	end:
401	1.10k	if (err) {
402	441	if (ret != NULL && ret != in) {
403	0	BN_clear_free(ret);
404	0	}
405	441	ret = NULL;
406	441	}
407	1.10k	BN_CTX_end(ctx);
408	1.10k	bn_check_top(ret);
409	1.10k	return ret;
410	1.10k	}