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.
31 * NOTE: This file is licensed pursuant to the OpenSSL license below and may
32 * be modified; but after modifications, the above covenant may no longer
33 * apply! In such cases, the corresponding paragraph ["In addition, Sun
34 * covenants ... causes the infringement."] and this note can be edited out;
35 * but please keep the Sun copyright notice and attribution.
38 /* ====================================================================
39 * Copyright (c) 1998-2002 The OpenSSL Project. All rights reserved.
41 * Redistribution and use in source and binary forms, with or without
42 * modification, are permitted provided that the following conditions
45 * 1. Redistributions of source code must retain the above copyright
46 * notice, this list of conditions and the following disclaimer.
48 * 2. Redistributions in binary form must reproduce the above copyright
49 * notice, this list of conditions and the following disclaimer in
50 * the documentation and/or other materials provided with the
53 * 3. All advertising materials mentioning features or use of this
54 * software must display the following acknowledgment:
55 * "This product includes software developed by the OpenSSL Project
56 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
58 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
59 * endorse or promote products derived from this software without
60 * prior written permission. For written permission, please contact
61 * openssl-core@openssl.org.
63 * 5. Products derived from this software may not be called "OpenSSL"
64 * nor may "OpenSSL" appear in their names without prior written
65 * permission of the OpenSSL Project.
67 * 6. Redistributions of any form whatsoever must retain the following
69 * "This product includes software developed by the OpenSSL Project
70 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
72 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
73 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
74 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
75 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
76 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
77 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
78 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
79 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
80 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
81 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
82 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
83 * OF THE POSSIBILITY OF SUCH DAMAGE.
84 * ====================================================================
86 * This product includes cryptographic software written by Eric Young
87 * (eay@cryptsoft.com). This product includes software written by Tim
88 * Hudson (tjh@cryptsoft.com).
95 #include "internal/cryptlib.h"
98 #ifndef OPENSSL_NO_EC2M
101 * Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should
104 # define MAX_ITERATIONS 50
106 static const BN_ULONG SQR_tb
[16] = { 0, 1, 4, 5, 16, 17, 20, 21,
107 64, 65, 68, 69, 80, 81, 84, 85
110 /* Platform-specific macros to accelerate squaring. */
111 # if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
113 SQR_tb[(w) >> 60 & 0xF] << 56 | SQR_tb[(w) >> 56 & 0xF] << 48 | \
114 SQR_tb[(w) >> 52 & 0xF] << 40 | SQR_tb[(w) >> 48 & 0xF] << 32 | \
115 SQR_tb[(w) >> 44 & 0xF] << 24 | SQR_tb[(w) >> 40 & 0xF] << 16 | \
116 SQR_tb[(w) >> 36 & 0xF] << 8 | SQR_tb[(w) >> 32 & 0xF]
118 SQR_tb[(w) >> 28 & 0xF] << 56 | SQR_tb[(w) >> 24 & 0xF] << 48 | \
119 SQR_tb[(w) >> 20 & 0xF] << 40 | SQR_tb[(w) >> 16 & 0xF] << 32 | \
120 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
121 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
123 # ifdef THIRTY_TWO_BIT
125 SQR_tb[(w) >> 28 & 0xF] << 24 | SQR_tb[(w) >> 24 & 0xF] << 16 | \
126 SQR_tb[(w) >> 20 & 0xF] << 8 | SQR_tb[(w) >> 16 & 0xF]
128 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
129 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
132 # if !defined(OPENSSL_BN_ASM_GF2m)
134 * Product of two polynomials a, b each with degree < BN_BITS2 - 1, result is
135 * a polynomial r with degree < 2 * BN_BITS - 1 The caller MUST ensure that
136 * the variables have the right amount of space allocated.
138 # ifdef THIRTY_TWO_BIT
139 static void bn_GF2m_mul_1x1(BN_ULONG
*r1
, BN_ULONG
*r0
, const BN_ULONG a
,
142 register BN_ULONG h
, l
, s
;
143 BN_ULONG tab
[8], top2b
= a
>> 30;
144 register BN_ULONG a1
, a2
, a4
;
146 a1
= a
& (0x3FFFFFFF);
157 tab
[7] = a1
^ a2
^ a4
;
161 s
= tab
[b
>> 3 & 0x7];
164 s
= tab
[b
>> 6 & 0x7];
167 s
= tab
[b
>> 9 & 0x7];
170 s
= tab
[b
>> 12 & 0x7];
173 s
= tab
[b
>> 15 & 0x7];
176 s
= tab
[b
>> 18 & 0x7];
179 s
= tab
[b
>> 21 & 0x7];
182 s
= tab
[b
>> 24 & 0x7];
185 s
= tab
[b
>> 27 & 0x7];
192 /* compensate for the top two bits of a */
207 # if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
208 static void bn_GF2m_mul_1x1(BN_ULONG
*r1
, BN_ULONG
*r0
, const BN_ULONG a
,
211 register BN_ULONG h
, l
, s
;
212 BN_ULONG tab
[16], top3b
= a
>> 61;
213 register BN_ULONG a1
, a2
, a4
, a8
;
215 a1
= a
& (0x1FFFFFFFFFFFFFFFULL
);
227 tab
[7] = a1
^ a2
^ a4
;
231 tab
[11] = a1
^ a2
^ a8
;
233 tab
[13] = a1
^ a4
^ a8
;
234 tab
[14] = a2
^ a4
^ a8
;
235 tab
[15] = a1
^ a2
^ a4
^ a8
;
239 s
= tab
[b
>> 4 & 0xF];
242 s
= tab
[b
>> 8 & 0xF];
245 s
= tab
[b
>> 12 & 0xF];
248 s
= tab
[b
>> 16 & 0xF];
251 s
= tab
[b
>> 20 & 0xF];
254 s
= tab
[b
>> 24 & 0xF];
257 s
= tab
[b
>> 28 & 0xF];
260 s
= tab
[b
>> 32 & 0xF];
263 s
= tab
[b
>> 36 & 0xF];
266 s
= tab
[b
>> 40 & 0xF];
269 s
= tab
[b
>> 44 & 0xF];
272 s
= tab
[b
>> 48 & 0xF];
275 s
= tab
[b
>> 52 & 0xF];
278 s
= tab
[b
>> 56 & 0xF];
285 /* compensate for the top three bits of a */
306 * Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1,
307 * result is a polynomial r with degree < 4 * BN_BITS2 - 1 The caller MUST
308 * ensure that the variables have the right amount of space allocated.
310 static void bn_GF2m_mul_2x2(BN_ULONG
*r
, const BN_ULONG a1
, const BN_ULONG a0
,
311 const BN_ULONG b1
, const BN_ULONG b0
)
314 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
315 bn_GF2m_mul_1x1(r
+ 3, r
+ 2, a1
, b1
);
316 bn_GF2m_mul_1x1(r
+ 1, r
, a0
, b0
);
317 bn_GF2m_mul_1x1(&m1
, &m0
, a0
^ a1
, b0
^ b1
);
318 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
319 r
[2] ^= m1
^ r
[1] ^ r
[3]; /* h0 ^= m1 ^ l1 ^ h1; */
320 r
[1] = r
[3] ^ r
[2] ^ r
[0] ^ m1
^ m0
; /* l1 ^= l0 ^ h0 ^ m0; */
323 void bn_GF2m_mul_2x2(BN_ULONG
*r
, BN_ULONG a1
, BN_ULONG a0
, BN_ULONG b1
,
328 * Add polynomials a and b and store result in r; r could be a or b, a and b
329 * could be equal; r is the bitwise XOR of a and b.
331 int BN_GF2m_add(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
)
334 const BIGNUM
*at
, *bt
;
339 if (a
->top
< b
->top
) {
347 if (bn_wexpand(r
, at
->top
) == NULL
)
350 for (i
= 0; i
< bt
->top
; i
++) {
351 r
->d
[i
] = at
->d
[i
] ^ bt
->d
[i
];
353 for (; i
< at
->top
; i
++) {
364 * Some functions allow for representation of the irreducible polynomials
365 * as an int[], say p. The irreducible f(t) is then of the form:
366 * t^p[0] + t^p[1] + ... + t^p[k]
367 * where m = p[0] > p[1] > ... > p[k] = 0.
370 /* Performs modular reduction of a and store result in r. r could be a. */
371 int BN_GF2m_mod_arr(BIGNUM
*r
, const BIGNUM
*a
, const int p
[])
380 /* reduction mod 1 => return 0 */
386 * Since the algorithm does reduction in the r value, if a != r, copy the
387 * contents of a into r so we can do reduction in r.
390 if (!bn_wexpand(r
, a
->top
))
392 for (j
= 0; j
< a
->top
; j
++) {
399 /* start reduction */
400 dN
= p
[0] / BN_BITS2
;
401 for (j
= r
->top
- 1; j
> dN
;) {
409 for (k
= 1; p
[k
] != 0; k
++) {
410 /* reducing component t^p[k] */
415 z
[j
- n
] ^= (zz
>> d0
);
417 z
[j
- n
- 1] ^= (zz
<< d1
);
420 /* reducing component t^0 */
422 d0
= p
[0] % BN_BITS2
;
424 z
[j
- n
] ^= (zz
>> d0
);
426 z
[j
- n
- 1] ^= (zz
<< d1
);
429 /* final round of reduction */
432 d0
= p
[0] % BN_BITS2
;
438 /* clear up the top d1 bits */
440 z
[dN
] = (z
[dN
] << d1
) >> d1
;
443 z
[0] ^= zz
; /* reduction t^0 component */
445 for (k
= 1; p
[k
] != 0; k
++) {
448 /* reducing component t^p[k] */
450 d0
= p
[k
] % BN_BITS2
;
453 if (d0
&& (tmp_ulong
= zz
>> d1
))
454 z
[n
+ 1] ^= tmp_ulong
;
464 * Performs modular reduction of a by p and store result in r. r could be a.
465 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
466 * function is only provided for convenience; for best performance, use the
467 * BN_GF2m_mod_arr function.
469 int BN_GF2m_mod(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
)
475 ret
= BN_GF2m_poly2arr(p
, arr
, OSSL_NELEM(arr
));
476 if (!ret
|| ret
> (int)OSSL_NELEM(arr
)) {
477 BNerr(BN_F_BN_GF2M_MOD
, BN_R_INVALID_LENGTH
);
480 ret
= BN_GF2m_mod_arr(r
, a
, arr
);
486 * Compute the product of two polynomials a and b, reduce modulo p, and store
487 * the result in r. r could be a or b; a could be b.
489 int BN_GF2m_mod_mul_arr(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
,
490 const int p
[], BN_CTX
*ctx
)
492 int zlen
, i
, j
, k
, ret
= 0;
494 BN_ULONG x1
, x0
, y1
, y0
, zz
[4];
500 return BN_GF2m_mod_sqr_arr(r
, a
, p
, ctx
);
504 if ((s
= BN_CTX_get(ctx
)) == NULL
)
507 zlen
= a
->top
+ b
->top
+ 4;
508 if (!bn_wexpand(s
, zlen
))
512 for (i
= 0; i
< zlen
; i
++)
515 for (j
= 0; j
< b
->top
; j
+= 2) {
517 y1
= ((j
+ 1) == b
->top
) ? 0 : b
->d
[j
+ 1];
518 for (i
= 0; i
< a
->top
; i
+= 2) {
520 x1
= ((i
+ 1) == a
->top
) ? 0 : a
->d
[i
+ 1];
521 bn_GF2m_mul_2x2(zz
, x1
, x0
, y1
, y0
);
522 for (k
= 0; k
< 4; k
++)
523 s
->d
[i
+ j
+ k
] ^= zz
[k
];
528 if (BN_GF2m_mod_arr(r
, s
, p
))
538 * Compute the product of two polynomials a and b, reduce modulo p, and store
539 * the result in r. r could be a or b; a could equal b. This function calls
540 * down to the BN_GF2m_mod_mul_arr implementation; this wrapper function is
541 * only provided for convenience; for best performance, use the
542 * BN_GF2m_mod_mul_arr function.
544 int BN_GF2m_mod_mul(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
,
545 const BIGNUM
*p
, BN_CTX
*ctx
)
548 const int max
= BN_num_bits(p
) + 1;
553 if ((arr
= OPENSSL_malloc(sizeof(*arr
) * max
)) == NULL
)
555 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
556 if (!ret
|| ret
> max
) {
557 BNerr(BN_F_BN_GF2M_MOD_MUL
, BN_R_INVALID_LENGTH
);
560 ret
= BN_GF2m_mod_mul_arr(r
, a
, b
, arr
, ctx
);
567 /* Square a, reduce the result mod p, and store it in a. r could be a. */
568 int BN_GF2m_mod_sqr_arr(BIGNUM
*r
, const BIGNUM
*a
, const int p
[],
576 if ((s
= BN_CTX_get(ctx
)) == NULL
)
578 if (!bn_wexpand(s
, 2 * a
->top
))
581 for (i
= a
->top
- 1; i
>= 0; i
--) {
582 s
->d
[2 * i
+ 1] = SQR1(a
->d
[i
]);
583 s
->d
[2 * i
] = SQR0(a
->d
[i
]);
588 if (!BN_GF2m_mod_arr(r
, s
, p
))
598 * Square a, reduce the result mod p, and store it in a. r could be a. This
599 * function calls down to the BN_GF2m_mod_sqr_arr implementation; this
600 * wrapper function is only provided for convenience; for best performance,
601 * use the BN_GF2m_mod_sqr_arr function.
603 int BN_GF2m_mod_sqr(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
606 const int max
= BN_num_bits(p
) + 1;
611 if ((arr
= OPENSSL_malloc(sizeof(*arr
) * max
)) == NULL
)
613 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
614 if (!ret
|| ret
> max
) {
615 BNerr(BN_F_BN_GF2M_MOD_SQR
, BN_R_INVALID_LENGTH
);
618 ret
= BN_GF2m_mod_sqr_arr(r
, a
, arr
, ctx
);
626 * Invert a, reduce modulo p, and store the result in r. r could be a. Uses
627 * Modified Almost Inverse Algorithm (Algorithm 10) from Hankerson, D.,
628 * Hernandez, J.L., and Menezes, A. "Software Implementation of Elliptic
629 * Curve Cryptography Over Binary Fields".
631 int BN_GF2m_mod_inv(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
633 BIGNUM
*b
, *c
= NULL
, *u
= NULL
, *v
= NULL
, *tmp
;
641 if ((b
= BN_CTX_get(ctx
)) == NULL
)
643 if ((c
= BN_CTX_get(ctx
)) == NULL
)
645 if ((u
= BN_CTX_get(ctx
)) == NULL
)
647 if ((v
= BN_CTX_get(ctx
)) == NULL
)
650 if (!BN_GF2m_mod(u
, a
, p
))
662 while (!BN_is_odd(u
)) {
665 if (!BN_rshift1(u
, u
))
668 if (!BN_GF2m_add(b
, b
, p
))
671 if (!BN_rshift1(b
, b
))
675 if (BN_abs_is_word(u
, 1))
678 if (BN_num_bits(u
) < BN_num_bits(v
)) {
687 if (!BN_GF2m_add(u
, u
, v
))
689 if (!BN_GF2m_add(b
, b
, c
))
694 int i
, ubits
= BN_num_bits(u
), vbits
= BN_num_bits(v
), /* v is copy
697 BN_ULONG
*udp
, *bdp
, *vdp
, *cdp
;
701 for (i
= u
->top
; i
< top
; i
++)
707 for (i
= 1; i
< top
; i
++)
712 for (i
= 0; i
< top
; i
++)
715 vdp
= v
->d
; /* It pays off to "cache" *->d pointers,
716 * because it allows optimizer to be more
717 * aggressive. But we don't have to "cache"
718 * p->d, because *p is declared 'const'... */
720 while (ubits
&& !(udp
[0] & 1)) {
721 BN_ULONG u0
, u1
, b0
, b1
, mask
;
725 mask
= (BN_ULONG
)0 - (b0
& 1);
726 b0
^= p
->d
[0] & mask
;
727 for (i
= 0; i
< top
- 1; i
++) {
729 udp
[i
] = ((u0
>> 1) | (u1
<< (BN_BITS2
- 1))) & BN_MASK2
;
731 b1
= bdp
[i
+ 1] ^ (p
->d
[i
+ 1] & mask
);
732 bdp
[i
] = ((b0
>> 1) | (b1
<< (BN_BITS2
- 1))) & BN_MASK2
;
740 if (ubits
<= BN_BITS2
&& udp
[0] == 1)
758 for (i
= 0; i
< top
; i
++) {
762 if (ubits
== vbits
) {
764 int utop
= (ubits
- 1) / BN_BITS2
;
766 while ((ul
= udp
[utop
]) == 0 && utop
)
768 ubits
= utop
* BN_BITS2
+ BN_num_bits_word(ul
);
781 # ifdef BN_DEBUG /* BN_CTX_end would complain about the
792 * Invert xx, reduce modulo p, and store the result in r. r could be xx.
793 * This function calls down to the BN_GF2m_mod_inv implementation; this
794 * wrapper function is only provided for convenience; for best performance,
795 * use the BN_GF2m_mod_inv function.
797 int BN_GF2m_mod_inv_arr(BIGNUM
*r
, const BIGNUM
*xx
, const int p
[],
805 if ((field
= BN_CTX_get(ctx
)) == NULL
)
807 if (!BN_GF2m_arr2poly(p
, field
))
810 ret
= BN_GF2m_mod_inv(r
, xx
, field
, ctx
);
818 # ifndef OPENSSL_SUN_GF2M_DIV
820 * Divide y by x, reduce modulo p, and store the result in r. r could be x
821 * or y, x could equal y.
823 int BN_GF2m_mod_div(BIGNUM
*r
, const BIGNUM
*y
, const BIGNUM
*x
,
824 const BIGNUM
*p
, BN_CTX
*ctx
)
834 xinv
= BN_CTX_get(ctx
);
838 if (!BN_GF2m_mod_inv(xinv
, x
, p
, ctx
))
840 if (!BN_GF2m_mod_mul(r
, y
, xinv
, p
, ctx
))
851 * Divide y by x, reduce modulo p, and store the result in r. r could be x
852 * or y, x could equal y. Uses algorithm Modular_Division_GF(2^m) from
853 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to the
856 int BN_GF2m_mod_div(BIGNUM
*r
, const BIGNUM
*y
, const BIGNUM
*x
,
857 const BIGNUM
*p
, BN_CTX
*ctx
)
859 BIGNUM
*a
, *b
, *u
, *v
;
875 /* reduce x and y mod p */
876 if (!BN_GF2m_mod(u
, y
, p
))
878 if (!BN_GF2m_mod(a
, x
, p
))
883 while (!BN_is_odd(a
)) {
884 if (!BN_rshift1(a
, a
))
887 if (!BN_GF2m_add(u
, u
, p
))
889 if (!BN_rshift1(u
, u
))
894 if (BN_GF2m_cmp(b
, a
) > 0) {
895 if (!BN_GF2m_add(b
, b
, a
))
897 if (!BN_GF2m_add(v
, v
, u
))
900 if (!BN_rshift1(b
, b
))
903 if (!BN_GF2m_add(v
, v
, p
))
905 if (!BN_rshift1(v
, v
))
907 } while (!BN_is_odd(b
));
908 } else if (BN_abs_is_word(a
, 1))
911 if (!BN_GF2m_add(a
, a
, b
))
913 if (!BN_GF2m_add(u
, u
, v
))
916 if (!BN_rshift1(a
, a
))
919 if (!BN_GF2m_add(u
, u
, p
))
921 if (!BN_rshift1(u
, u
))
923 } while (!BN_is_odd(a
));
939 * Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
940 * * or yy, xx could equal yy. This function calls down to the
941 * BN_GF2m_mod_div implementation; this wrapper function is only provided for
942 * convenience; for best performance, use the BN_GF2m_mod_div function.
944 int BN_GF2m_mod_div_arr(BIGNUM
*r
, const BIGNUM
*yy
, const BIGNUM
*xx
,
945 const int p
[], BN_CTX
*ctx
)
954 if ((field
= BN_CTX_get(ctx
)) == NULL
)
956 if (!BN_GF2m_arr2poly(p
, field
))
959 ret
= BN_GF2m_mod_div(r
, yy
, xx
, field
, ctx
);
968 * Compute the bth power of a, reduce modulo p, and store the result in r. r
969 * could be a. Uses simple square-and-multiply algorithm A.5.1 from IEEE
972 int BN_GF2m_mod_exp_arr(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
,
973 const int p
[], BN_CTX
*ctx
)
984 if (BN_abs_is_word(b
, 1))
985 return (BN_copy(r
, a
) != NULL
);
988 if ((u
= BN_CTX_get(ctx
)) == NULL
)
991 if (!BN_GF2m_mod_arr(u
, a
, p
))
994 n
= BN_num_bits(b
) - 1;
995 for (i
= n
- 1; i
>= 0; i
--) {
996 if (!BN_GF2m_mod_sqr_arr(u
, u
, p
, ctx
))
998 if (BN_is_bit_set(b
, i
)) {
999 if (!BN_GF2m_mod_mul_arr(u
, u
, a
, p
, ctx
))
1013 * Compute the bth power of a, reduce modulo p, and store the result in r. r
1014 * could be a. This function calls down to the BN_GF2m_mod_exp_arr
1015 * implementation; this wrapper function is only provided for convenience;
1016 * for best performance, use the BN_GF2m_mod_exp_arr function.
1018 int BN_GF2m_mod_exp(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
,
1019 const BIGNUM
*p
, BN_CTX
*ctx
)
1022 const int max
= BN_num_bits(p
) + 1;
1027 if ((arr
= OPENSSL_malloc(sizeof(*arr
) * max
)) == NULL
)
1029 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
1030 if (!ret
|| ret
> max
) {
1031 BNerr(BN_F_BN_GF2M_MOD_EXP
, BN_R_INVALID_LENGTH
);
1034 ret
= BN_GF2m_mod_exp_arr(r
, a
, b
, arr
, ctx
);
1042 * Compute the square root of a, reduce modulo p, and store the result in r.
1043 * r could be a. Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
1045 int BN_GF2m_mod_sqrt_arr(BIGNUM
*r
, const BIGNUM
*a
, const int p
[],
1054 /* reduction mod 1 => return 0 */
1060 if ((u
= BN_CTX_get(ctx
)) == NULL
)
1063 if (!BN_set_bit(u
, p
[0] - 1))
1065 ret
= BN_GF2m_mod_exp_arr(r
, a
, u
, p
, ctx
);
1074 * Compute the square root of a, reduce modulo p, and store the result in r.
1075 * r could be a. This function calls down to the BN_GF2m_mod_sqrt_arr
1076 * implementation; this wrapper function is only provided for convenience;
1077 * for best performance, use the BN_GF2m_mod_sqrt_arr function.
1079 int BN_GF2m_mod_sqrt(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
1082 const int max
= BN_num_bits(p
) + 1;
1086 if ((arr
= OPENSSL_malloc(sizeof(*arr
) * max
)) == NULL
)
1088 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
1089 if (!ret
|| ret
> max
) {
1090 BNerr(BN_F_BN_GF2M_MOD_SQRT
, BN_R_INVALID_LENGTH
);
1093 ret
= BN_GF2m_mod_sqrt_arr(r
, a
, arr
, ctx
);
1101 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns
1102 * 0. Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
1104 int BN_GF2m_mod_solve_quad_arr(BIGNUM
*r
, const BIGNUM
*a_
, const int p
[],
1107 int ret
= 0, count
= 0, j
;
1108 BIGNUM
*a
, *z
, *rho
, *w
, *w2
, *tmp
;
1113 /* reduction mod 1 => return 0 */
1119 a
= BN_CTX_get(ctx
);
1120 z
= BN_CTX_get(ctx
);
1121 w
= BN_CTX_get(ctx
);
1125 if (!BN_GF2m_mod_arr(a
, a_
, p
))
1128 if (BN_is_zero(a
)) {
1134 if (p
[0] & 0x1) { /* m is odd */
1135 /* compute half-trace of a */
1138 for (j
= 1; j
<= (p
[0] - 1) / 2; j
++) {
1139 if (!BN_GF2m_mod_sqr_arr(z
, z
, p
, ctx
))
1141 if (!BN_GF2m_mod_sqr_arr(z
, z
, p
, ctx
))
1143 if (!BN_GF2m_add(z
, z
, a
))
1147 } else { /* m is even */
1149 rho
= BN_CTX_get(ctx
);
1150 w2
= BN_CTX_get(ctx
);
1151 tmp
= BN_CTX_get(ctx
);
1155 if (!BN_rand(rho
, p
[0], 0, 0))
1157 if (!BN_GF2m_mod_arr(rho
, rho
, p
))
1160 if (!BN_copy(w
, rho
))
1162 for (j
= 1; j
<= p
[0] - 1; j
++) {
1163 if (!BN_GF2m_mod_sqr_arr(z
, z
, p
, ctx
))
1165 if (!BN_GF2m_mod_sqr_arr(w2
, w
, p
, ctx
))
1167 if (!BN_GF2m_mod_mul_arr(tmp
, w2
, a
, p
, ctx
))
1169 if (!BN_GF2m_add(z
, z
, tmp
))
1171 if (!BN_GF2m_add(w
, w2
, rho
))
1175 } while (BN_is_zero(w
) && (count
< MAX_ITERATIONS
));
1176 if (BN_is_zero(w
)) {
1177 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR
, BN_R_TOO_MANY_ITERATIONS
);
1182 if (!BN_GF2m_mod_sqr_arr(w
, z
, p
, ctx
))
1184 if (!BN_GF2m_add(w
, z
, w
))
1186 if (BN_GF2m_cmp(w
, a
)) {
1187 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR
, BN_R_NO_SOLUTION
);
1203 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns
1204 * 0. This function calls down to the BN_GF2m_mod_solve_quad_arr
1205 * implementation; this wrapper function is only provided for convenience;
1206 * for best performance, use the BN_GF2m_mod_solve_quad_arr function.
1208 int BN_GF2m_mod_solve_quad(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
,
1212 const int max
= BN_num_bits(p
) + 1;
1216 if ((arr
= OPENSSL_malloc(sizeof(*arr
) * max
)) == NULL
)
1218 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
1219 if (!ret
|| ret
> max
) {
1220 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD
, BN_R_INVALID_LENGTH
);
1223 ret
= BN_GF2m_mod_solve_quad_arr(r
, a
, arr
, ctx
);
1231 * Convert the bit-string representation of a polynomial ( \sum_{i=0}^n a_i *
1232 * x^i) into an array of integers corresponding to the bits with non-zero
1233 * coefficient. Array is terminated with -1. Up to max elements of the array
1234 * will be filled. Return value is total number of array elements that would
1235 * be filled if array was large enough.
1237 int BN_GF2m_poly2arr(const BIGNUM
*a
, int p
[], int max
)
1245 for (i
= a
->top
- 1; i
>= 0; i
--) {
1247 /* skip word if a->d[i] == 0 */
1250 for (j
= BN_BITS2
- 1; j
>= 0; j
--) {
1251 if (a
->d
[i
] & mask
) {
1253 p
[k
] = BN_BITS2
* i
+ j
;
1269 * Convert the coefficient array representation of a polynomial to a
1270 * bit-string. The array must be terminated by -1.
1272 int BN_GF2m_arr2poly(const int p
[], BIGNUM
*a
)
1278 for (i
= 0; p
[i
] != -1; i
++) {
1279 if (BN_set_bit(a
, p
[i
]) == 0)