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 %

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^e*q  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 := (2*a)*b^2
167
		 * we have
168
		 *     i^2 = (2*a)^((1 + (|p|-5)/4)*2)
169
		 *         = (2*a)^((p-1)/2)
170
		 *         = -1;
171
		 * so if we set
172
		 *      x := a*b*(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)*(2*a*b^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 := 2*a */
186
58
		if (!BN_mod_lshift1_quick(t, A, p))
187
			goto end;
188
189
		/* b := (2*a)^((|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 := (2*a)*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 = a*b*t */
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 := a*x^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 := a*x    (= 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
}