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1 /* crypto/bn/bn_gf2m.c */
2 /* ====================================================================
3  * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
4  *
5  * The Elliptic Curve Public-Key Crypto Library (ECC Code) included
6  * herein is developed by SUN MICROSYSTEMS, INC., and is contributed
7  * to the OpenSSL project.
8  *
9  * The ECC Code is licensed pursuant to the OpenSSL open source
10  * license provided below.
11  *
12  * In addition, Sun covenants to all licensees who provide a reciprocal
13  * covenant with respect to their own patents if any, not to sue under
14  * current and future patent claims necessarily infringed by the making,
15  * using, practicing, selling, offering for sale and/or otherwise
16  * disposing of the ECC Code as delivered hereunder (or portions thereof),
17  * provided that such covenant shall not apply:
18  *  1) for code that a licensee deletes from the ECC Code;
19  *  2) separates from the ECC Code; or
20  *  3) for infringements caused by:
21  *       i) the modification of the ECC Code or
22  *      ii) the combination of the ECC Code with other software or
23  *          devices where such combination causes the infringement.
24  *
25  * The software is originally written by Sheueling Chang Shantz and
26  * Douglas Stebila of Sun Microsystems Laboratories.
27  *
28  */
29 
30 /* NOTE: This file is licensed pursuant to the OpenSSL license below
31  * and may be modified; but after modifications, the above covenant
32  * may no longer apply!  In such cases, the corresponding paragraph
33  * ["In addition, Sun covenants ... causes the infringement."] and
34  * this note can be edited out; but please keep the Sun copyright
35  * notice and attribution. */
36 
37 /* ====================================================================
38  * Copyright (c) 1998-2002 The OpenSSL Project.  All rights reserved.
39  *
40  * Redistribution and use in source and binary forms, with or without
41  * modification, are permitted provided that the following conditions
42  * are met:
43  *
44  * 1. Redistributions of source code must retain the above copyright
45  *    notice, this list of conditions and the following disclaimer.
46  *
47  * 2. Redistributions in binary form must reproduce the above copyright
48  *    notice, this list of conditions and the following disclaimer in
49  *    the documentation and/or other materials provided with the
50  *    distribution.
51  *
52  * 3. All advertising materials mentioning features or use of this
53  *    software must display the following acknowledgment:
54  *    "This product includes software developed by the OpenSSL Project
55  *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
56  *
57  * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
58  *    endorse or promote products derived from this software without
59  *    prior written permission. For written permission, please contact
60  *    openssl-core@openssl.org.
61  *
62  * 5. Products derived from this software may not be called "OpenSSL"
63  *    nor may "OpenSSL" appear in their names without prior written
64  *    permission of the OpenSSL Project.
65  *
66  * 6. Redistributions of any form whatsoever must retain the following
67  *    acknowledgment:
68  *    "This product includes software developed by the OpenSSL Project
69  *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
70  *
71  * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
72  * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
73  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
74  * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
75  * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
76  * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
77  * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
78  * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
79  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
80  * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
81  * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
82  * OF THE POSSIBILITY OF SUCH DAMAGE.
83  * ====================================================================
84  *
85  * This product includes cryptographic software written by Eric Young
86  * (eay@cryptsoft.com).  This product includes software written by Tim
87  * Hudson (tjh@cryptsoft.com).
88  *
89  */
90 
91 #include <assert.h>
92 #include <limits.h>
93 #include <stdio.h>
94 #include "cryptlib.h"
95 #include "bn_lcl.h"
96 
97 /* Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should fail. */
98 #define MAX_ITERATIONS 50
99 
100 static const BN_ULONG SQR_tb[16] =
101   {     0,     1,     4,     5,    16,    17,    20,    21,
102        64,    65,    68,    69,    80,    81,    84,    85 };
103 /* Platform-specific macros to accelerate squaring. */
104 #if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
105 #define SQR1(w) \
106     SQR_tb[(w) >> 60 & 0xF] << 56 | SQR_tb[(w) >> 56 & 0xF] << 48 | \
107     SQR_tb[(w) >> 52 & 0xF] << 40 | SQR_tb[(w) >> 48 & 0xF] << 32 | \
108     SQR_tb[(w) >> 44 & 0xF] << 24 | SQR_tb[(w) >> 40 & 0xF] << 16 | \
109     SQR_tb[(w) >> 36 & 0xF] <<  8 | SQR_tb[(w) >> 32 & 0xF]
110 #define SQR0(w) \
111     SQR_tb[(w) >> 28 & 0xF] << 56 | SQR_tb[(w) >> 24 & 0xF] << 48 | \
112     SQR_tb[(w) >> 20 & 0xF] << 40 | SQR_tb[(w) >> 16 & 0xF] << 32 | \
113     SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >>  8 & 0xF] << 16 | \
114     SQR_tb[(w) >>  4 & 0xF] <<  8 | SQR_tb[(w)       & 0xF]
115 #endif
116 #ifdef THIRTY_TWO_BIT
117 #define SQR1(w) \
118     SQR_tb[(w) >> 28 & 0xF] << 24 | SQR_tb[(w) >> 24 & 0xF] << 16 | \
119     SQR_tb[(w) >> 20 & 0xF] <<  8 | SQR_tb[(w) >> 16 & 0xF]
120 #define SQR0(w) \
121     SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >>  8 & 0xF] << 16 | \
122     SQR_tb[(w) >>  4 & 0xF] <<  8 | SQR_tb[(w)       & 0xF]
123 #endif
124 
125 /* Product of two polynomials a, b each with degree < BN_BITS2 - 1,
126  * result is a polynomial r with degree < 2 * BN_BITS - 1
127  * The caller MUST ensure that the variables have the right amount
128  * of space allocated.
129  */
130 #ifdef THIRTY_TWO_BIT
bn_GF2m_mul_1x1(BN_ULONG * r1,BN_ULONG * r0,const BN_ULONG a,const BN_ULONG b)131 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
132 	{
133 	register BN_ULONG h, l, s;
134 	BN_ULONG tab[8], top2b = a >> 30;
135 	register BN_ULONG a1, a2, a4;
136 
137 	a1 = a & (0x3FFFFFFF); a2 = a1 << 1; a4 = a2 << 1;
138 
139 	tab[0] =  0; tab[1] = a1;    tab[2] = a2;    tab[3] = a1^a2;
140 	tab[4] = a4; tab[5] = a1^a4; tab[6] = a2^a4; tab[7] = a1^a2^a4;
141 
142 	s = tab[b       & 0x7]; l  = s;
143 	s = tab[b >>  3 & 0x7]; l ^= s <<  3; h  = s >> 29;
144 	s = tab[b >>  6 & 0x7]; l ^= s <<  6; h ^= s >> 26;
145 	s = tab[b >>  9 & 0x7]; l ^= s <<  9; h ^= s >> 23;
146 	s = tab[b >> 12 & 0x7]; l ^= s << 12; h ^= s >> 20;
147 	s = tab[b >> 15 & 0x7]; l ^= s << 15; h ^= s >> 17;
148 	s = tab[b >> 18 & 0x7]; l ^= s << 18; h ^= s >> 14;
149 	s = tab[b >> 21 & 0x7]; l ^= s << 21; h ^= s >> 11;
150 	s = tab[b >> 24 & 0x7]; l ^= s << 24; h ^= s >>  8;
151 	s = tab[b >> 27 & 0x7]; l ^= s << 27; h ^= s >>  5;
152 	s = tab[b >> 30      ]; l ^= s << 30; h ^= s >>  2;
153 
154 	/* compensate for the top two bits of a */
155 
156 	if (top2b & 01) { l ^= b << 30; h ^= b >> 2; }
157 	if (top2b & 02) { l ^= b << 31; h ^= b >> 1; }
158 
159 	*r1 = h; *r0 = l;
160 	}
161 #endif
162 #if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
bn_GF2m_mul_1x1(BN_ULONG * r1,BN_ULONG * r0,const BN_ULONG a,const BN_ULONG b)163 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
164 	{
165 	register BN_ULONG h, l, s;
166 	BN_ULONG tab[16], top3b = a >> 61;
167 	register BN_ULONG a1, a2, a4, a8;
168 
169 	a1 = a & (0x1FFFFFFFFFFFFFFFULL); a2 = a1 << 1; a4 = a2 << 1; a8 = a4 << 1;
170 
171 	tab[ 0] = 0;     tab[ 1] = a1;       tab[ 2] = a2;       tab[ 3] = a1^a2;
172 	tab[ 4] = a4;    tab[ 5] = a1^a4;    tab[ 6] = a2^a4;    tab[ 7] = a1^a2^a4;
173 	tab[ 8] = a8;    tab[ 9] = a1^a8;    tab[10] = a2^a8;    tab[11] = a1^a2^a8;
174 	tab[12] = a4^a8; tab[13] = a1^a4^a8; tab[14] = a2^a4^a8; tab[15] = a1^a2^a4^a8;
175 
176 	s = tab[b       & 0xF]; l  = s;
177 	s = tab[b >>  4 & 0xF]; l ^= s <<  4; h  = s >> 60;
178 	s = tab[b >>  8 & 0xF]; l ^= s <<  8; h ^= s >> 56;
179 	s = tab[b >> 12 & 0xF]; l ^= s << 12; h ^= s >> 52;
180 	s = tab[b >> 16 & 0xF]; l ^= s << 16; h ^= s >> 48;
181 	s = tab[b >> 20 & 0xF]; l ^= s << 20; h ^= s >> 44;
182 	s = tab[b >> 24 & 0xF]; l ^= s << 24; h ^= s >> 40;
183 	s = tab[b >> 28 & 0xF]; l ^= s << 28; h ^= s >> 36;
184 	s = tab[b >> 32 & 0xF]; l ^= s << 32; h ^= s >> 32;
185 	s = tab[b >> 36 & 0xF]; l ^= s << 36; h ^= s >> 28;
186 	s = tab[b >> 40 & 0xF]; l ^= s << 40; h ^= s >> 24;
187 	s = tab[b >> 44 & 0xF]; l ^= s << 44; h ^= s >> 20;
188 	s = tab[b >> 48 & 0xF]; l ^= s << 48; h ^= s >> 16;
189 	s = tab[b >> 52 & 0xF]; l ^= s << 52; h ^= s >> 12;
190 	s = tab[b >> 56 & 0xF]; l ^= s << 56; h ^= s >>  8;
191 	s = tab[b >> 60      ]; l ^= s << 60; h ^= s >>  4;
192 
193 	/* compensate for the top three bits of a */
194 
195 	if (top3b & 01) { l ^= b << 61; h ^= b >> 3; }
196 	if (top3b & 02) { l ^= b << 62; h ^= b >> 2; }
197 	if (top3b & 04) { l ^= b << 63; h ^= b >> 1; }
198 
199 	*r1 = h; *r0 = l;
200 	}
201 #endif
202 
203 /* Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1,
204  * result is a polynomial r with degree < 4 * BN_BITS2 - 1
205  * The caller MUST ensure that the variables have the right amount
206  * of space allocated.
207  */
bn_GF2m_mul_2x2(BN_ULONG * r,const BN_ULONG a1,const BN_ULONG a0,const BN_ULONG b1,const BN_ULONG b0)208 static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0, const BN_ULONG b1, const BN_ULONG b0)
209 	{
210 	BN_ULONG m1, m0;
211 	/* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
212 	bn_GF2m_mul_1x1(r+3, r+2, a1, b1);
213 	bn_GF2m_mul_1x1(r+1, r, a0, b0);
214 	bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1);
215 	/* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
216 	r[2] ^= m1 ^ r[1] ^ r[3];  /* h0 ^= m1 ^ l1 ^ h1; */
217 	r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0;  /* l1 ^= l0 ^ h0 ^ m0; */
218 	}
219 
220 
221 /* Add polynomials a and b and store result in r; r could be a or b, a and b
222  * could be equal; r is the bitwise XOR of a and b.
223  */
BN_GF2m_add(BIGNUM * r,const BIGNUM * a,const BIGNUM * b)224 int	BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b)
225 	{
226 	int i;
227 	const BIGNUM *at, *bt;
228 
229 	bn_check_top(a);
230 	bn_check_top(b);
231 
232 	if (a->top < b->top) { at = b; bt = a; }
233 	else { at = a; bt = b; }
234 
235 	if(bn_wexpand(r, at->top) == NULL)
236 		return 0;
237 
238 	for (i = 0; i < bt->top; i++)
239 		{
240 		r->d[i] = at->d[i] ^ bt->d[i];
241 		}
242 	for (; i < at->top; i++)
243 		{
244 		r->d[i] = at->d[i];
245 		}
246 
247 	r->top = at->top;
248 	bn_correct_top(r);
249 
250 	return 1;
251 	}
252 
253 
254 /* Some functions allow for representation of the irreducible polynomials
255  * as an int[], say p.  The irreducible f(t) is then of the form:
256  *     t^p[0] + t^p[1] + ... + t^p[k]
257  * where m = p[0] > p[1] > ... > p[k] = 0.
258  */
259 
260 
261 /* Performs modular reduction of a and store result in r.  r could be a. */
BN_GF2m_mod_arr(BIGNUM * r,const BIGNUM * a,const int p[])262 int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[])
263 	{
264 	int j, k;
265 	int n, dN, d0, d1;
266 	BN_ULONG zz, *z;
267 
268 	bn_check_top(a);
269 
270 	if (!p[0])
271 		{
272 		/* reduction mod 1 => return 0 */
273 		BN_zero(r);
274 		return 1;
275 		}
276 
277 	/* Since the algorithm does reduction in the r value, if a != r, copy
278 	 * the contents of a into r so we can do reduction in r.
279 	 */
280 	if (a != r)
281 		{
282 		if (!bn_wexpand(r, a->top)) return 0;
283 		for (j = 0; j < a->top; j++)
284 			{
285 			r->d[j] = a->d[j];
286 			}
287 		r->top = a->top;
288 		}
289 	z = r->d;
290 
291 	/* start reduction */
292 	dN = p[0] / BN_BITS2;
293 	for (j = r->top - 1; j > dN;)
294 		{
295 		zz = z[j];
296 		if (z[j] == 0) { j--; continue; }
297 		z[j] = 0;
298 
299 		for (k = 1; p[k] != 0; k++)
300 			{
301 			/* reducing component t^p[k] */
302 			n = p[0] - p[k];
303 			d0 = n % BN_BITS2;  d1 = BN_BITS2 - d0;
304 			n /= BN_BITS2;
305 			z[j-n] ^= (zz>>d0);
306 			if (d0) z[j-n-1] ^= (zz<<d1);
307 			}
308 
309 		/* reducing component t^0 */
310 		n = dN;
311 		d0 = p[0] % BN_BITS2;
312 		d1 = BN_BITS2 - d0;
313 		z[j-n] ^= (zz >> d0);
314 		if (d0) z[j-n-1] ^= (zz << d1);
315 		}
316 
317 	/* final round of reduction */
318 	while (j == dN)
319 		{
320 
321 		d0 = p[0] % BN_BITS2;
322 		zz = z[dN] >> d0;
323 		if (zz == 0) break;
324 		d1 = BN_BITS2 - d0;
325 
326 		/* clear up the top d1 bits */
327 		if (d0)
328 			z[dN] = (z[dN] << d1) >> d1;
329 		else
330 			z[dN] = 0;
331 		z[0] ^= zz; /* reduction t^0 component */
332 
333 		for (k = 1; p[k] != 0; k++)
334 			{
335 			BN_ULONG tmp_ulong;
336 
337 			/* reducing component t^p[k]*/
338 			n = p[k] / BN_BITS2;
339 			d0 = p[k] % BN_BITS2;
340 			d1 = BN_BITS2 - d0;
341 			z[n] ^= (zz << d0);
342 			tmp_ulong = zz >> d1;
343                         if (d0 && tmp_ulong)
344                                 z[n+1] ^= tmp_ulong;
345 			}
346 
347 
348 		}
349 
350 	bn_correct_top(r);
351 	return 1;
352 	}
353 
354 /* Performs modular reduction of a by p and store result in r.  r could be a.
355  *
356  * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
357  * function is only provided for convenience; for best performance, use the
358  * BN_GF2m_mod_arr function.
359  */
BN_GF2m_mod(BIGNUM * r,const BIGNUM * a,const BIGNUM * p)360 int	BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
361 	{
362 	int ret = 0;
363 	const int max = BN_num_bits(p) + 1;
364 	int *arr=NULL;
365 	bn_check_top(a);
366 	bn_check_top(p);
367 	if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
368 	ret = BN_GF2m_poly2arr(p, arr, max);
369 	if (!ret || ret > max)
370 		{
371 		BNerr(BN_F_BN_GF2M_MOD,BN_R_INVALID_LENGTH);
372 		goto err;
373 		}
374 	ret = BN_GF2m_mod_arr(r, a, arr);
375 	bn_check_top(r);
376 err:
377 	if (arr) OPENSSL_free(arr);
378 	return ret;
379 	}
380 
381 
382 /* Compute the product of two polynomials a and b, reduce modulo p, and store
383  * the result in r.  r could be a or b; a could be b.
384  */
BN_GF2m_mod_mul_arr(BIGNUM * r,const BIGNUM * a,const BIGNUM * b,const int p[],BN_CTX * ctx)385 int	BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx)
386 	{
387 	int zlen, i, j, k, ret = 0;
388 	BIGNUM *s;
389 	BN_ULONG x1, x0, y1, y0, zz[4];
390 
391 	bn_check_top(a);
392 	bn_check_top(b);
393 
394 	if (a == b)
395 		{
396 		return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
397 		}
398 
399 	BN_CTX_start(ctx);
400 	if ((s = BN_CTX_get(ctx)) == NULL) goto err;
401 
402 	zlen = a->top + b->top + 4;
403 	if (!bn_wexpand(s, zlen)) goto err;
404 	s->top = zlen;
405 
406 	for (i = 0; i < zlen; i++) s->d[i] = 0;
407 
408 	for (j = 0; j < b->top; j += 2)
409 		{
410 		y0 = b->d[j];
411 		y1 = ((j+1) == b->top) ? 0 : b->d[j+1];
412 		for (i = 0; i < a->top; i += 2)
413 			{
414 			x0 = a->d[i];
415 			x1 = ((i+1) == a->top) ? 0 : a->d[i+1];
416 			bn_GF2m_mul_2x2(zz, x1, x0, y1, y0);
417 			for (k = 0; k < 4; k++) s->d[i+j+k] ^= zz[k];
418 			}
419 		}
420 
421 	bn_correct_top(s);
422 	if (BN_GF2m_mod_arr(r, s, p))
423 		ret = 1;
424 	bn_check_top(r);
425 
426 err:
427 	BN_CTX_end(ctx);
428 	return ret;
429 	}
430 
431 /* Compute the product of two polynomials a and b, reduce modulo p, and store
432  * the result in r.  r could be a or b; a could equal b.
433  *
434  * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper
435  * function is only provided for convenience; for best performance, use the
436  * BN_GF2m_mod_mul_arr function.
437  */
BN_GF2m_mod_mul(BIGNUM * r,const BIGNUM * a,const BIGNUM * b,const BIGNUM * p,BN_CTX * ctx)438 int	BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
439 	{
440 	int ret = 0;
441 	const int max = BN_num_bits(p) + 1;
442 	int *arr=NULL;
443 	bn_check_top(a);
444 	bn_check_top(b);
445 	bn_check_top(p);
446 	if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
447 	ret = BN_GF2m_poly2arr(p, arr, max);
448 	if (!ret || ret > max)
449 		{
450 		BNerr(BN_F_BN_GF2M_MOD_MUL,BN_R_INVALID_LENGTH);
451 		goto err;
452 		}
453 	ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx);
454 	bn_check_top(r);
455 err:
456 	if (arr) OPENSSL_free(arr);
457 	return ret;
458 	}
459 
460 
461 /* Square a, reduce the result mod p, and store it in a.  r could be a. */
BN_GF2m_mod_sqr_arr(BIGNUM * r,const BIGNUM * a,const int p[],BN_CTX * ctx)462 int	BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx)
463 	{
464 	int i, ret = 0;
465 	BIGNUM *s;
466 
467 	bn_check_top(a);
468 	BN_CTX_start(ctx);
469 	if ((s = BN_CTX_get(ctx)) == NULL) return 0;
470 	if (!bn_wexpand(s, 2 * a->top)) goto err;
471 
472 	for (i = a->top - 1; i >= 0; i--)
473 		{
474 		s->d[2*i+1] = SQR1(a->d[i]);
475 		s->d[2*i  ] = SQR0(a->d[i]);
476 		}
477 
478 	s->top = 2 * a->top;
479 	bn_correct_top(s);
480 	if (!BN_GF2m_mod_arr(r, s, p)) goto err;
481 	bn_check_top(r);
482 	ret = 1;
483 err:
484 	BN_CTX_end(ctx);
485 	return ret;
486 	}
487 
488 /* Square a, reduce the result mod p, and store it in a.  r could be a.
489  *
490  * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper
491  * function is only provided for convenience; for best performance, use the
492  * BN_GF2m_mod_sqr_arr function.
493  */
BN_GF2m_mod_sqr(BIGNUM * r,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx)494 int	BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
495 	{
496 	int ret = 0;
497 	const int max = BN_num_bits(p) + 1;
498 	int *arr=NULL;
499 
500 	bn_check_top(a);
501 	bn_check_top(p);
502 	if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
503 	ret = BN_GF2m_poly2arr(p, arr, max);
504 	if (!ret || ret > max)
505 		{
506 		BNerr(BN_F_BN_GF2M_MOD_SQR,BN_R_INVALID_LENGTH);
507 		goto err;
508 		}
509 	ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx);
510 	bn_check_top(r);
511 err:
512 	if (arr) OPENSSL_free(arr);
513 	return ret;
514 	}
515 
516 
517 /* Invert a, reduce modulo p, and store the result in r. r could be a.
518  * Uses Modified Almost Inverse Algorithm (Algorithm 10) from
519  *     Hankerson, D., Hernandez, J.L., and Menezes, A.  "Software Implementation
520  *     of Elliptic Curve Cryptography Over Binary Fields".
521  */
BN_GF2m_mod_inv(BIGNUM * r,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx)522 int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
523 	{
524 	BIGNUM *b, *c, *u, *v, *tmp;
525 	int ret = 0;
526 
527 	bn_check_top(a);
528 	bn_check_top(p);
529 
530 	BN_CTX_start(ctx);
531 
532 	b = BN_CTX_get(ctx);
533 	c = BN_CTX_get(ctx);
534 	u = BN_CTX_get(ctx);
535 	v = BN_CTX_get(ctx);
536 	if (v == NULL) goto err;
537 
538 	if (!BN_one(b)) goto err;
539 	if (!BN_GF2m_mod(u, a, p)) goto err;
540 	if (!BN_copy(v, p)) goto err;
541 
542 	if (BN_is_zero(u)) goto err;
543 
544 	while (1)
545 		{
546 		while (!BN_is_odd(u))
547 			{
548 			if (!BN_rshift1(u, u)) goto err;
549 			if (BN_is_odd(b))
550 				{
551 				if (!BN_GF2m_add(b, b, p)) goto err;
552 				}
553 			if (!BN_rshift1(b, b)) goto err;
554 			}
555 
556 		if (BN_abs_is_word(u, 1)) break;
557 
558 		if (BN_num_bits(u) < BN_num_bits(v))
559 			{
560 			tmp = u; u = v; v = tmp;
561 			tmp = b; b = c; c = tmp;
562 			}
563 
564 		if (!BN_GF2m_add(u, u, v)) goto err;
565 		if (!BN_GF2m_add(b, b, c)) goto err;
566 		}
567 
568 
569 	if (!BN_copy(r, b)) goto err;
570 	bn_check_top(r);
571 	ret = 1;
572 
573 err:
574   	BN_CTX_end(ctx);
575 	return ret;
576 	}
577 
578 /* Invert xx, reduce modulo p, and store the result in r. r could be xx.
579  *
580  * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper
581  * function is only provided for convenience; for best performance, use the
582  * BN_GF2m_mod_inv function.
583  */
BN_GF2m_mod_inv_arr(BIGNUM * r,const BIGNUM * xx,const int p[],BN_CTX * ctx)584 int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], BN_CTX *ctx)
585 	{
586 	BIGNUM *field;
587 	int ret = 0;
588 
589 	bn_check_top(xx);
590 	BN_CTX_start(ctx);
591 	if ((field = BN_CTX_get(ctx)) == NULL) goto err;
592 	if (!BN_GF2m_arr2poly(p, field)) goto err;
593 
594 	ret = BN_GF2m_mod_inv(r, xx, field, ctx);
595 	bn_check_top(r);
596 
597 err:
598 	BN_CTX_end(ctx);
599 	return ret;
600 	}
601 
602 
603 #ifndef OPENSSL_SUN_GF2M_DIV
604 /* Divide y by x, reduce modulo p, and store the result in r. r could be x
605  * or y, x could equal y.
606  */
BN_GF2m_mod_div(BIGNUM * r,const BIGNUM * y,const BIGNUM * x,const BIGNUM * p,BN_CTX * ctx)607 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
608 	{
609 	BIGNUM *xinv = NULL;
610 	int ret = 0;
611 
612 	bn_check_top(y);
613 	bn_check_top(x);
614 	bn_check_top(p);
615 
616 	BN_CTX_start(ctx);
617 	xinv = BN_CTX_get(ctx);
618 	if (xinv == NULL) goto err;
619 
620 	if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err;
621 	if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err;
622 	bn_check_top(r);
623 	ret = 1;
624 
625 err:
626 	BN_CTX_end(ctx);
627 	return ret;
628 	}
629 #else
630 /* Divide y by x, reduce modulo p, and store the result in r. r could be x
631  * or y, x could equal y.
632  * Uses algorithm Modular_Division_GF(2^m) from
633  *     Chang-Shantz, S.  "From Euclid's GCD to Montgomery Multiplication to
634  *     the Great Divide".
635  */
BN_GF2m_mod_div(BIGNUM * r,const BIGNUM * y,const BIGNUM * x,const BIGNUM * p,BN_CTX * ctx)636 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
637 	{
638 	BIGNUM *a, *b, *u, *v;
639 	int ret = 0;
640 
641 	bn_check_top(y);
642 	bn_check_top(x);
643 	bn_check_top(p);
644 
645 	BN_CTX_start(ctx);
646 
647 	a = BN_CTX_get(ctx);
648 	b = BN_CTX_get(ctx);
649 	u = BN_CTX_get(ctx);
650 	v = BN_CTX_get(ctx);
651 	if (v == NULL) goto err;
652 
653 	/* reduce x and y mod p */
654 	if (!BN_GF2m_mod(u, y, p)) goto err;
655 	if (!BN_GF2m_mod(a, x, p)) goto err;
656 	if (!BN_copy(b, p)) goto err;
657 
658 	while (!BN_is_odd(a))
659 		{
660 		if (!BN_rshift1(a, a)) goto err;
661 		if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
662 		if (!BN_rshift1(u, u)) goto err;
663 		}
664 
665 	do
666 		{
667 		if (BN_GF2m_cmp(b, a) > 0)
668 			{
669 			if (!BN_GF2m_add(b, b, a)) goto err;
670 			if (!BN_GF2m_add(v, v, u)) goto err;
671 			do
672 				{
673 				if (!BN_rshift1(b, b)) goto err;
674 				if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err;
675 				if (!BN_rshift1(v, v)) goto err;
676 				} while (!BN_is_odd(b));
677 			}
678 		else if (BN_abs_is_word(a, 1))
679 			break;
680 		else
681 			{
682 			if (!BN_GF2m_add(a, a, b)) goto err;
683 			if (!BN_GF2m_add(u, u, v)) goto err;
684 			do
685 				{
686 				if (!BN_rshift1(a, a)) goto err;
687 				if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
688 				if (!BN_rshift1(u, u)) goto err;
689 				} while (!BN_is_odd(a));
690 			}
691 		} while (1);
692 
693 	if (!BN_copy(r, u)) goto err;
694 	bn_check_top(r);
695 	ret = 1;
696 
697 err:
698   	BN_CTX_end(ctx);
699 	return ret;
700 	}
701 #endif
702 
703 /* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
704  * or yy, xx could equal yy.
705  *
706  * This function calls down to the BN_GF2m_mod_div implementation; this wrapper
707  * function is only provided for convenience; for best performance, use the
708  * BN_GF2m_mod_div function.
709  */
BN_GF2m_mod_div_arr(BIGNUM * r,const BIGNUM * yy,const BIGNUM * xx,const int p[],BN_CTX * ctx)710 int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const int p[], BN_CTX *ctx)
711 	{
712 	BIGNUM *field;
713 	int ret = 0;
714 
715 	bn_check_top(yy);
716 	bn_check_top(xx);
717 
718 	BN_CTX_start(ctx);
719 	if ((field = BN_CTX_get(ctx)) == NULL) goto err;
720 	if (!BN_GF2m_arr2poly(p, field)) goto err;
721 
722 	ret = BN_GF2m_mod_div(r, yy, xx, field, ctx);
723 	bn_check_top(r);
724 
725 err:
726 	BN_CTX_end(ctx);
727 	return ret;
728 	}
729 
730 
731 /* Compute the bth power of a, reduce modulo p, and store
732  * the result in r.  r could be a.
733  * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363.
734  */
BN_GF2m_mod_exp_arr(BIGNUM * r,const BIGNUM * a,const BIGNUM * b,const int p[],BN_CTX * ctx)735 int	BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx)
736 	{
737 	int ret = 0, i, n;
738 	BIGNUM *u;
739 
740 	bn_check_top(a);
741 	bn_check_top(b);
742 
743 	if (BN_is_zero(b))
744 		return(BN_one(r));
745 
746 	if (BN_abs_is_word(b, 1))
747 		return (BN_copy(r, a) != NULL);
748 
749 	BN_CTX_start(ctx);
750 	if ((u = BN_CTX_get(ctx)) == NULL) goto err;
751 
752 	if (!BN_GF2m_mod_arr(u, a, p)) goto err;
753 
754 	n = BN_num_bits(b) - 1;
755 	for (i = n - 1; i >= 0; i--)
756 		{
757 		if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err;
758 		if (BN_is_bit_set(b, i))
759 			{
760 			if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err;
761 			}
762 		}
763 	if (!BN_copy(r, u)) goto err;
764 	bn_check_top(r);
765 	ret = 1;
766 err:
767 	BN_CTX_end(ctx);
768 	return ret;
769 	}
770 
771 /* Compute the bth power of a, reduce modulo p, and store
772  * the result in r.  r could be a.
773  *
774  * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper
775  * function is only provided for convenience; for best performance, use the
776  * BN_GF2m_mod_exp_arr function.
777  */
BN_GF2m_mod_exp(BIGNUM * r,const BIGNUM * a,const BIGNUM * b,const BIGNUM * p,BN_CTX * ctx)778 int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
779 	{
780 	int ret = 0;
781 	const int max = BN_num_bits(p) + 1;
782 	int *arr=NULL;
783 	bn_check_top(a);
784 	bn_check_top(b);
785 	bn_check_top(p);
786 	if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
787 	ret = BN_GF2m_poly2arr(p, arr, max);
788 	if (!ret || ret > max)
789 		{
790 		BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH);
791 		goto err;
792 		}
793 	ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx);
794 	bn_check_top(r);
795 err:
796 	if (arr) OPENSSL_free(arr);
797 	return ret;
798 	}
799 
800 /* Compute the square root of a, reduce modulo p, and store
801  * the result in r.  r could be a.
802  * Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
803  */
BN_GF2m_mod_sqrt_arr(BIGNUM * r,const BIGNUM * a,const int p[],BN_CTX * ctx)804 int	BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx)
805 	{
806 	int ret = 0;
807 	BIGNUM *u;
808 
809 	bn_check_top(a);
810 
811 	if (!p[0])
812 		{
813 		/* reduction mod 1 => return 0 */
814 		BN_zero(r);
815 		return 1;
816 		}
817 
818 	BN_CTX_start(ctx);
819 	if ((u = BN_CTX_get(ctx)) == NULL) goto err;
820 
821 	if (!BN_set_bit(u, p[0] - 1)) goto err;
822 	ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx);
823 	bn_check_top(r);
824 
825 err:
826 	BN_CTX_end(ctx);
827 	return ret;
828 	}
829 
830 /* Compute the square root of a, reduce modulo p, and store
831  * the result in r.  r could be a.
832  *
833  * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper
834  * function is only provided for convenience; for best performance, use the
835  * BN_GF2m_mod_sqrt_arr function.
836  */
BN_GF2m_mod_sqrt(BIGNUM * r,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx)837 int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
838 	{
839 	int ret = 0;
840 	const int max = BN_num_bits(p) + 1;
841 	int *arr=NULL;
842 	bn_check_top(a);
843 	bn_check_top(p);
844 	if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
845 	ret = BN_GF2m_poly2arr(p, arr, max);
846 	if (!ret || ret > max)
847 		{
848 		BNerr(BN_F_BN_GF2M_MOD_SQRT,BN_R_INVALID_LENGTH);
849 		goto err;
850 		}
851 	ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx);
852 	bn_check_top(r);
853 err:
854 	if (arr) OPENSSL_free(arr);
855 	return ret;
856 	}
857 
858 /* Find r such that r^2 + r = a mod p.  r could be a. If no r exists returns 0.
859  * Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
860  */
BN_GF2m_mod_solve_quad_arr(BIGNUM * r,const BIGNUM * a_,const int p[],BN_CTX * ctx)861 int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], BN_CTX *ctx)
862 	{
863 	int ret = 0, count = 0, j;
864 	BIGNUM *a, *z, *rho, *w, *w2, *tmp;
865 
866 	bn_check_top(a_);
867 
868 	if (!p[0])
869 		{
870 		/* reduction mod 1 => return 0 */
871 		BN_zero(r);
872 		return 1;
873 		}
874 
875 	BN_CTX_start(ctx);
876 	a = BN_CTX_get(ctx);
877 	z = BN_CTX_get(ctx);
878 	w = BN_CTX_get(ctx);
879 	if (w == NULL) goto err;
880 
881 	if (!BN_GF2m_mod_arr(a, a_, p)) goto err;
882 
883 	if (BN_is_zero(a))
884 		{
885 		BN_zero(r);
886 		ret = 1;
887 		goto err;
888 		}
889 
890 	if (p[0] & 0x1) /* m is odd */
891 		{
892 		/* compute half-trace of a */
893 		if (!BN_copy(z, a)) goto err;
894 		for (j = 1; j <= (p[0] - 1) / 2; j++)
895 			{
896 			if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
897 			if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
898 			if (!BN_GF2m_add(z, z, a)) goto err;
899 			}
900 
901 		}
902 	else /* m is even */
903 		{
904 		rho = BN_CTX_get(ctx);
905 		w2 = BN_CTX_get(ctx);
906 		tmp = BN_CTX_get(ctx);
907 		if (tmp == NULL) goto err;
908 		do
909 			{
910 			if (!BN_rand(rho, p[0], 0, 0)) goto err;
911 			if (!BN_GF2m_mod_arr(rho, rho, p)) goto err;
912 			BN_zero(z);
913 			if (!BN_copy(w, rho)) goto err;
914 			for (j = 1; j <= p[0] - 1; j++)
915 				{
916 				if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
917 				if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err;
918 				if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err;
919 				if (!BN_GF2m_add(z, z, tmp)) goto err;
920 				if (!BN_GF2m_add(w, w2, rho)) goto err;
921 				}
922 			count++;
923 			} while (BN_is_zero(w) && (count < MAX_ITERATIONS));
924 		if (BN_is_zero(w))
925 			{
926 			BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS);
927 			goto err;
928 			}
929 		}
930 
931 	if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err;
932 	if (!BN_GF2m_add(w, z, w)) goto err;
933 	if (BN_GF2m_cmp(w, a))
934 		{
935 		BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION);
936 		goto err;
937 		}
938 
939 	if (!BN_copy(r, z)) goto err;
940 	bn_check_top(r);
941 
942 	ret = 1;
943 
944 err:
945 	BN_CTX_end(ctx);
946 	return ret;
947 	}
948 
949 /* Find r such that r^2 + r = a mod p.  r could be a. If no r exists returns 0.
950  *
951  * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper
952  * function is only provided for convenience; for best performance, use the
953  * BN_GF2m_mod_solve_quad_arr function.
954  */
BN_GF2m_mod_solve_quad(BIGNUM * r,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx)955 int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
956 	{
957 	int ret = 0;
958 	const int max = BN_num_bits(p) + 1;
959 	int *arr=NULL;
960 	bn_check_top(a);
961 	bn_check_top(p);
962 	if ((arr = (int *)OPENSSL_malloc(sizeof(int) *
963 						max)) == NULL) goto err;
964 	ret = BN_GF2m_poly2arr(p, arr, max);
965 	if (!ret || ret > max)
966 		{
967 		BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH);
968 		goto err;
969 		}
970 	ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx);
971 	bn_check_top(r);
972 err:
973 	if (arr) OPENSSL_free(arr);
974 	return ret;
975 	}
976 
977 /* Convert the bit-string representation of a polynomial
978  * ( \sum_{i=0}^n a_i * x^i) into an array of integers corresponding
979  * to the bits with non-zero coefficient.  Array is terminated with -1.
980  * Up to max elements of the array will be filled.  Return value is total
981  * number of array elements that would be filled if array was large enough.
982  */
BN_GF2m_poly2arr(const BIGNUM * a,int p[],int max)983 int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max)
984 	{
985 	int i, j, k = 0;
986 	BN_ULONG mask;
987 
988 	if (BN_is_zero(a))
989 		return 0;
990 
991 	for (i = a->top - 1; i >= 0; i--)
992 		{
993 		if (!a->d[i])
994 			/* skip word if a->d[i] == 0 */
995 			continue;
996 		mask = BN_TBIT;
997 		for (j = BN_BITS2 - 1; j >= 0; j--)
998 			{
999 			if (a->d[i] & mask)
1000 				{
1001 				if (k < max) p[k] = BN_BITS2 * i + j;
1002 				k++;
1003 				}
1004 			mask >>= 1;
1005 			}
1006 		}
1007 
1008 	if (k < max) {
1009 		p[k] = -1;
1010 		k++;
1011 	}
1012 
1013 	return k;
1014 	}
1015 
1016 /* Convert the coefficient array representation of a polynomial to a
1017  * bit-string.  The array must be terminated by -1.
1018  */
BN_GF2m_arr2poly(const int p[],BIGNUM * a)1019 int BN_GF2m_arr2poly(const int p[], BIGNUM *a)
1020 	{
1021 	int i;
1022 
1023 	bn_check_top(a);
1024 	BN_zero(a);
1025 	for (i = 0; p[i] != -1; i++)
1026 		{
1027 		if (BN_set_bit(a, p[i]) == 0)
1028 			return 0;
1029 		}
1030 	bn_check_top(a);
1031 
1032 	return 1;
1033 	}
1034 
1035