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 /* 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. */
37 /* ====================================================================
38 * Copyright (c) 1998-2002 The OpenSSL Project. All rights reserved.
40 * Redistribution and use in source and binary forms, with or without
41 * modification, are permitted provided that the following conditions
44 * 1. Redistributions of source code must retain the above copyright
45 * notice, this list of conditions and the following disclaimer.
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
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/)"
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.
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.
66 * 6. Redistributions of any form whatsoever must retain the following
68 * "This product includes software developed by the OpenSSL Project
69 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
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 * ====================================================================
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).
97 /* Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should fail. */
98 #define MAX_ITERATIONS 50
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)
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]
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]
116 #ifdef THIRTY_TWO_BIT
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]
121 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
122 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
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.
130 #ifdef THIRTY_TWO_BIT
131 static void bn_GF2m_mul_1x1(BN_ULONG
*r1
, BN_ULONG
*r0
, const BN_ULONG a
, const BN_ULONG b
)
133 register BN_ULONG h
, l
, s
;
134 BN_ULONG tab
[8], top2b
= a
>> 30;
135 register BN_ULONG a1
, a2
, a4
;
137 a1
= a
& (0x3FFFFFFF); a2
= a1
<< 1; a4
= a2
<< 1;
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
;
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;
154 /* compensate for the top two bits of a */
156 if (top2b
& 01) { l
^= b
<< 30; h
^= b
>> 2; }
157 if (top2b
& 02) { l
^= b
<< 31; h
^= b
>> 1; }
162 #if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
163 static void bn_GF2m_mul_1x1(BN_ULONG
*r1
, BN_ULONG
*r0
, const BN_ULONG a
, const BN_ULONG b
)
165 register BN_ULONG h
, l
, s
;
166 BN_ULONG tab
[16], top3b
= a
>> 61;
167 register BN_ULONG a1
, a2
, a4
, a8
;
169 a1
= a
& (0x1FFFFFFFFFFFFFFFULL
); a2
= a1
<< 1; a4
= a2
<< 1; a8
= a4
<< 1;
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
;
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;
193 /* compensate for the top three bits of a */
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; }
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.
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
)
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; */
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.
224 int BN_GF2m_add(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
)
227 const BIGNUM
*at
, *bt
;
232 if (a
->top
< b
->top
) { at
= b
; bt
= a
; }
233 else { at
= a
; bt
= b
; }
235 bn_wexpand(r
, at
->top
);
237 for (i
= 0; i
< bt
->top
; i
++)
239 r
->d
[i
] = at
->d
[i
] ^ bt
->d
[i
];
241 for (; i
< at
->top
; i
++)
253 /* Some functions allow for representation of the irreducible polynomials
254 * as an int[], say p. The irreducible f(t) is then of the form:
255 * t^p[0] + t^p[1] + ... + t^p[k]
256 * where m = p[0] > p[1] > ... > p[k] = 0.
260 /* Performs modular reduction of a and store result in r. r could be a. */
261 int BN_GF2m_mod_arr(BIGNUM
*r
, const BIGNUM
*a
, const unsigned int p
[])
271 /* reduction mod 1 => return 0 */
276 /* Since the algorithm does reduction in the r value, if a != r, copy
277 * the contents of a into r so we can do reduction in r.
281 if (!bn_wexpand(r
, a
->top
)) return 0;
282 for (j
= 0; j
< a
->top
; j
++)
290 /* start reduction */
291 dN
= p
[0] / BN_BITS2
;
292 for (j
= r
->top
- 1; j
> dN
;)
295 if (z
[j
] == 0) { j
--; continue; }
298 for (k
= 1; p
[k
] != 0; k
++)
300 /* reducing component t^p[k] */
302 d0
= n
% BN_BITS2
; d1
= BN_BITS2
- d0
;
305 if (d0
) z
[j
-n
-1] ^= (zz
<<d1
);
308 /* reducing component t^0 */
310 d0
= p
[0] % BN_BITS2
;
312 z
[j
-n
] ^= (zz
>> d0
);
313 if (d0
) z
[j
-n
-1] ^= (zz
<< d1
);
316 /* final round of reduction */
320 d0
= p
[0] % BN_BITS2
;
325 if (d0
) z
[dN
] = (z
[dN
] << d1
) >> d1
; /* clear up the top d1 bits */
326 z
[0] ^= zz
; /* reduction t^0 component */
328 for (k
= 1; p
[k
] != 0; k
++)
332 /* reducing component t^p[k]*/
334 d0
= p
[k
] % BN_BITS2
;
337 tmp_ulong
= zz
>> d1
;
349 /* Performs modular reduction of a by p and store result in r. r could be a.
351 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
352 * function is only provided for convenience; for best performance, use the
353 * BN_GF2m_mod_arr function.
355 int BN_GF2m_mod(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
)
358 const int max
= BN_num_bits(p
);
359 unsigned int *arr
=NULL
;
362 if ((arr
= (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max
)) == NULL
) goto err
;
363 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
364 if (!ret
|| ret
> max
)
366 BNerr(BN_F_BN_GF2M_MOD
,BN_R_INVALID_LENGTH
);
369 ret
= BN_GF2m_mod_arr(r
, a
, arr
);
372 if (arr
) OPENSSL_free(arr
);
377 /* Compute the product of two polynomials a and b, reduce modulo p, and store
378 * the result in r. r could be a or b; a could be b.
380 int BN_GF2m_mod_mul_arr(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
, const unsigned int p
[], BN_CTX
*ctx
)
382 int zlen
, i
, j
, k
, ret
= 0;
384 BN_ULONG x1
, x0
, y1
, y0
, zz
[4];
391 return BN_GF2m_mod_sqr_arr(r
, a
, p
, ctx
);
395 if ((s
= BN_CTX_get(ctx
)) == NULL
) goto err
;
397 zlen
= a
->top
+ b
->top
+ 4;
398 if (!bn_wexpand(s
, zlen
)) goto err
;
401 for (i
= 0; i
< zlen
; i
++) s
->d
[i
] = 0;
403 for (j
= 0; j
< b
->top
; j
+= 2)
406 y1
= ((j
+1) == b
->top
) ? 0 : b
->d
[j
+1];
407 for (i
= 0; i
< a
->top
; i
+= 2)
410 x1
= ((i
+1) == a
->top
) ? 0 : a
->d
[i
+1];
411 bn_GF2m_mul_2x2(zz
, x1
, x0
, y1
, y0
);
412 for (k
= 0; k
< 4; k
++) s
->d
[i
+j
+k
] ^= zz
[k
];
417 if (BN_GF2m_mod_arr(r
, s
, p
))
426 /* Compute the product of two polynomials a and b, reduce modulo p, and store
427 * the result in r. r could be a or b; a could equal b.
429 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper
430 * function is only provided for convenience; for best performance, use the
431 * BN_GF2m_mod_mul_arr function.
433 int BN_GF2m_mod_mul(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
, const BIGNUM
*p
, BN_CTX
*ctx
)
436 const int max
= BN_num_bits(p
);
437 unsigned int *arr
=NULL
;
441 if ((arr
= (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max
)) == NULL
) goto err
;
442 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
443 if (!ret
|| ret
> max
)
445 BNerr(BN_F_BN_GF2M_MOD_MUL
,BN_R_INVALID_LENGTH
);
448 ret
= BN_GF2m_mod_mul_arr(r
, a
, b
, arr
, ctx
);
451 if (arr
) OPENSSL_free(arr
);
456 /* Square a, reduce the result mod p, and store it in a. r could be a. */
457 int BN_GF2m_mod_sqr_arr(BIGNUM
*r
, const BIGNUM
*a
, const unsigned int p
[], BN_CTX
*ctx
)
464 if ((s
= BN_CTX_get(ctx
)) == NULL
) return 0;
465 if (!bn_wexpand(s
, 2 * a
->top
)) goto err
;
467 for (i
= a
->top
- 1; i
>= 0; i
--)
469 s
->d
[2*i
+1] = SQR1(a
->d
[i
]);
470 s
->d
[2*i
] = SQR0(a
->d
[i
]);
475 if (!BN_GF2m_mod_arr(r
, s
, p
)) goto err
;
483 /* Square a, reduce the result mod p, and store it in a. r could be a.
485 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper
486 * function is only provided for convenience; for best performance, use the
487 * BN_GF2m_mod_sqr_arr function.
489 int BN_GF2m_mod_sqr(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
492 const int max
= BN_num_bits(p
);
493 unsigned int *arr
=NULL
;
497 if ((arr
= (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max
)) == NULL
) goto err
;
498 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
499 if (!ret
|| ret
> max
)
501 BNerr(BN_F_BN_GF2M_MOD_SQR
,BN_R_INVALID_LENGTH
);
504 ret
= BN_GF2m_mod_sqr_arr(r
, a
, arr
, ctx
);
507 if (arr
) OPENSSL_free(arr
);
512 /* Invert a, reduce modulo p, and store the result in r. r could be a.
513 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from
514 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation
515 * of Elliptic Curve Cryptography Over Binary Fields".
517 int BN_GF2m_mod_inv(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
519 BIGNUM
*b
, *c
, *u
, *v
, *tmp
;
531 if (v
== NULL
) goto err
;
533 if (!BN_one(b
)) goto err
;
534 if (!BN_GF2m_mod(u
, a
, p
)) goto err
;
535 if (!BN_copy(v
, p
)) goto err
;
537 if (BN_is_zero(u
)) goto err
;
541 while (!BN_is_odd(u
))
543 if (!BN_rshift1(u
, u
)) goto err
;
546 if (!BN_GF2m_add(b
, b
, p
)) goto err
;
548 if (!BN_rshift1(b
, b
)) goto err
;
551 if (BN_abs_is_word(u
, 1)) break;
553 if (BN_num_bits(u
) < BN_num_bits(v
))
555 tmp
= u
; u
= v
; v
= tmp
;
556 tmp
= b
; b
= c
; c
= tmp
;
559 if (!BN_GF2m_add(u
, u
, v
)) goto err
;
560 if (!BN_GF2m_add(b
, b
, c
)) goto err
;
564 if (!BN_copy(r
, b
)) goto err
;
573 /* Invert xx, reduce modulo p, and store the result in r. r could be xx.
575 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper
576 * function is only provided for convenience; for best performance, use the
577 * BN_GF2m_mod_inv function.
579 int BN_GF2m_mod_inv_arr(BIGNUM
*r
, const BIGNUM
*xx
, const unsigned int p
[], BN_CTX
*ctx
)
586 if ((field
= BN_CTX_get(ctx
)) == NULL
) goto err
;
587 if (!BN_GF2m_arr2poly(p
, field
)) goto err
;
589 ret
= BN_GF2m_mod_inv(r
, xx
, field
, ctx
);
598 #ifndef OPENSSL_SUN_GF2M_DIV
599 /* Divide y by x, reduce modulo p, and store the result in r. r could be x
600 * or y, x could equal y.
602 int BN_GF2m_mod_div(BIGNUM
*r
, const BIGNUM
*y
, const BIGNUM
*x
, const BIGNUM
*p
, BN_CTX
*ctx
)
612 xinv
= BN_CTX_get(ctx
);
613 if (xinv
== NULL
) goto err
;
615 if (!BN_GF2m_mod_inv(xinv
, x
, p
, ctx
)) goto err
;
616 if (!BN_GF2m_mod_mul(r
, y
, xinv
, p
, ctx
)) goto err
;
625 /* Divide y by x, reduce modulo p, and store the result in r. r could be x
626 * or y, x could equal y.
627 * Uses algorithm Modular_Division_GF(2^m) from
628 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to
631 int BN_GF2m_mod_div(BIGNUM
*r
, const BIGNUM
*y
, const BIGNUM
*x
, const BIGNUM
*p
, BN_CTX
*ctx
)
633 BIGNUM
*a
, *b
, *u
, *v
;
646 if (v
== NULL
) goto err
;
648 /* reduce x and y mod p */
649 if (!BN_GF2m_mod(u
, y
, p
)) goto err
;
650 if (!BN_GF2m_mod(a
, x
, p
)) goto err
;
651 if (!BN_copy(b
, p
)) goto err
;
653 while (!BN_is_odd(a
))
655 if (!BN_rshift1(a
, a
)) goto err
;
656 if (BN_is_odd(u
)) if (!BN_GF2m_add(u
, u
, p
)) goto err
;
657 if (!BN_rshift1(u
, u
)) goto err
;
662 if (BN_GF2m_cmp(b
, a
) > 0)
664 if (!BN_GF2m_add(b
, b
, a
)) goto err
;
665 if (!BN_GF2m_add(v
, v
, u
)) goto err
;
668 if (!BN_rshift1(b
, b
)) goto err
;
669 if (BN_is_odd(v
)) if (!BN_GF2m_add(v
, v
, p
)) goto err
;
670 if (!BN_rshift1(v
, v
)) goto err
;
671 } while (!BN_is_odd(b
));
673 else if (BN_abs_is_word(a
, 1))
677 if (!BN_GF2m_add(a
, a
, b
)) goto err
;
678 if (!BN_GF2m_add(u
, u
, v
)) goto err
;
681 if (!BN_rshift1(a
, a
)) goto err
;
682 if (BN_is_odd(u
)) if (!BN_GF2m_add(u
, u
, p
)) goto err
;
683 if (!BN_rshift1(u
, u
)) goto err
;
684 } while (!BN_is_odd(a
));
688 if (!BN_copy(r
, u
)) goto err
;
698 /* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
699 * or yy, xx could equal yy.
701 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper
702 * function is only provided for convenience; for best performance, use the
703 * BN_GF2m_mod_div function.
705 int BN_GF2m_mod_div_arr(BIGNUM
*r
, const BIGNUM
*yy
, const BIGNUM
*xx
, const unsigned int p
[], BN_CTX
*ctx
)
714 if ((field
= BN_CTX_get(ctx
)) == NULL
) goto err
;
715 if (!BN_GF2m_arr2poly(p
, field
)) goto err
;
717 ret
= BN_GF2m_mod_div(r
, yy
, xx
, field
, ctx
);
726 /* Compute the bth power of a, reduce modulo p, and store
727 * the result in r. r could be a.
728 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363.
730 int BN_GF2m_mod_exp_arr(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
, const unsigned int p
[], BN_CTX
*ctx
)
741 if (BN_abs_is_word(b
, 1))
742 return (BN_copy(r
, a
) != NULL
);
745 if ((u
= BN_CTX_get(ctx
)) == NULL
) goto err
;
747 if (!BN_GF2m_mod_arr(u
, a
, p
)) goto err
;
749 n
= BN_num_bits(b
) - 1;
750 for (i
= n
- 1; i
>= 0; i
--)
752 if (!BN_GF2m_mod_sqr_arr(u
, u
, p
, ctx
)) goto err
;
753 if (BN_is_bit_set(b
, i
))
755 if (!BN_GF2m_mod_mul_arr(u
, u
, a
, p
, ctx
)) goto err
;
758 if (!BN_copy(r
, u
)) goto err
;
766 /* Compute the bth power of a, reduce modulo p, and store
767 * the result in r. r could be a.
769 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper
770 * function is only provided for convenience; for best performance, use the
771 * BN_GF2m_mod_exp_arr function.
773 int BN_GF2m_mod_exp(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
, const BIGNUM
*p
, BN_CTX
*ctx
)
776 const int max
= BN_num_bits(p
);
777 unsigned int *arr
=NULL
;
781 if ((arr
= (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max
)) == NULL
) goto err
;
782 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
783 if (!ret
|| ret
> max
)
785 BNerr(BN_F_BN_GF2M_MOD_EXP
,BN_R_INVALID_LENGTH
);
788 ret
= BN_GF2m_mod_exp_arr(r
, a
, b
, arr
, ctx
);
791 if (arr
) OPENSSL_free(arr
);
795 /* Compute the square root of a, reduce modulo p, and store
796 * the result in r. r could be a.
797 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
799 int BN_GF2m_mod_sqrt_arr(BIGNUM
*r
, const BIGNUM
*a
, const unsigned int p
[], BN_CTX
*ctx
)
808 /* reduction mod 1 => return 0 */
814 if ((u
= BN_CTX_get(ctx
)) == NULL
) goto err
;
816 if (!BN_set_bit(u
, p
[0] - 1)) goto err
;
817 ret
= BN_GF2m_mod_exp_arr(r
, a
, u
, p
, ctx
);
825 /* Compute the square root of a, reduce modulo p, and store
826 * the result in r. r could be a.
828 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper
829 * function is only provided for convenience; for best performance, use the
830 * BN_GF2m_mod_sqrt_arr function.
832 int BN_GF2m_mod_sqrt(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
835 const int max
= BN_num_bits(p
);
836 unsigned int *arr
=NULL
;
839 if ((arr
= (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max
)) == NULL
) goto err
;
840 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
841 if (!ret
|| ret
> max
)
843 BNerr(BN_F_BN_GF2M_MOD_SQRT
,BN_R_INVALID_LENGTH
);
846 ret
= BN_GF2m_mod_sqrt_arr(r
, a
, arr
, ctx
);
849 if (arr
) OPENSSL_free(arr
);
853 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
854 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
856 int BN_GF2m_mod_solve_quad_arr(BIGNUM
*r
, const BIGNUM
*a_
, const unsigned int p
[], BN_CTX
*ctx
)
858 int ret
= 0, count
= 0;
860 BIGNUM
*a
, *z
, *rho
, *w
, *w2
, *tmp
;
866 /* reduction mod 1 => return 0 */
875 if (w
== NULL
) goto err
;
877 if (!BN_GF2m_mod_arr(a
, a_
, p
)) goto err
;
886 if (p
[0] & 0x1) /* m is odd */
888 /* compute half-trace of a */
889 if (!BN_copy(z
, a
)) goto err
;
890 for (j
= 1; j
<= (p
[0] - 1) / 2; j
++)
892 if (!BN_GF2m_mod_sqr_arr(z
, z
, p
, ctx
)) goto err
;
893 if (!BN_GF2m_mod_sqr_arr(z
, z
, p
, ctx
)) goto err
;
894 if (!BN_GF2m_add(z
, z
, a
)) goto err
;
900 rho
= BN_CTX_get(ctx
);
901 w2
= BN_CTX_get(ctx
);
902 tmp
= BN_CTX_get(ctx
);
903 if (tmp
== NULL
) goto err
;
906 if (!BN_rand(rho
, p
[0], 0, 0)) goto err
;
907 if (!BN_GF2m_mod_arr(rho
, rho
, p
)) goto err
;
909 if (!BN_copy(w
, rho
)) goto err
;
910 for (j
= 1; j
<= p
[0] - 1; j
++)
912 if (!BN_GF2m_mod_sqr_arr(z
, z
, p
, ctx
)) goto err
;
913 if (!BN_GF2m_mod_sqr_arr(w2
, w
, p
, ctx
)) goto err
;
914 if (!BN_GF2m_mod_mul_arr(tmp
, w2
, a
, p
, ctx
)) goto err
;
915 if (!BN_GF2m_add(z
, z
, tmp
)) goto err
;
916 if (!BN_GF2m_add(w
, w2
, rho
)) goto err
;
919 } while (BN_is_zero(w
) && (count
< MAX_ITERATIONS
));
922 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR
,BN_R_TOO_MANY_ITERATIONS
);
927 if (!BN_GF2m_mod_sqr_arr(w
, z
, p
, ctx
)) goto err
;
928 if (!BN_GF2m_add(w
, z
, w
)) goto err
;
929 if (BN_GF2m_cmp(w
, a
))
931 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR
, BN_R_NO_SOLUTION
);
935 if (!BN_copy(r
, z
)) goto err
;
945 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
947 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper
948 * function is only provided for convenience; for best performance, use the
949 * BN_GF2m_mod_solve_quad_arr function.
951 int BN_GF2m_mod_solve_quad(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
954 const int max
= BN_num_bits(p
);
955 unsigned int *arr
=NULL
;
958 if ((arr
= (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) *
959 max
)) == NULL
) goto err
;
960 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
961 if (!ret
|| ret
> max
)
963 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD
,BN_R_INVALID_LENGTH
);
966 ret
= BN_GF2m_mod_solve_quad_arr(r
, a
, arr
, ctx
);
969 if (arr
) OPENSSL_free(arr
);
973 /* Convert the bit-string representation of a polynomial
974 * ( \sum_{i=0}^n a_i * x^i , where a_0 is *not* zero) into an array
975 * of integers corresponding to the bits with non-zero coefficient.
976 * Up to max elements of the array will be filled. Return value is total
977 * number of coefficients that would be extracted if array was large enough.
979 int BN_GF2m_poly2arr(const BIGNUM
*a
, unsigned int p
[], int max
)
984 if (BN_is_zero(a
) || !BN_is_bit_set(a
, 0))
985 /* a_0 == 0 => return error (the unsigned int array
986 * must be terminated by 0)
990 for (i
= a
->top
- 1; i
>= 0; i
--)
993 /* skip word if a->d[i] == 0 */
996 for (j
= BN_BITS2
- 1; j
>= 0; j
--)
1000 if (k
< max
) p
[k
] = BN_BITS2
* i
+ j
;
1010 /* Convert the coefficient array representation of a polynomial to a
1011 * bit-string. The array must be terminated by 0.
1013 int BN_GF2m_arr2poly(const unsigned int p
[], BIGNUM
*a
)
1019 for (i
= 0; p
[i
] != 0; i
++)
1021 if (BN_set_bit(a
, p
[i
]) == 0)