1*de0e0e4dSAntonio Huete Jimenez /* $OpenBSD: bn_sqrt.c,v 1.11 2022/06/20 15:02:21 tb Exp $ */
2f5b1c8a1SJohn Marino /* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
3f5b1c8a1SJohn Marino * and Bodo Moeller for the OpenSSL project. */
4f5b1c8a1SJohn Marino /* ====================================================================
5f5b1c8a1SJohn Marino * Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved.
6f5b1c8a1SJohn Marino *
7f5b1c8a1SJohn Marino * Redistribution and use in source and binary forms, with or without
8f5b1c8a1SJohn Marino * modification, are permitted provided that the following conditions
9f5b1c8a1SJohn Marino * are met:
10f5b1c8a1SJohn Marino *
11f5b1c8a1SJohn Marino * 1. Redistributions of source code must retain the above copyright
12f5b1c8a1SJohn Marino * notice, this list of conditions and the following disclaimer.
13f5b1c8a1SJohn Marino *
14f5b1c8a1SJohn Marino * 2. Redistributions in binary form must reproduce the above copyright
15f5b1c8a1SJohn Marino * notice, this list of conditions and the following disclaimer in
16f5b1c8a1SJohn Marino * the documentation and/or other materials provided with the
17f5b1c8a1SJohn Marino * distribution.
18f5b1c8a1SJohn Marino *
19f5b1c8a1SJohn Marino * 3. All advertising materials mentioning features or use of this
20f5b1c8a1SJohn Marino * software must display the following acknowledgment:
21f5b1c8a1SJohn Marino * "This product includes software developed by the OpenSSL Project
22f5b1c8a1SJohn Marino * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
23f5b1c8a1SJohn Marino *
24f5b1c8a1SJohn Marino * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
25f5b1c8a1SJohn Marino * endorse or promote products derived from this software without
26f5b1c8a1SJohn Marino * prior written permission. For written permission, please contact
27f5b1c8a1SJohn Marino * openssl-core@openssl.org.
28f5b1c8a1SJohn Marino *
29f5b1c8a1SJohn Marino * 5. Products derived from this software may not be called "OpenSSL"
30f5b1c8a1SJohn Marino * nor may "OpenSSL" appear in their names without prior written
31f5b1c8a1SJohn Marino * permission of the OpenSSL Project.
32f5b1c8a1SJohn Marino *
33f5b1c8a1SJohn Marino * 6. Redistributions of any form whatsoever must retain the following
34f5b1c8a1SJohn Marino * acknowledgment:
35f5b1c8a1SJohn Marino * "This product includes software developed by the OpenSSL Project
36f5b1c8a1SJohn Marino * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
37f5b1c8a1SJohn Marino *
38f5b1c8a1SJohn Marino * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
39f5b1c8a1SJohn Marino * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
40f5b1c8a1SJohn Marino * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
41f5b1c8a1SJohn Marino * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
42f5b1c8a1SJohn Marino * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
43f5b1c8a1SJohn Marino * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
44f5b1c8a1SJohn Marino * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
45f5b1c8a1SJohn Marino * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
46f5b1c8a1SJohn Marino * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
47f5b1c8a1SJohn Marino * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
48f5b1c8a1SJohn Marino * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
49f5b1c8a1SJohn Marino * OF THE POSSIBILITY OF SUCH DAMAGE.
50f5b1c8a1SJohn Marino * ====================================================================
51f5b1c8a1SJohn Marino *
52f5b1c8a1SJohn Marino * This product includes cryptographic software written by Eric Young
53f5b1c8a1SJohn Marino * (eay@cryptsoft.com). This product includes software written by Tim
54f5b1c8a1SJohn Marino * Hudson (tjh@cryptsoft.com).
55f5b1c8a1SJohn Marino *
56f5b1c8a1SJohn Marino */
57f5b1c8a1SJohn Marino
58f5b1c8a1SJohn Marino #include <openssl/err.h>
59f5b1c8a1SJohn Marino
60f5b1c8a1SJohn Marino #include "bn_lcl.h"
61f5b1c8a1SJohn Marino
62f5b1c8a1SJohn Marino BIGNUM *
BN_mod_sqrt(BIGNUM * in,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx)63f5b1c8a1SJohn Marino BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
64f5b1c8a1SJohn Marino /* Returns 'ret' such that
65f5b1c8a1SJohn Marino * ret^2 == a (mod p),
66f5b1c8a1SJohn Marino * using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
67f5b1c8a1SJohn Marino * in Algebraic Computational Number Theory", algorithm 1.5.1).
68f5b1c8a1SJohn Marino * 'p' must be prime!
69f5b1c8a1SJohn Marino */
70f5b1c8a1SJohn Marino {
71f5b1c8a1SJohn Marino BIGNUM *ret = in;
72f5b1c8a1SJohn Marino int err = 1;
73f5b1c8a1SJohn Marino int r;
74f5b1c8a1SJohn Marino BIGNUM *A, *b, *q, *t, *x, *y;
75f5b1c8a1SJohn Marino int e, i, j;
76f5b1c8a1SJohn Marino
77f5b1c8a1SJohn Marino if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
78f5b1c8a1SJohn Marino if (BN_abs_is_word(p, 2)) {
79f5b1c8a1SJohn Marino if (ret == NULL)
80f5b1c8a1SJohn Marino ret = BN_new();
81f5b1c8a1SJohn Marino if (ret == NULL)
82f5b1c8a1SJohn Marino goto end;
83f5b1c8a1SJohn Marino if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
84f5b1c8a1SJohn Marino if (ret != in)
85f5b1c8a1SJohn Marino BN_free(ret);
86f5b1c8a1SJohn Marino return NULL;
87f5b1c8a1SJohn Marino }
88f5b1c8a1SJohn Marino bn_check_top(ret);
89f5b1c8a1SJohn Marino return ret;
90f5b1c8a1SJohn Marino }
91f5b1c8a1SJohn Marino
9272c33676SMaxim Ag BNerror(BN_R_P_IS_NOT_PRIME);
93f5b1c8a1SJohn Marino return (NULL);
94f5b1c8a1SJohn Marino }
95f5b1c8a1SJohn Marino
96f5b1c8a1SJohn Marino if (BN_is_zero(a) || BN_is_one(a)) {
97f5b1c8a1SJohn Marino if (ret == NULL)
98f5b1c8a1SJohn Marino ret = BN_new();
99f5b1c8a1SJohn Marino if (ret == NULL)
100f5b1c8a1SJohn Marino goto end;
101f5b1c8a1SJohn Marino if (!BN_set_word(ret, BN_is_one(a))) {
102f5b1c8a1SJohn Marino if (ret != in)
103f5b1c8a1SJohn Marino BN_free(ret);
104f5b1c8a1SJohn Marino return NULL;
105f5b1c8a1SJohn Marino }
106f5b1c8a1SJohn Marino bn_check_top(ret);
107f5b1c8a1SJohn Marino return ret;
108f5b1c8a1SJohn Marino }
109f5b1c8a1SJohn Marino
110f5b1c8a1SJohn Marino BN_CTX_start(ctx);
111f5b1c8a1SJohn Marino if ((A = BN_CTX_get(ctx)) == NULL)
112f5b1c8a1SJohn Marino goto end;
113f5b1c8a1SJohn Marino if ((b = BN_CTX_get(ctx)) == NULL)
114f5b1c8a1SJohn Marino goto end;
115f5b1c8a1SJohn Marino if ((q = BN_CTX_get(ctx)) == NULL)
116f5b1c8a1SJohn Marino goto end;
117f5b1c8a1SJohn Marino if ((t = BN_CTX_get(ctx)) == NULL)
118f5b1c8a1SJohn Marino goto end;
119f5b1c8a1SJohn Marino if ((x = BN_CTX_get(ctx)) == NULL)
120f5b1c8a1SJohn Marino goto end;
121f5b1c8a1SJohn Marino if ((y = BN_CTX_get(ctx)) == NULL)
122f5b1c8a1SJohn Marino goto end;
123f5b1c8a1SJohn Marino
124f5b1c8a1SJohn Marino if (ret == NULL)
125f5b1c8a1SJohn Marino ret = BN_new();
126f5b1c8a1SJohn Marino if (ret == NULL)
127f5b1c8a1SJohn Marino goto end;
128f5b1c8a1SJohn Marino
129f5b1c8a1SJohn Marino /* A = a mod p */
130f5b1c8a1SJohn Marino if (!BN_nnmod(A, a, p, ctx))
131f5b1c8a1SJohn Marino goto end;
132f5b1c8a1SJohn Marino
133f5b1c8a1SJohn Marino /* now write |p| - 1 as 2^e*q where q is odd */
134f5b1c8a1SJohn Marino e = 1;
135f5b1c8a1SJohn Marino while (!BN_is_bit_set(p, e))
136f5b1c8a1SJohn Marino e++;
137f5b1c8a1SJohn Marino /* we'll set q later (if needed) */
138f5b1c8a1SJohn Marino
139f5b1c8a1SJohn Marino if (e == 1) {
140f5b1c8a1SJohn Marino /* The easy case: (|p|-1)/2 is odd, so 2 has an inverse
141f5b1c8a1SJohn Marino * modulo (|p|-1)/2, and square roots can be computed
142f5b1c8a1SJohn Marino * directly by modular exponentiation.
143f5b1c8a1SJohn Marino * We have
144f5b1c8a1SJohn Marino * 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2),
145f5b1c8a1SJohn Marino * so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1.
146f5b1c8a1SJohn Marino */
147f5b1c8a1SJohn Marino if (!BN_rshift(q, p, 2))
148f5b1c8a1SJohn Marino goto end;
149f5b1c8a1SJohn Marino q->neg = 0;
150f5b1c8a1SJohn Marino if (!BN_add_word(q, 1))
151f5b1c8a1SJohn Marino goto end;
15272c33676SMaxim Ag if (!BN_mod_exp_ct(ret, A, q, p, ctx))
153f5b1c8a1SJohn Marino goto end;
154f5b1c8a1SJohn Marino err = 0;
155f5b1c8a1SJohn Marino goto vrfy;
156f5b1c8a1SJohn Marino }
157f5b1c8a1SJohn Marino
158f5b1c8a1SJohn Marino if (e == 2) {
159f5b1c8a1SJohn Marino /* |p| == 5 (mod 8)
160f5b1c8a1SJohn Marino *
161f5b1c8a1SJohn Marino * In this case 2 is always a non-square since
162f5b1c8a1SJohn Marino * Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
163f5b1c8a1SJohn Marino * So if a really is a square, then 2*a is a non-square.
164f5b1c8a1SJohn Marino * Thus for
165f5b1c8a1SJohn Marino * b := (2*a)^((|p|-5)/8),
166f5b1c8a1SJohn Marino * i := (2*a)*b^2
167f5b1c8a1SJohn Marino * we have
168f5b1c8a1SJohn Marino * i^2 = (2*a)^((1 + (|p|-5)/4)*2)
169f5b1c8a1SJohn Marino * = (2*a)^((p-1)/2)
170f5b1c8a1SJohn Marino * = -1;
171f5b1c8a1SJohn Marino * so if we set
172f5b1c8a1SJohn Marino * x := a*b*(i-1),
173f5b1c8a1SJohn Marino * then
174f5b1c8a1SJohn Marino * x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
175f5b1c8a1SJohn Marino * = a^2 * b^2 * (-2*i)
176f5b1c8a1SJohn Marino * = a*(-i)*(2*a*b^2)
177f5b1c8a1SJohn Marino * = a*(-i)*i
178f5b1c8a1SJohn Marino * = a.
179f5b1c8a1SJohn Marino *
180f5b1c8a1SJohn Marino * (This is due to A.O.L. Atkin,
181f5b1c8a1SJohn Marino * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
182f5b1c8a1SJohn Marino * November 1992.)
183f5b1c8a1SJohn Marino */
184f5b1c8a1SJohn Marino
185f5b1c8a1SJohn Marino /* t := 2*a */
186f5b1c8a1SJohn Marino if (!BN_mod_lshift1_quick(t, A, p))
187f5b1c8a1SJohn Marino goto end;
188f5b1c8a1SJohn Marino
189f5b1c8a1SJohn Marino /* b := (2*a)^((|p|-5)/8) */
190f5b1c8a1SJohn Marino if (!BN_rshift(q, p, 3))
191f5b1c8a1SJohn Marino goto end;
192f5b1c8a1SJohn Marino q->neg = 0;
19372c33676SMaxim Ag if (!BN_mod_exp_ct(b, t, q, p, ctx))
194f5b1c8a1SJohn Marino goto end;
195f5b1c8a1SJohn Marino
196f5b1c8a1SJohn Marino /* y := b^2 */
197f5b1c8a1SJohn Marino if (!BN_mod_sqr(y, b, p, ctx))
198f5b1c8a1SJohn Marino goto end;
199f5b1c8a1SJohn Marino
200f5b1c8a1SJohn Marino /* t := (2*a)*b^2 - 1*/
201f5b1c8a1SJohn Marino if (!BN_mod_mul(t, t, y, p, ctx))
202f5b1c8a1SJohn Marino goto end;
203f5b1c8a1SJohn Marino if (!BN_sub_word(t, 1))
204f5b1c8a1SJohn Marino goto end;
205f5b1c8a1SJohn Marino
206f5b1c8a1SJohn Marino /* x = a*b*t */
207f5b1c8a1SJohn Marino if (!BN_mod_mul(x, A, b, p, ctx))
208f5b1c8a1SJohn Marino goto end;
209f5b1c8a1SJohn Marino if (!BN_mod_mul(x, x, t, p, ctx))
210f5b1c8a1SJohn Marino goto end;
211f5b1c8a1SJohn Marino
212f5b1c8a1SJohn Marino if (!BN_copy(ret, x))
213f5b1c8a1SJohn Marino goto end;
214f5b1c8a1SJohn Marino err = 0;
215f5b1c8a1SJohn Marino goto vrfy;
216f5b1c8a1SJohn Marino }
217f5b1c8a1SJohn Marino
218f5b1c8a1SJohn Marino /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
219f5b1c8a1SJohn Marino * First, find some y that is not a square. */
220*de0e0e4dSAntonio Huete Jimenez if (!BN_copy(q, p)) /* use 'q' as temp */
221*de0e0e4dSAntonio Huete Jimenez goto end;
222f5b1c8a1SJohn Marino q->neg = 0;
223f5b1c8a1SJohn Marino i = 2;
224f5b1c8a1SJohn Marino do {
225f5b1c8a1SJohn Marino /* For efficiency, try small numbers first;
226f5b1c8a1SJohn Marino * if this fails, try random numbers.
227f5b1c8a1SJohn Marino */
228f5b1c8a1SJohn Marino if (i < 22) {
229f5b1c8a1SJohn Marino if (!BN_set_word(y, i))
230f5b1c8a1SJohn Marino goto end;
231f5b1c8a1SJohn Marino } else {
232f5b1c8a1SJohn Marino if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0))
233f5b1c8a1SJohn Marino goto end;
234f5b1c8a1SJohn Marino if (BN_ucmp(y, p) >= 0) {
23572c33676SMaxim Ag if (p->neg) {
23672c33676SMaxim Ag if (!BN_add(y, y, p))
237f5b1c8a1SJohn Marino goto end;
23872c33676SMaxim Ag } else {
23972c33676SMaxim Ag if (!BN_sub(y, y, p))
24072c33676SMaxim Ag goto end;
24172c33676SMaxim Ag }
242f5b1c8a1SJohn Marino }
243f5b1c8a1SJohn Marino /* now 0 <= y < |p| */
244f5b1c8a1SJohn Marino if (BN_is_zero(y))
245f5b1c8a1SJohn Marino if (!BN_set_word(y, i))
246f5b1c8a1SJohn Marino goto end;
247f5b1c8a1SJohn Marino }
248f5b1c8a1SJohn Marino
249f5b1c8a1SJohn Marino r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
250f5b1c8a1SJohn Marino if (r < -1)
251f5b1c8a1SJohn Marino goto end;
252f5b1c8a1SJohn Marino if (r == 0) {
253f5b1c8a1SJohn Marino /* m divides p */
25472c33676SMaxim Ag BNerror(BN_R_P_IS_NOT_PRIME);
255f5b1c8a1SJohn Marino goto end;
256f5b1c8a1SJohn Marino }
257*de0e0e4dSAntonio Huete Jimenez } while (r == 1 && ++i < 82);
258f5b1c8a1SJohn Marino
259f5b1c8a1SJohn Marino if (r != -1) {
260f5b1c8a1SJohn Marino /* Many rounds and still no non-square -- this is more likely
261f5b1c8a1SJohn Marino * a bug than just bad luck.
262f5b1c8a1SJohn Marino * Even if p is not prime, we should have found some y
263f5b1c8a1SJohn Marino * such that r == -1.
264f5b1c8a1SJohn Marino */
26572c33676SMaxim Ag BNerror(BN_R_TOO_MANY_ITERATIONS);
266f5b1c8a1SJohn Marino goto end;
267f5b1c8a1SJohn Marino }
268f5b1c8a1SJohn Marino
269f5b1c8a1SJohn Marino /* Here's our actual 'q': */
270f5b1c8a1SJohn Marino if (!BN_rshift(q, q, e))
271f5b1c8a1SJohn Marino goto end;
272f5b1c8a1SJohn Marino
273f5b1c8a1SJohn Marino /* Now that we have some non-square, we can find an element
274f5b1c8a1SJohn Marino * of order 2^e by computing its q'th power. */
27572c33676SMaxim Ag if (!BN_mod_exp_ct(y, y, q, p, ctx))
276f5b1c8a1SJohn Marino goto end;
277f5b1c8a1SJohn Marino if (BN_is_one(y)) {
27872c33676SMaxim Ag BNerror(BN_R_P_IS_NOT_PRIME);
279f5b1c8a1SJohn Marino goto end;
280f5b1c8a1SJohn Marino }
281f5b1c8a1SJohn Marino
282f5b1c8a1SJohn Marino /* Now we know that (if p is indeed prime) there is an integer
283f5b1c8a1SJohn Marino * k, 0 <= k < 2^e, such that
284f5b1c8a1SJohn Marino *
285f5b1c8a1SJohn Marino * a^q * y^k == 1 (mod p).
286f5b1c8a1SJohn Marino *
287f5b1c8a1SJohn Marino * As a^q is a square and y is not, k must be even.
288f5b1c8a1SJohn Marino * q+1 is even, too, so there is an element
289f5b1c8a1SJohn Marino *
290f5b1c8a1SJohn Marino * X := a^((q+1)/2) * y^(k/2),
291f5b1c8a1SJohn Marino *
292f5b1c8a1SJohn Marino * and it satisfies
293f5b1c8a1SJohn Marino *
294f5b1c8a1SJohn Marino * X^2 = a^q * a * y^k
295f5b1c8a1SJohn Marino * = a,
296f5b1c8a1SJohn Marino *
297f5b1c8a1SJohn Marino * so it is the square root that we are looking for.
298f5b1c8a1SJohn Marino */
299f5b1c8a1SJohn Marino
300f5b1c8a1SJohn Marino /* t := (q-1)/2 (note that q is odd) */
301f5b1c8a1SJohn Marino if (!BN_rshift1(t, q))
302f5b1c8a1SJohn Marino goto end;
303f5b1c8a1SJohn Marino
304f5b1c8a1SJohn Marino /* x := a^((q-1)/2) */
305*de0e0e4dSAntonio Huete Jimenez if (BN_is_zero(t)) { /* special case: p = 2^e + 1 */
306f5b1c8a1SJohn Marino if (!BN_nnmod(t, A, p, ctx))
307f5b1c8a1SJohn Marino goto end;
308f5b1c8a1SJohn Marino if (BN_is_zero(t)) {
309f5b1c8a1SJohn Marino /* special case: a == 0 (mod p) */
310f5b1c8a1SJohn Marino BN_zero(ret);
311f5b1c8a1SJohn Marino err = 0;
312f5b1c8a1SJohn Marino goto end;
313f5b1c8a1SJohn Marino } else if (!BN_one(x))
314f5b1c8a1SJohn Marino goto end;
315f5b1c8a1SJohn Marino } else {
31672c33676SMaxim Ag if (!BN_mod_exp_ct(x, A, t, p, ctx))
317f5b1c8a1SJohn Marino goto end;
318f5b1c8a1SJohn Marino if (BN_is_zero(x)) {
319f5b1c8a1SJohn Marino /* special case: a == 0 (mod p) */
320f5b1c8a1SJohn Marino BN_zero(ret);
321f5b1c8a1SJohn Marino err = 0;
322f5b1c8a1SJohn Marino goto end;
323f5b1c8a1SJohn Marino }
324f5b1c8a1SJohn Marino }
325f5b1c8a1SJohn Marino
326f5b1c8a1SJohn Marino /* b := a*x^2 (= a^q) */
327f5b1c8a1SJohn Marino if (!BN_mod_sqr(b, x, p, ctx))
328f5b1c8a1SJohn Marino goto end;
329f5b1c8a1SJohn Marino if (!BN_mod_mul(b, b, A, p, ctx))
330f5b1c8a1SJohn Marino goto end;
331f5b1c8a1SJohn Marino
332f5b1c8a1SJohn Marino /* x := a*x (= a^((q+1)/2)) */
333f5b1c8a1SJohn Marino if (!BN_mod_mul(x, x, A, p, ctx))
334f5b1c8a1SJohn Marino goto end;
335f5b1c8a1SJohn Marino
336f5b1c8a1SJohn Marino while (1) {
337f5b1c8a1SJohn Marino /* Now b is a^q * y^k for some even k (0 <= k < 2^E
338f5b1c8a1SJohn Marino * where E refers to the original value of e, which we
339f5b1c8a1SJohn Marino * don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
340f5b1c8a1SJohn Marino *
341f5b1c8a1SJohn Marino * We have a*b = x^2,
342f5b1c8a1SJohn Marino * y^2^(e-1) = -1,
343f5b1c8a1SJohn Marino * b^2^(e-1) = 1.
344f5b1c8a1SJohn Marino */
345f5b1c8a1SJohn Marino
346f5b1c8a1SJohn Marino if (BN_is_one(b)) {
347f5b1c8a1SJohn Marino if (!BN_copy(ret, x))
348f5b1c8a1SJohn Marino goto end;
349f5b1c8a1SJohn Marino err = 0;
350f5b1c8a1SJohn Marino goto vrfy;
351f5b1c8a1SJohn Marino }
352f5b1c8a1SJohn Marino
353*de0e0e4dSAntonio Huete Jimenez /* Find the smallest i with 0 < i < e such that b^(2^i) = 1. */
354*de0e0e4dSAntonio Huete Jimenez for (i = 1; i < e; i++) {
355*de0e0e4dSAntonio Huete Jimenez if (i == 1) {
356f5b1c8a1SJohn Marino if (!BN_mod_sqr(t, b, p, ctx))
357f5b1c8a1SJohn Marino goto end;
358*de0e0e4dSAntonio Huete Jimenez } else {
359*de0e0e4dSAntonio Huete Jimenez if (!BN_mod_sqr(t, t, p, ctx))
360*de0e0e4dSAntonio Huete Jimenez goto end;
361*de0e0e4dSAntonio Huete Jimenez }
362*de0e0e4dSAntonio Huete Jimenez if (BN_is_one(t))
363*de0e0e4dSAntonio Huete Jimenez break;
364*de0e0e4dSAntonio Huete Jimenez }
365*de0e0e4dSAntonio Huete Jimenez if (i >= e) {
36672c33676SMaxim Ag BNerror(BN_R_NOT_A_SQUARE);
367f5b1c8a1SJohn Marino goto end;
368f5b1c8a1SJohn Marino }
369f5b1c8a1SJohn Marino
370f5b1c8a1SJohn Marino /* t := y^2^(e - i - 1) */
371f5b1c8a1SJohn Marino if (!BN_copy(t, y))
372f5b1c8a1SJohn Marino goto end;
373f5b1c8a1SJohn Marino for (j = e - i - 1; j > 0; j--) {
374f5b1c8a1SJohn Marino if (!BN_mod_sqr(t, t, p, ctx))
375f5b1c8a1SJohn Marino goto end;
376f5b1c8a1SJohn Marino }
377f5b1c8a1SJohn Marino if (!BN_mod_mul(y, t, t, p, ctx))
378f5b1c8a1SJohn Marino goto end;
379f5b1c8a1SJohn Marino if (!BN_mod_mul(x, x, t, p, ctx))
380f5b1c8a1SJohn Marino goto end;
381f5b1c8a1SJohn Marino if (!BN_mod_mul(b, b, y, p, ctx))
382f5b1c8a1SJohn Marino goto end;
383f5b1c8a1SJohn Marino e = i;
384f5b1c8a1SJohn Marino }
385f5b1c8a1SJohn Marino
386f5b1c8a1SJohn Marino vrfy:
387f5b1c8a1SJohn Marino if (!err) {
388f5b1c8a1SJohn Marino /* verify the result -- the input might have been not a square
389f5b1c8a1SJohn Marino * (test added in 0.9.8) */
390f5b1c8a1SJohn Marino
391f5b1c8a1SJohn Marino if (!BN_mod_sqr(x, ret, p, ctx))
392f5b1c8a1SJohn Marino err = 1;
393f5b1c8a1SJohn Marino
394f5b1c8a1SJohn Marino if (!err && 0 != BN_cmp(x, A)) {
39572c33676SMaxim Ag BNerror(BN_R_NOT_A_SQUARE);
396f5b1c8a1SJohn Marino err = 1;
397f5b1c8a1SJohn Marino }
398f5b1c8a1SJohn Marino }
399f5b1c8a1SJohn Marino
400f5b1c8a1SJohn Marino end:
401f5b1c8a1SJohn Marino if (err) {
402f5b1c8a1SJohn Marino if (ret != NULL && ret != in) {
403f5b1c8a1SJohn Marino BN_clear_free(ret);
404f5b1c8a1SJohn Marino }
405f5b1c8a1SJohn Marino ret = NULL;
406f5b1c8a1SJohn Marino }
407f5b1c8a1SJohn Marino BN_CTX_end(ctx);
408f5b1c8a1SJohn Marino bn_check_top(ret);
409f5b1c8a1SJohn Marino return ret;
410f5b1c8a1SJohn Marino }
411