1 /* crypto/bn/bn_gf2m.c */
2 /* ====================================================================
3 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
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.
9 * The ECC Code is licensed pursuant to the OpenSSL open source
10 * license provided below.
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.
25 * The software is originally written by Sheueling Chang Shantz and
26 * Douglas Stebila of Sun Microsystems Laboratories.
30 /* ====================================================================
31 * Copyright (c) 1998-2002 The OpenSSL Project. All rights reserved.
33 * Redistribution and use in source and binary forms, with or without
34 * modification, are permitted provided that the following conditions
37 * 1. Redistributions of source code must retain the above copyright
38 * notice, this list of conditions and the following disclaimer.
40 * 2. Redistributions in binary form must reproduce the above copyright
41 * notice, this list of conditions and the following disclaimer in
42 * the documentation and/or other materials provided with the
45 * 3. All advertising materials mentioning features or use of this
46 * software must display the following acknowledgment:
47 * "This product includes software developed by the OpenSSL Project
48 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
50 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
51 * endorse or promote products derived from this software without
52 * prior written permission. For written permission, please contact
53 * openssl-core@openssl.org.
55 * 5. Products derived from this software may not be called "OpenSSL"
56 * nor may "OpenSSL" appear in their names without prior written
57 * permission of the OpenSSL Project.
59 * 6. Redistributions of any form whatsoever must retain the following
61 * "This product includes software developed by the OpenSSL Project
62 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
64 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
65 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
66 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
67 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
68 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
69 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
70 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
71 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
72 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
73 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
74 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
75 * OF THE POSSIBILITY OF SUCH DAMAGE.
76 * ====================================================================
78 * This product includes cryptographic software written by Eric Young
79 * (eay@cryptsoft.com). This product includes software written by Tim
80 * Hudson (tjh@cryptsoft.com).
90 /* Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should fail. */
91 #define MAX_ITERATIONS 50
93 static const BN_ULONG SQR_tb
[16] =
94 { 0, 1, 4, 5, 16, 17, 20, 21,
95 64, 65, 68, 69, 80, 81, 84, 85 };
96 /* Platform-specific macros to accelerate squaring. */
97 #if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
99 SQR_tb[(w) >> 60 & 0xF] << 56 | SQR_tb[(w) >> 56 & 0xF] << 48 | \
100 SQR_tb[(w) >> 52 & 0xF] << 40 | SQR_tb[(w) >> 48 & 0xF] << 32 | \
101 SQR_tb[(w) >> 44 & 0xF] << 24 | SQR_tb[(w) >> 40 & 0xF] << 16 | \
102 SQR_tb[(w) >> 36 & 0xF] << 8 | SQR_tb[(w) >> 32 & 0xF]
104 SQR_tb[(w) >> 28 & 0xF] << 56 | SQR_tb[(w) >> 24 & 0xF] << 48 | \
105 SQR_tb[(w) >> 20 & 0xF] << 40 | SQR_tb[(w) >> 16 & 0xF] << 32 | \
106 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
107 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
109 #ifdef THIRTY_TWO_BIT
111 SQR_tb[(w) >> 28 & 0xF] << 24 | SQR_tb[(w) >> 24 & 0xF] << 16 | \
112 SQR_tb[(w) >> 20 & 0xF] << 8 | SQR_tb[(w) >> 16 & 0xF]
114 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
115 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
119 SQR_tb[(w) >> 12 & 0xF] << 8 | SQR_tb[(w) >> 8 & 0xF]
121 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
125 SQR_tb[(w) >> 4 & 0xF]
130 /* Product of two polynomials a, b each with degree < BN_BITS2 - 1,
131 * result is a polynomial r with degree < 2 * BN_BITS - 1
132 * The caller MUST ensure that the variables have the right amount
133 * of space allocated.
136 static void bn_GF2m_mul_1x1(BN_ULONG
*r1
, BN_ULONG
*r0
, const BN_ULONG a
, const BN_ULONG b
)
138 register BN_ULONG h
, l
, s
;
139 BN_ULONG tab
[4], top1b
= a
>> 7;
140 register BN_ULONG a1
, a2
;
142 a1
= a
& (0x7F); a2
= a1
<< 1;
144 tab
[0] = 0; tab
[1] = a1
; tab
[2] = a2
; tab
[3] = a1
^a2
;
146 s
= tab
[b
& 0x3]; l
= s
;
147 s
= tab
[b
>> 2 & 0x3]; l
^= s
<< 2; h
= s
>> 6;
148 s
= tab
[b
>> 4 & 0x3]; l
^= s
<< 4; h
^= s
>> 4;
149 s
= tab
[b
>> 6 ]; l
^= s
<< 6; h
^= s
>> 2;
151 /* compensate for the top bit of a */
153 if (top1b
& 01) { l
^= b
<< 7; h
^= b
>> 1; }
159 static void bn_GF2m_mul_1x1(BN_ULONG
*r1
, BN_ULONG
*r0
, const BN_ULONG a
, const BN_ULONG b
)
161 register BN_ULONG h
, l
, s
;
162 BN_ULONG tab
[4], top1b
= a
>> 15;
163 register BN_ULONG a1
, a2
;
165 a1
= a
& (0x7FFF); a2
= a1
<< 1;
167 tab
[0] = 0; tab
[1] = a1
; tab
[2] = a2
; tab
[3] = a1
^a2
;
169 s
= tab
[b
& 0x3]; l
= s
;
170 s
= tab
[b
>> 2 & 0x3]; l
^= s
<< 2; h
= s
>> 14;
171 s
= tab
[b
>> 4 & 0x3]; l
^= s
<< 4; h
^= s
>> 12;
172 s
= tab
[b
>> 6 & 0x3]; l
^= s
<< 6; h
^= s
>> 10;
173 s
= tab
[b
>> 8 & 0x3]; l
^= s
<< 8; h
^= s
>> 8;
174 s
= tab
[b
>>10 & 0x3]; l
^= s
<< 10; h
^= s
>> 6;
175 s
= tab
[b
>>12 & 0x3]; l
^= s
<< 12; h
^= s
>> 4;
176 s
= tab
[b
>>14 ]; l
^= s
<< 14; h
^= s
>> 2;
178 /* compensate for the top bit of a */
180 if (top1b
& 01) { l
^= b
<< 15; h
^= b
>> 1; }
185 #ifdef THIRTY_TWO_BIT
186 static void bn_GF2m_mul_1x1(BN_ULONG
*r1
, BN_ULONG
*r0
, const BN_ULONG a
, const BN_ULONG b
)
188 register BN_ULONG h
, l
, s
;
189 BN_ULONG tab
[8], top2b
= a
>> 30;
190 register BN_ULONG a1
, a2
, a4
;
192 a1
= a
& (0x3FFFFFFF); a2
= a1
<< 1; a4
= a2
<< 1;
194 tab
[0] = 0; tab
[1] = a1
; tab
[2] = a2
; tab
[3] = a1
^a2
;
195 tab
[4] = a4
; tab
[5] = a1
^a4
; tab
[6] = a2
^a4
; tab
[7] = a1
^a2
^a4
;
197 s
= tab
[b
& 0x7]; l
= s
;
198 s
= tab
[b
>> 3 & 0x7]; l
^= s
<< 3; h
= s
>> 29;
199 s
= tab
[b
>> 6 & 0x7]; l
^= s
<< 6; h
^= s
>> 26;
200 s
= tab
[b
>> 9 & 0x7]; l
^= s
<< 9; h
^= s
>> 23;
201 s
= tab
[b
>> 12 & 0x7]; l
^= s
<< 12; h
^= s
>> 20;
202 s
= tab
[b
>> 15 & 0x7]; l
^= s
<< 15; h
^= s
>> 17;
203 s
= tab
[b
>> 18 & 0x7]; l
^= s
<< 18; h
^= s
>> 14;
204 s
= tab
[b
>> 21 & 0x7]; l
^= s
<< 21; h
^= s
>> 11;
205 s
= tab
[b
>> 24 & 0x7]; l
^= s
<< 24; h
^= s
>> 8;
206 s
= tab
[b
>> 27 & 0x7]; l
^= s
<< 27; h
^= s
>> 5;
207 s
= tab
[b
>> 30 ]; l
^= s
<< 30; h
^= s
>> 2;
209 /* compensate for the top two bits of a */
211 if (top2b
& 01) { l
^= b
<< 30; h
^= b
>> 2; }
212 if (top2b
& 02) { l
^= b
<< 31; h
^= b
>> 1; }
217 #if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
218 static void bn_GF2m_mul_1x1(BN_ULONG
*r1
, BN_ULONG
*r0
, const BN_ULONG a
, const BN_ULONG b
)
220 register BN_ULONG h
, l
, s
;
221 BN_ULONG tab
[16], top3b
= a
>> 61;
222 register BN_ULONG a1
, a2
, a4
, a8
;
224 a1
= a
& (0x1FFFFFFFFFFFFFFF); a2
= a1
<< 1; a4
= a2
<< 1; a8
= a4
<< 1;
226 tab
[ 0] = 0; tab
[ 1] = a1
; tab
[ 2] = a2
; tab
[ 3] = a1
^a2
;
227 tab
[ 4] = a4
; tab
[ 5] = a1
^a4
; tab
[ 6] = a2
^a4
; tab
[ 7] = a1
^a2
^a4
;
228 tab
[ 8] = a8
; tab
[ 9] = a1
^a8
; tab
[10] = a2
^a8
; tab
[11] = a1
^a2
^a8
;
229 tab
[12] = a4
^a8
; tab
[13] = a1
^a4
^a8
; tab
[14] = a2
^a4
^a8
; tab
[15] = a1
^a2
^a4
^a8
;
231 s
= tab
[b
& 0xF]; l
= s
;
232 s
= tab
[b
>> 4 & 0xF]; l
^= s
<< 4; h
= s
>> 60;
233 s
= tab
[b
>> 8 & 0xF]; l
^= s
<< 8; h
^= s
>> 56;
234 s
= tab
[b
>> 12 & 0xF]; l
^= s
<< 12; h
^= s
>> 52;
235 s
= tab
[b
>> 16 & 0xF]; l
^= s
<< 16; h
^= s
>> 48;
236 s
= tab
[b
>> 20 & 0xF]; l
^= s
<< 20; h
^= s
>> 44;
237 s
= tab
[b
>> 24 & 0xF]; l
^= s
<< 24; h
^= s
>> 40;
238 s
= tab
[b
>> 28 & 0xF]; l
^= s
<< 28; h
^= s
>> 36;
239 s
= tab
[b
>> 32 & 0xF]; l
^= s
<< 32; h
^= s
>> 32;
240 s
= tab
[b
>> 36 & 0xF]; l
^= s
<< 36; h
^= s
>> 28;
241 s
= tab
[b
>> 40 & 0xF]; l
^= s
<< 40; h
^= s
>> 24;
242 s
= tab
[b
>> 44 & 0xF]; l
^= s
<< 44; h
^= s
>> 20;
243 s
= tab
[b
>> 48 & 0xF]; l
^= s
<< 48; h
^= s
>> 16;
244 s
= tab
[b
>> 52 & 0xF]; l
^= s
<< 52; h
^= s
>> 12;
245 s
= tab
[b
>> 56 & 0xF]; l
^= s
<< 56; h
^= s
>> 8;
246 s
= tab
[b
>> 60 ]; l
^= s
<< 60; h
^= s
>> 4;
248 /* compensate for the top three bits of a */
250 if (top3b
& 01) { l
^= b
<< 61; h
^= b
>> 3; }
251 if (top3b
& 02) { l
^= b
<< 62; h
^= b
>> 2; }
252 if (top3b
& 04) { l
^= b
<< 63; h
^= b
>> 1; }
258 /* Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1,
259 * result is a polynomial r with degree < 4 * BN_BITS2 - 1
260 * The caller MUST ensure that the variables have the right amount
261 * of space allocated.
263 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
)
266 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
267 bn_GF2m_mul_1x1(r
+3, r
+2, a1
, b1
);
268 bn_GF2m_mul_1x1(r
+1, r
, a0
, b0
);
269 bn_GF2m_mul_1x1(&m1
, &m0
, a0
^ a1
, b0
^ b1
);
270 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
271 r
[2] ^= m1
^ r
[1] ^ r
[3]; /* h0 ^= m1 ^ l1 ^ h1; */
272 r
[1] = r
[3] ^ r
[2] ^ r
[0] ^ m1
^ m0
; /* l1 ^= l0 ^ h0 ^ m0; */
276 /* Add polynomials a and b and store result in r; r could be a or b, a and b
277 * could be equal; r is the bitwise XOR of a and b.
279 int BN_GF2m_add(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
)
282 const BIGNUM
*at
, *bt
;
284 if (a
->top
< b
->top
) { at
= b
; bt
= a
; }
285 else { at
= a
; bt
= b
; }
287 bn_wexpand(r
, at
->top
);
289 for (i
= 0; i
< bt
->top
; i
++)
291 r
->d
[i
] = at
->d
[i
] ^ bt
->d
[i
];
293 for (; i
< at
->top
; i
++)
305 /* Some functions allow for representation of the irreducible polynomials
306 * as an int[], say p. The irreducible f(t) is then of the form:
307 * t^p[0] + t^p[1] + ... + t^p[k]
308 * where m = p[0] > p[1] > ... > p[k] = 0.
312 /* Performs modular reduction of a and store result in r. r could be a. */
313 int BN_GF2m_mod_arr(BIGNUM
*r
, const BIGNUM
*a
, const unsigned int p
[])
319 /* Since the algorithm does reduction in place, if a == r, copy the
320 * contents of a into r so we can do reduction in r.
322 if ((a
!= NULL
) && (a
->d
!= r
->d
))
324 if (!bn_wexpand(r
, a
->top
)) return 0;
325 for (j
= 0; j
< a
->top
; j
++)
333 /* start reduction */
334 dN
= p
[0] / BN_BITS2
;
335 for (j
= r
->top
- 1; j
> dN
;)
338 if (z
[j
] == 0) { j
--; continue; }
341 for (k
= 1; p
[k
] > 0; k
++)
343 /* reducing component t^p[k] */
345 d0
= n
% BN_BITS2
; d1
= BN_BITS2
- d0
;
348 if (d0
) z
[j
-n
-1] ^= (zz
<<d1
);
351 /* reducing component t^0 */
353 d0
= p
[0] % BN_BITS2
;
355 z
[j
-n
] ^= (zz
>> d0
);
356 if (d0
) z
[j
-n
-1] ^= (zz
<< d1
);
359 /* final round of reduction */
363 d0
= p
[0] % BN_BITS2
;
368 if (d0
) z
[dN
] = (z
[dN
] << d1
) >> d1
; /* clear up the top d1 bits */
369 z
[0] ^= zz
; /* reduction t^0 component */
371 for (k
= 1; p
[k
] > 0; k
++)
373 /* reducing component t^p[k]*/
375 d0
= p
[k
] % BN_BITS2
;
378 if (d0
) z
[n
+1] ^= (zz
>> d1
);
389 /* Performs modular reduction of a by p and store result in r. r could be a.
391 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
392 * function is only provided for convenience; for best performance, use the
393 * BN_GF2m_mod_arr function.
395 int BN_GF2m_mod(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
)
397 const int max
= BN_num_bits(p
);
398 unsigned int *arr
=NULL
, ret
= 0;
399 if ((arr
= (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max
)) == NULL
) goto err
;
400 if (BN_GF2m_poly2arr(p
, arr
, max
) > max
)
402 BNerr(BN_F_BN_GF2M_MOD
,BN_R_INVALID_LENGTH
);
405 ret
= BN_GF2m_mod_arr(r
, a
, arr
);
407 if (arr
) OPENSSL_free(arr
);
412 /* Compute the product of two polynomials a and b, reduce modulo p, and store
413 * the result in r. r could be a or b; a could be b.
415 int BN_GF2m_mod_mul_arr(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
, const unsigned int p
[], BN_CTX
*ctx
)
417 int zlen
, i
, j
, k
, ret
= 0;
419 BN_ULONG x1
, x0
, y1
, y0
, zz
[4];
423 return BN_GF2m_mod_sqr_arr(r
, a
, p
, ctx
);
428 if ((s
= BN_CTX_get(ctx
)) == NULL
) goto err
;
430 zlen
= a
->top
+ b
->top
;
431 if (!bn_wexpand(s
, zlen
)) goto err
;
434 for (i
= 0; i
< zlen
; i
++) s
->d
[i
] = 0;
436 for (j
= 0; j
< b
->top
; j
+= 2)
439 y1
= ((j
+1) == b
->top
) ? 0 : b
->d
[j
+1];
440 for (i
= 0; i
< a
->top
; i
+= 2)
443 x1
= ((i
+1) == a
->top
) ? 0 : a
->d
[i
+1];
444 bn_GF2m_mul_2x2(zz
, x1
, x0
, y1
, y0
);
445 for (k
= 0; k
< 4; k
++) s
->d
[i
+j
+k
] ^= zz
[k
];
450 BN_GF2m_mod_arr(r
, s
, p
);
459 /* Compute the product of two polynomials a and b, reduce modulo p, and store
460 * the result in r. r could be a or b; a could equal b.
462 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper
463 * function is only provided for convenience; for best performance, use the
464 * BN_GF2m_mod_mul_arr function.
466 int BN_GF2m_mod_mul(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
, const BIGNUM
*p
, BN_CTX
*ctx
)
468 const int max
= BN_num_bits(p
);
469 unsigned int *arr
=NULL
, ret
= 0;
470 if ((arr
= (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max
)) == NULL
) goto err
;
471 if (BN_GF2m_poly2arr(p
, arr
, max
) > max
)
473 BNerr(BN_F_BN_GF2M_MOD_MUL
,BN_R_INVALID_LENGTH
);
476 ret
= BN_GF2m_mod_mul_arr(r
, a
, b
, arr
, ctx
);
478 if (arr
) OPENSSL_free(arr
);
483 /* Square a, reduce the result mod p, and store it in a. r could be a. */
484 int BN_GF2m_mod_sqr_arr(BIGNUM
*r
, const BIGNUM
*a
, const unsigned int p
[], BN_CTX
*ctx
)
490 if ((s
= BN_CTX_get(ctx
)) == NULL
) return 0;
491 if (!bn_wexpand(s
, 2 * a
->top
)) goto err
;
493 for (i
= a
->top
- 1; i
>= 0; i
--)
495 s
->d
[2*i
+1] = SQR1(a
->d
[i
]);
496 s
->d
[2*i
] = SQR0(a
->d
[i
]);
501 if (!BN_GF2m_mod_arr(r
, s
, p
)) goto err
;
508 /* Square a, reduce the result mod p, and store it in a. r could be a.
510 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper
511 * function is only provided for convenience; for best performance, use the
512 * BN_GF2m_mod_sqr_arr function.
514 int BN_GF2m_mod_sqr(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
516 const int max
= BN_num_bits(p
);
517 unsigned int *arr
=NULL
, ret
= 0;
518 if ((arr
= (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max
)) == NULL
) goto err
;
519 if (BN_GF2m_poly2arr(p
, arr
, max
) > max
)
521 BNerr(BN_F_BN_GF2M_MOD_SQR
,BN_R_INVALID_LENGTH
);
524 ret
= BN_GF2m_mod_sqr_arr(r
, a
, arr
, ctx
);
526 if (arr
) OPENSSL_free(arr
);
531 /* Invert a, reduce modulo p, and store the result in r. r could be a.
532 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from
533 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation
534 * of Elliptic Curve Cryptography Over Binary Fields".
536 int BN_GF2m_mod_inv(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
538 BIGNUM
*b
, *c
, *u
, *v
, *tmp
;
547 if (v
== NULL
) goto err
;
549 if (!BN_one(b
)) goto err
;
550 if (!BN_zero(c
)) goto err
;
551 if (!BN_GF2m_mod(u
, a
, p
)) goto err
;
552 if (!BN_copy(v
, p
)) goto err
;
554 u
->neg
= 0; /* Need to set u->neg = 0 because BN_is_one(u) checks
555 * the neg flag of the bignum.
558 if (BN_is_zero(u
)) goto err
;
562 while (!BN_is_odd(u
))
564 if (!BN_rshift1(u
, u
)) goto err
;
567 if (!BN_GF2m_add(b
, b
, p
)) goto err
;
569 if (!BN_rshift1(b
, b
)) goto err
;
572 if (BN_is_one(u
)) break;
574 if (BN_num_bits(u
) < BN_num_bits(v
))
576 tmp
= u
; u
= v
; v
= tmp
;
577 tmp
= b
; b
= c
; c
= tmp
;
580 if (!BN_GF2m_add(u
, u
, v
)) goto err
;
581 if (!BN_GF2m_add(b
, b
, c
)) goto err
;
585 if (!BN_copy(r
, b
)) goto err
;
593 /* Invert xx, reduce modulo p, and store the result in r. r could be xx.
595 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper
596 * function is only provided for convenience; for best performance, use the
597 * BN_GF2m_mod_inv function.
599 int BN_GF2m_mod_inv_arr(BIGNUM
*r
, const BIGNUM
*xx
, const unsigned int p
[], BN_CTX
*ctx
)
605 if ((field
= BN_CTX_get(ctx
)) == NULL
) goto err
;
606 if (!BN_GF2m_arr2poly(p
, field
)) goto err
;
608 ret
= BN_GF2m_mod_inv(r
, xx
, field
, ctx
);
616 #ifndef OPENSSL_SUN_GF2M_DIV
617 /* Divide y by x, reduce modulo p, and store the result in r. r could be x
618 * or y, x could equal y.
620 int BN_GF2m_mod_div(BIGNUM
*r
, const BIGNUM
*y
, const BIGNUM
*x
, const BIGNUM
*p
, BN_CTX
*ctx
)
626 xinv
= BN_CTX_get(ctx
);
627 if (xinv
== NULL
) goto err
;
629 if (!BN_GF2m_mod_inv(xinv
, x
, p
, ctx
)) goto err
;
630 if (!BN_GF2m_mod_mul(r
, y
, xinv
, p
, ctx
)) goto err
;
638 /* Divide y by x, reduce modulo p, and store the result in r. r could be x
639 * or y, x could equal y.
640 * Uses algorithm Modular_Division_GF(2^m) from
641 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to
644 int BN_GF2m_mod_div(BIGNUM
*r
, const BIGNUM
*y
, const BIGNUM
*x
, const BIGNUM
*p
, BN_CTX
*ctx
)
646 BIGNUM
*a
, *b
, *u
, *v
;
655 if (v
== NULL
) goto err
;
657 /* reduce x and y mod p */
658 if (!BN_GF2m_mod(u
, y
, p
)) goto err
;
659 if (!BN_GF2m_mod(a
, x
, p
)) goto err
;
660 if (!BN_copy(b
, p
)) goto err
;
661 if (!BN_zero(v
)) goto err
;
663 a
->neg
= 0; /* Need to set a->neg = 0 because BN_is_one(a) checks
664 * the neg flag of the bignum.
667 while (!BN_is_odd(a
))
669 if (!BN_rshift1(a
, a
)) goto err
;
670 if (BN_is_odd(u
)) if (!BN_GF2m_add(u
, u
, p
)) goto err
;
671 if (!BN_rshift1(u
, u
)) goto err
;
676 if (BN_GF2m_cmp(b
, a
) > 0)
678 if (!BN_GF2m_add(b
, b
, a
)) goto err
;
679 if (!BN_GF2m_add(v
, v
, u
)) goto err
;
682 if (!BN_rshift1(b
, b
)) goto err
;
683 if (BN_is_odd(v
)) if (!BN_GF2m_add(v
, v
, p
)) goto err
;
684 if (!BN_rshift1(v
, v
)) goto err
;
685 } while (!BN_is_odd(b
));
687 else if (BN_is_one(a
))
691 if (!BN_GF2m_add(a
, a
, b
)) goto err
;
692 if (!BN_GF2m_add(u
, u
, v
)) goto err
;
695 if (!BN_rshift1(a
, a
)) goto err
;
696 if (BN_is_odd(u
)) if (!BN_GF2m_add(u
, u
, p
)) goto err
;
697 if (!BN_rshift1(u
, u
)) goto err
;
698 } while (!BN_is_odd(a
));
702 if (!BN_copy(r
, u
)) goto err
;
711 /* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
712 * or yy, xx could equal yy.
714 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper
715 * function is only provided for convenience; for best performance, use the
716 * BN_GF2m_mod_div function.
718 int BN_GF2m_mod_div_arr(BIGNUM
*r
, const BIGNUM
*yy
, const BIGNUM
*xx
, const unsigned int p
[], BN_CTX
*ctx
)
724 if ((field
= BN_CTX_get(ctx
)) == NULL
) goto err
;
725 if (!BN_GF2m_arr2poly(p
, field
)) goto err
;
727 ret
= BN_GF2m_mod_div(r
, yy
, xx
, field
, ctx
);
735 /* Compute the bth power of a, reduce modulo p, and store
736 * the result in r. r could be a.
737 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363.
739 int BN_GF2m_mod_exp_arr(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
, const unsigned int p
[], BN_CTX
*ctx
)
751 if ((u
= BN_CTX_get(ctx
)) == NULL
) goto err
;
753 if (!BN_GF2m_mod_arr(u
, a
, p
)) goto err
;
755 n
= BN_num_bits(b
) - 1;
756 for (i
= n
- 1; i
>= 0; i
--)
758 if (!BN_GF2m_mod_sqr_arr(u
, u
, p
, ctx
)) goto err
;
759 if (BN_is_bit_set(b
, i
))
761 if (!BN_GF2m_mod_mul_arr(u
, u
, a
, p
, ctx
)) goto err
;
764 if (!BN_copy(r
, u
)) goto err
;
773 /* Compute the bth power of a, reduce modulo p, and store
774 * the result in r. r could be a.
776 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper
777 * function is only provided for convenience; for best performance, use the
778 * BN_GF2m_mod_exp_arr function.
780 int BN_GF2m_mod_exp(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
, const BIGNUM
*p
, BN_CTX
*ctx
)
782 const int max
= BN_num_bits(p
);
783 unsigned int *arr
=NULL
, ret
= 0;
784 if ((arr
= (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max
)) == NULL
) goto err
;
785 if (BN_GF2m_poly2arr(p
, arr
, max
) > max
)
787 BNerr(BN_F_BN_GF2M_MOD_EXP
,BN_R_INVALID_LENGTH
);
790 ret
= BN_GF2m_mod_exp_arr(r
, a
, b
, arr
, ctx
);
792 if (arr
) OPENSSL_free(arr
);
796 /* Compute the square root of a, reduce modulo p, and store
797 * the result in r. r could be a.
798 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
800 int BN_GF2m_mod_sqrt_arr(BIGNUM
*r
, const BIGNUM
*a
, const unsigned int p
[], BN_CTX
*ctx
)
806 if ((u
= BN_CTX_get(ctx
)) == NULL
) goto err
;
808 if (!BN_zero(u
)) goto err
;
809 if (!BN_set_bit(u
, p
[0] - 1)) goto err
;
810 ret
= BN_GF2m_mod_exp_arr(r
, a
, u
, p
, ctx
);
817 /* Compute the square root of a, reduce modulo p, and store
818 * the result in r. r could be a.
820 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper
821 * function is only provided for convenience; for best performance, use the
822 * BN_GF2m_mod_sqrt_arr function.
824 int BN_GF2m_mod_sqrt(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
826 const int max
= BN_num_bits(p
);
827 unsigned int *arr
=NULL
, ret
= 0;
828 if ((arr
= (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max
)) == NULL
) goto err
;
829 if (BN_GF2m_poly2arr(p
, arr
, max
) > max
)
831 BNerr(BN_F_BN_GF2M_MOD_EXP
,BN_R_INVALID_LENGTH
);
834 ret
= BN_GF2m_mod_sqrt_arr(r
, a
, arr
, ctx
);
836 if (arr
) OPENSSL_free(arr
);
840 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
841 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
843 int BN_GF2m_mod_solve_quad_arr(BIGNUM
*r
, const BIGNUM
*a_
, const unsigned int p
[], BN_CTX
*ctx
)
845 int ret
= 0, i
, count
= 0;
846 BIGNUM
*a
, *z
, *rho
, *w
, *w2
, *tmp
;
852 if (w
== NULL
) goto err
;
854 if (!BN_GF2m_mod_arr(a
, a_
, p
)) goto err
;
862 if (p
[0] & 0x1) /* m is odd */
864 /* compute half-trace of a */
865 if (!BN_copy(z
, a
)) goto err
;
866 for (i
= 1; i
<= (p
[0] - 1) / 2; i
++)
868 if (!BN_GF2m_mod_sqr_arr(z
, z
, p
, ctx
)) goto err
;
869 if (!BN_GF2m_mod_sqr_arr(z
, z
, p
, ctx
)) goto err
;
870 if (!BN_GF2m_add(z
, z
, a
)) goto err
;
876 rho
= BN_CTX_get(ctx
);
877 w2
= BN_CTX_get(ctx
);
878 tmp
= BN_CTX_get(ctx
);
879 if (tmp
== NULL
) goto err
;
882 if (!BN_rand(rho
, p
[0], 0, 0)) goto err
;
883 if (!BN_GF2m_mod_arr(rho
, rho
, p
)) goto err
;
884 if (!BN_zero(z
)) goto err
;
885 if (!BN_copy(w
, rho
)) goto err
;
886 for (i
= 1; i
<= p
[0] - 1; i
++)
888 if (!BN_GF2m_mod_sqr_arr(z
, z
, p
, ctx
)) goto err
;
889 if (!BN_GF2m_mod_sqr_arr(w2
, w
, p
, ctx
)) goto err
;
890 if (!BN_GF2m_mod_mul_arr(tmp
, w2
, a
, p
, ctx
)) goto err
;
891 if (!BN_GF2m_add(z
, z
, tmp
)) goto err
;
892 if (!BN_GF2m_add(w
, w2
, rho
)) goto err
;
895 } while (BN_is_zero(w
) && (count
< MAX_ITERATIONS
));
898 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR
,BN_R_TOO_MANY_ITERATIONS
);
903 if (!BN_GF2m_mod_sqr_arr(w
, z
, p
, ctx
)) goto err
;
904 if (!BN_GF2m_add(w
, z
, w
)) goto err
;
905 if (BN_GF2m_cmp(w
, a
)) goto err
;
907 if (!BN_copy(r
, z
)) goto err
;
916 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
918 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper
919 * function is only provided for convenience; for best performance, use the
920 * BN_GF2m_mod_solve_quad_arr function.
922 int BN_GF2m_mod_solve_quad(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
924 const int max
= BN_num_bits(p
);
925 unsigned int *arr
=NULL
, ret
= 0;
926 if ((arr
= (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max
)) == NULL
) goto err
;
927 if (BN_GF2m_poly2arr(p
, arr
, max
) > max
)
929 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD
,BN_R_INVALID_LENGTH
);
932 ret
= BN_GF2m_mod_solve_quad_arr(r
, a
, arr
, ctx
);
934 if (arr
) OPENSSL_free(arr
);
938 /* Convert the bit-string representation of a polynomial a into an array
939 * of integers corresponding to the bits with non-zero coefficient.
940 * Up to max elements of the array will be filled. Return value is total
941 * number of coefficients that would be extracted if array was large enough.
943 int BN_GF2m_poly2arr(const BIGNUM
*a
, unsigned int p
[], int max
)
948 for (k
= 0; k
< max
; k
++) p
[k
] = 0;
951 for (i
= a
->top
- 1; i
>= 0; i
--)
954 for (j
= BN_BITS2
- 1; j
>= 0; j
--)
958 if (k
< max
) p
[k
] = BN_BITS2
* i
+ j
;
968 /* Convert the coefficient array representation of a polynomial to a
969 * bit-string. The array must be terminated by 0.
971 int BN_GF2m_arr2poly(const unsigned int p
[], BIGNUM
*a
)
976 for (i
= 0; p
[i
] > 0; i
++)