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 #ifndef OPENSSL_NO_EC2M
99 /* Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should fail. */
100 #define MAX_ITERATIONS 50
102 static const BN_ULONG SQR_tb
[16] =
103 { 0, 1, 4, 5, 16, 17, 20, 21,
104 64, 65, 68, 69, 80, 81, 84, 85 };
105 /* Platform-specific macros to accelerate squaring. */
106 #if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
108 SQR_tb[(w) >> 60 & 0xF] << 56 | SQR_tb[(w) >> 56 & 0xF] << 48 | \
109 SQR_tb[(w) >> 52 & 0xF] << 40 | SQR_tb[(w) >> 48 & 0xF] << 32 | \
110 SQR_tb[(w) >> 44 & 0xF] << 24 | SQR_tb[(w) >> 40 & 0xF] << 16 | \
111 SQR_tb[(w) >> 36 & 0xF] << 8 | SQR_tb[(w) >> 32 & 0xF]
113 SQR_tb[(w) >> 28 & 0xF] << 56 | SQR_tb[(w) >> 24 & 0xF] << 48 | \
114 SQR_tb[(w) >> 20 & 0xF] << 40 | SQR_tb[(w) >> 16 & 0xF] << 32 | \
115 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
116 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
118 #ifdef THIRTY_TWO_BIT
120 SQR_tb[(w) >> 28 & 0xF] << 24 | SQR_tb[(w) >> 24 & 0xF] << 16 | \
121 SQR_tb[(w) >> 20 & 0xF] << 8 | SQR_tb[(w) >> 16 & 0xF]
123 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
124 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
127 /* Product of two polynomials a, b each with degree < BN_BITS2 - 1,
128 * result is a polynomial r with degree < 2 * BN_BITS - 1
129 * The caller MUST ensure that the variables have the right amount
130 * of space allocated.
132 #ifdef THIRTY_TWO_BIT
133 static void bn_GF2m_mul_1x1(BN_ULONG
*r1
, BN_ULONG
*r0
, const BN_ULONG a
, const BN_ULONG b
)
135 register BN_ULONG h
, l
, s
;
136 BN_ULONG tab
[8], top2b
= a
>> 30;
137 register BN_ULONG a1
, a2
, a4
;
139 a1
= a
& (0x3FFFFFFF); a2
= a1
<< 1; a4
= a2
<< 1;
141 tab
[0] = 0; tab
[1] = a1
; tab
[2] = a2
; tab
[3] = a1
^a2
;
142 tab
[4] = a4
; tab
[5] = a1
^a4
; tab
[6] = a2
^a4
; tab
[7] = a1
^a2
^a4
;
144 s
= tab
[b
& 0x7]; l
= s
;
145 s
= tab
[b
>> 3 & 0x7]; l
^= s
<< 3; h
= s
>> 29;
146 s
= tab
[b
>> 6 & 0x7]; l
^= s
<< 6; h
^= s
>> 26;
147 s
= tab
[b
>> 9 & 0x7]; l
^= s
<< 9; h
^= s
>> 23;
148 s
= tab
[b
>> 12 & 0x7]; l
^= s
<< 12; h
^= s
>> 20;
149 s
= tab
[b
>> 15 & 0x7]; l
^= s
<< 15; h
^= s
>> 17;
150 s
= tab
[b
>> 18 & 0x7]; l
^= s
<< 18; h
^= s
>> 14;
151 s
= tab
[b
>> 21 & 0x7]; l
^= s
<< 21; h
^= s
>> 11;
152 s
= tab
[b
>> 24 & 0x7]; l
^= s
<< 24; h
^= s
>> 8;
153 s
= tab
[b
>> 27 & 0x7]; l
^= s
<< 27; h
^= s
>> 5;
154 s
= tab
[b
>> 30 ]; l
^= s
<< 30; h
^= s
>> 2;
156 /* compensate for the top two bits of a */
158 if (top2b
& 01) { l
^= b
<< 30; h
^= b
>> 2; }
159 if (top2b
& 02) { l
^= b
<< 31; h
^= b
>> 1; }
164 #if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
165 static void bn_GF2m_mul_1x1(BN_ULONG
*r1
, BN_ULONG
*r0
, const BN_ULONG a
, const BN_ULONG b
)
167 register BN_ULONG h
, l
, s
;
168 BN_ULONG tab
[16], top3b
= a
>> 61;
169 register BN_ULONG a1
, a2
, a4
, a8
;
171 a1
= a
& (0x1FFFFFFFFFFFFFFFULL
); a2
= a1
<< 1; a4
= a2
<< 1; a8
= a4
<< 1;
173 tab
[ 0] = 0; tab
[ 1] = a1
; tab
[ 2] = a2
; tab
[ 3] = a1
^a2
;
174 tab
[ 4] = a4
; tab
[ 5] = a1
^a4
; tab
[ 6] = a2
^a4
; tab
[ 7] = a1
^a2
^a4
;
175 tab
[ 8] = a8
; tab
[ 9] = a1
^a8
; tab
[10] = a2
^a8
; tab
[11] = a1
^a2
^a8
;
176 tab
[12] = a4
^a8
; tab
[13] = a1
^a4
^a8
; tab
[14] = a2
^a4
^a8
; tab
[15] = a1
^a2
^a4
^a8
;
178 s
= tab
[b
& 0xF]; l
= s
;
179 s
= tab
[b
>> 4 & 0xF]; l
^= s
<< 4; h
= s
>> 60;
180 s
= tab
[b
>> 8 & 0xF]; l
^= s
<< 8; h
^= s
>> 56;
181 s
= tab
[b
>> 12 & 0xF]; l
^= s
<< 12; h
^= s
>> 52;
182 s
= tab
[b
>> 16 & 0xF]; l
^= s
<< 16; h
^= s
>> 48;
183 s
= tab
[b
>> 20 & 0xF]; l
^= s
<< 20; h
^= s
>> 44;
184 s
= tab
[b
>> 24 & 0xF]; l
^= s
<< 24; h
^= s
>> 40;
185 s
= tab
[b
>> 28 & 0xF]; l
^= s
<< 28; h
^= s
>> 36;
186 s
= tab
[b
>> 32 & 0xF]; l
^= s
<< 32; h
^= s
>> 32;
187 s
= tab
[b
>> 36 & 0xF]; l
^= s
<< 36; h
^= s
>> 28;
188 s
= tab
[b
>> 40 & 0xF]; l
^= s
<< 40; h
^= s
>> 24;
189 s
= tab
[b
>> 44 & 0xF]; l
^= s
<< 44; h
^= s
>> 20;
190 s
= tab
[b
>> 48 & 0xF]; l
^= s
<< 48; h
^= s
>> 16;
191 s
= tab
[b
>> 52 & 0xF]; l
^= s
<< 52; h
^= s
>> 12;
192 s
= tab
[b
>> 56 & 0xF]; l
^= s
<< 56; h
^= s
>> 8;
193 s
= tab
[b
>> 60 ]; l
^= s
<< 60; h
^= s
>> 4;
195 /* compensate for the top three bits of a */
197 if (top3b
& 01) { l
^= b
<< 61; h
^= b
>> 3; }
198 if (top3b
& 02) { l
^= b
<< 62; h
^= b
>> 2; }
199 if (top3b
& 04) { l
^= b
<< 63; h
^= b
>> 1; }
205 /* Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1,
206 * result is a polynomial r with degree < 4 * BN_BITS2 - 1
207 * The caller MUST ensure that the variables have the right amount
208 * of space allocated.
210 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
)
213 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
214 bn_GF2m_mul_1x1(r
+3, r
+2, a1
, b1
);
215 bn_GF2m_mul_1x1(r
+1, r
, a0
, b0
);
216 bn_GF2m_mul_1x1(&m1
, &m0
, a0
^ a1
, b0
^ b1
);
217 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
218 r
[2] ^= m1
^ r
[1] ^ r
[3]; /* h0 ^= m1 ^ l1 ^ h1; */
219 r
[1] = r
[3] ^ r
[2] ^ r
[0] ^ m1
^ m0
; /* l1 ^= l0 ^ h0 ^ m0; */
223 /* Add polynomials a and b and store result in r; r could be a or b, a and b
224 * could be equal; r is the bitwise XOR of a and b.
226 int BN_GF2m_add(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
)
229 const BIGNUM
*at
, *bt
;
234 if (a
->top
< b
->top
) { at
= b
; bt
= a
; }
235 else { at
= a
; bt
= b
; }
237 if(bn_wexpand(r
, at
->top
) == NULL
)
240 for (i
= 0; i
< bt
->top
; i
++)
242 r
->d
[i
] = at
->d
[i
] ^ bt
->d
[i
];
244 for (; i
< at
->top
; i
++)
256 /* Some functions allow for representation of the irreducible polynomials
257 * as an int[], say p. The irreducible f(t) is then of the form:
258 * t^p[0] + t^p[1] + ... + t^p[k]
259 * where m = p[0] > p[1] > ... > p[k] = 0.
263 /* Performs modular reduction of a and store result in r. r could be a. */
264 int BN_GF2m_mod_arr(BIGNUM
*r
, const BIGNUM
*a
, const int p
[])
274 /* reduction mod 1 => return 0 */
279 /* Since the algorithm does reduction in the r value, if a != r, copy
280 * the contents of a into r so we can do reduction in r.
284 if (!bn_wexpand(r
, a
->top
)) return 0;
285 for (j
= 0; j
< a
->top
; j
++)
293 /* start reduction */
294 dN
= p
[0] / BN_BITS2
;
295 for (j
= r
->top
- 1; j
> dN
;)
298 if (z
[j
] == 0) { j
--; continue; }
301 for (k
= 1; p
[k
] != 0; k
++)
303 /* reducing component t^p[k] */
305 d0
= n
% BN_BITS2
; d1
= BN_BITS2
- d0
;
308 if (d0
) z
[j
-n
-1] ^= (zz
<<d1
);
311 /* reducing component t^0 */
313 d0
= p
[0] % BN_BITS2
;
315 z
[j
-n
] ^= (zz
>> d0
);
316 if (d0
) z
[j
-n
-1] ^= (zz
<< d1
);
319 /* final round of reduction */
323 d0
= p
[0] % BN_BITS2
;
328 /* clear up the top d1 bits */
330 z
[dN
] = (z
[dN
] << d1
) >> d1
;
333 z
[0] ^= zz
; /* reduction t^0 component */
335 for (k
= 1; p
[k
] != 0; k
++)
339 /* reducing component t^p[k]*/
341 d0
= p
[k
] % BN_BITS2
;
344 tmp_ulong
= zz
>> d1
;
356 /* Performs modular reduction of a by p and store result in r. r could be a.
358 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
359 * function is only provided for convenience; for best performance, use the
360 * BN_GF2m_mod_arr function.
362 int BN_GF2m_mod(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
)
365 const int max
= BN_num_bits(p
) + 1;
369 if ((arr
= (int *)OPENSSL_malloc(sizeof(int) * max
)) == NULL
) goto err
;
370 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
371 if (!ret
|| ret
> max
)
373 BNerr(BN_F_BN_GF2M_MOD
,BN_R_INVALID_LENGTH
);
376 ret
= BN_GF2m_mod_arr(r
, a
, arr
);
379 if (arr
) OPENSSL_free(arr
);
384 /* Compute the product of two polynomials a and b, reduce modulo p, and store
385 * the result in r. r could be a or b; a could be b.
387 int BN_GF2m_mod_mul_arr(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
, const int p
[], BN_CTX
*ctx
)
389 int zlen
, i
, j
, k
, ret
= 0;
391 BN_ULONG x1
, x0
, y1
, y0
, zz
[4];
398 return BN_GF2m_mod_sqr_arr(r
, a
, p
, ctx
);
402 if ((s
= BN_CTX_get(ctx
)) == NULL
) goto err
;
404 zlen
= a
->top
+ b
->top
+ 4;
405 if (!bn_wexpand(s
, zlen
)) goto err
;
408 for (i
= 0; i
< zlen
; i
++) s
->d
[i
] = 0;
410 for (j
= 0; j
< b
->top
; j
+= 2)
413 y1
= ((j
+1) == b
->top
) ? 0 : b
->d
[j
+1];
414 for (i
= 0; i
< a
->top
; i
+= 2)
417 x1
= ((i
+1) == a
->top
) ? 0 : a
->d
[i
+1];
418 bn_GF2m_mul_2x2(zz
, x1
, x0
, y1
, y0
);
419 for (k
= 0; k
< 4; k
++) s
->d
[i
+j
+k
] ^= zz
[k
];
424 if (BN_GF2m_mod_arr(r
, s
, p
))
433 /* Compute the product of two polynomials a and b, reduce modulo p, and store
434 * the result in r. r could be a or b; a could equal b.
436 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper
437 * function is only provided for convenience; for best performance, use the
438 * BN_GF2m_mod_mul_arr function.
440 int BN_GF2m_mod_mul(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
, const BIGNUM
*p
, BN_CTX
*ctx
)
443 const int max
= BN_num_bits(p
) + 1;
448 if ((arr
= (int *)OPENSSL_malloc(sizeof(int) * max
)) == NULL
) goto err
;
449 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
450 if (!ret
|| ret
> max
)
452 BNerr(BN_F_BN_GF2M_MOD_MUL
,BN_R_INVALID_LENGTH
);
455 ret
= BN_GF2m_mod_mul_arr(r
, a
, b
, arr
, ctx
);
458 if (arr
) OPENSSL_free(arr
);
463 /* Square a, reduce the result mod p, and store it in a. r could be a. */
464 int BN_GF2m_mod_sqr_arr(BIGNUM
*r
, const BIGNUM
*a
, const int p
[], BN_CTX
*ctx
)
471 if ((s
= BN_CTX_get(ctx
)) == NULL
) return 0;
472 if (!bn_wexpand(s
, 2 * a
->top
)) goto err
;
474 for (i
= a
->top
- 1; i
>= 0; i
--)
476 s
->d
[2*i
+1] = SQR1(a
->d
[i
]);
477 s
->d
[2*i
] = SQR0(a
->d
[i
]);
482 if (!BN_GF2m_mod_arr(r
, s
, p
)) goto err
;
490 /* Square a, reduce the result mod p, and store it in a. r could be a.
492 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper
493 * function is only provided for convenience; for best performance, use the
494 * BN_GF2m_mod_sqr_arr function.
496 int BN_GF2m_mod_sqr(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
499 const int max
= BN_num_bits(p
) + 1;
504 if ((arr
= (int *)OPENSSL_malloc(sizeof(int) * max
)) == NULL
) goto err
;
505 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
506 if (!ret
|| ret
> max
)
508 BNerr(BN_F_BN_GF2M_MOD_SQR
,BN_R_INVALID_LENGTH
);
511 ret
= BN_GF2m_mod_sqr_arr(r
, a
, arr
, ctx
);
514 if (arr
) OPENSSL_free(arr
);
519 /* Invert a, reduce modulo p, and store the result in r. r could be a.
520 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from
521 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation
522 * of Elliptic Curve Cryptography Over Binary Fields".
524 int BN_GF2m_mod_inv(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
526 BIGNUM
*b
, *c
, *u
, *v
, *tmp
;
538 if (v
== NULL
) goto err
;
540 if (!BN_one(b
)) goto err
;
541 if (!BN_GF2m_mod(u
, a
, p
)) goto err
;
542 if (!BN_copy(v
, p
)) goto err
;
544 if (BN_is_zero(u
)) goto err
;
548 while (!BN_is_odd(u
))
550 if (!BN_rshift1(u
, u
)) goto err
;
553 if (!BN_GF2m_add(b
, b
, p
)) goto err
;
555 if (!BN_rshift1(b
, b
)) goto err
;
558 if (BN_abs_is_word(u
, 1)) break;
560 if (BN_num_bits(u
) < BN_num_bits(v
))
562 tmp
= u
; u
= v
; v
= tmp
;
563 tmp
= b
; b
= c
; c
= tmp
;
566 if (!BN_GF2m_add(u
, u
, v
)) goto err
;
567 if (!BN_GF2m_add(b
, b
, c
)) goto err
;
571 if (!BN_copy(r
, b
)) goto err
;
580 /* Invert xx, reduce modulo p, and store the result in r. r could be xx.
582 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper
583 * function is only provided for convenience; for best performance, use the
584 * BN_GF2m_mod_inv function.
586 int BN_GF2m_mod_inv_arr(BIGNUM
*r
, const BIGNUM
*xx
, const int p
[], BN_CTX
*ctx
)
593 if ((field
= BN_CTX_get(ctx
)) == NULL
) goto err
;
594 if (!BN_GF2m_arr2poly(p
, field
)) goto err
;
596 ret
= BN_GF2m_mod_inv(r
, xx
, field
, ctx
);
605 #ifndef OPENSSL_SUN_GF2M_DIV
606 /* Divide y by x, reduce modulo p, and store the result in r. r could be x
607 * or y, x could equal y.
609 int BN_GF2m_mod_div(BIGNUM
*r
, const BIGNUM
*y
, const BIGNUM
*x
, const BIGNUM
*p
, BN_CTX
*ctx
)
619 xinv
= BN_CTX_get(ctx
);
620 if (xinv
== NULL
) goto err
;
622 if (!BN_GF2m_mod_inv(xinv
, x
, p
, ctx
)) goto err
;
623 if (!BN_GF2m_mod_mul(r
, y
, xinv
, p
, ctx
)) goto err
;
632 /* Divide y by x, reduce modulo p, and store the result in r. r could be x
633 * or y, x could equal y.
634 * Uses algorithm Modular_Division_GF(2^m) from
635 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to
638 int BN_GF2m_mod_div(BIGNUM
*r
, const BIGNUM
*y
, const BIGNUM
*x
, const BIGNUM
*p
, BN_CTX
*ctx
)
640 BIGNUM
*a
, *b
, *u
, *v
;
653 if (v
== NULL
) goto err
;
655 /* reduce x and y mod p */
656 if (!BN_GF2m_mod(u
, y
, p
)) goto err
;
657 if (!BN_GF2m_mod(a
, x
, p
)) goto err
;
658 if (!BN_copy(b
, p
)) goto err
;
660 while (!BN_is_odd(a
))
662 if (!BN_rshift1(a
, a
)) goto err
;
663 if (BN_is_odd(u
)) if (!BN_GF2m_add(u
, u
, p
)) goto err
;
664 if (!BN_rshift1(u
, u
)) goto err
;
669 if (BN_GF2m_cmp(b
, a
) > 0)
671 if (!BN_GF2m_add(b
, b
, a
)) goto err
;
672 if (!BN_GF2m_add(v
, v
, u
)) goto err
;
675 if (!BN_rshift1(b
, b
)) goto err
;
676 if (BN_is_odd(v
)) if (!BN_GF2m_add(v
, v
, p
)) goto err
;
677 if (!BN_rshift1(v
, v
)) goto err
;
678 } while (!BN_is_odd(b
));
680 else if (BN_abs_is_word(a
, 1))
684 if (!BN_GF2m_add(a
, a
, b
)) goto err
;
685 if (!BN_GF2m_add(u
, u
, v
)) goto err
;
688 if (!BN_rshift1(a
, a
)) goto err
;
689 if (BN_is_odd(u
)) if (!BN_GF2m_add(u
, u
, p
)) goto err
;
690 if (!BN_rshift1(u
, u
)) goto err
;
691 } while (!BN_is_odd(a
));
695 if (!BN_copy(r
, u
)) goto err
;
705 /* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
706 * or yy, xx could equal yy.
708 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper
709 * function is only provided for convenience; for best performance, use the
710 * BN_GF2m_mod_div function.
712 int BN_GF2m_mod_div_arr(BIGNUM
*r
, const BIGNUM
*yy
, const BIGNUM
*xx
, const int p
[], BN_CTX
*ctx
)
721 if ((field
= BN_CTX_get(ctx
)) == NULL
) goto err
;
722 if (!BN_GF2m_arr2poly(p
, field
)) goto err
;
724 ret
= BN_GF2m_mod_div(r
, yy
, xx
, field
, ctx
);
733 /* Compute the bth power of a, reduce modulo p, and store
734 * the result in r. r could be a.
735 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363.
737 int BN_GF2m_mod_exp_arr(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*b
, const int p
[], BN_CTX
*ctx
)
748 if (BN_abs_is_word(b
, 1))
749 return (BN_copy(r
, a
) != NULL
);
752 if ((u
= BN_CTX_get(ctx
)) == NULL
) goto err
;
754 if (!BN_GF2m_mod_arr(u
, a
, p
)) goto err
;
756 n
= BN_num_bits(b
) - 1;
757 for (i
= n
- 1; i
>= 0; i
--)
759 if (!BN_GF2m_mod_sqr_arr(u
, u
, p
, ctx
)) goto err
;
760 if (BN_is_bit_set(b
, i
))
762 if (!BN_GF2m_mod_mul_arr(u
, u
, a
, p
, ctx
)) goto err
;
765 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
)
783 const int max
= BN_num_bits(p
) + 1;
788 if ((arr
= (int *)OPENSSL_malloc(sizeof(int) * max
)) == NULL
) goto err
;
789 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
790 if (!ret
|| ret
> max
)
792 BNerr(BN_F_BN_GF2M_MOD_EXP
,BN_R_INVALID_LENGTH
);
795 ret
= BN_GF2m_mod_exp_arr(r
, a
, b
, arr
, ctx
);
798 if (arr
) OPENSSL_free(arr
);
802 /* Compute the square root of a, reduce modulo p, and store
803 * the result in r. r could be a.
804 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
806 int BN_GF2m_mod_sqrt_arr(BIGNUM
*r
, const BIGNUM
*a
, const int p
[], BN_CTX
*ctx
)
815 /* reduction mod 1 => return 0 */
821 if ((u
= BN_CTX_get(ctx
)) == NULL
) goto err
;
823 if (!BN_set_bit(u
, p
[0] - 1)) goto err
;
824 ret
= BN_GF2m_mod_exp_arr(r
, a
, u
, p
, ctx
);
832 /* Compute the square root of a, reduce modulo p, and store
833 * the result in r. r could be a.
835 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper
836 * function is only provided for convenience; for best performance, use the
837 * BN_GF2m_mod_sqrt_arr function.
839 int BN_GF2m_mod_sqrt(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
842 const int max
= BN_num_bits(p
) + 1;
846 if ((arr
= (int *)OPENSSL_malloc(sizeof(int) * max
)) == NULL
) goto err
;
847 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
848 if (!ret
|| ret
> max
)
850 BNerr(BN_F_BN_GF2M_MOD_SQRT
,BN_R_INVALID_LENGTH
);
853 ret
= BN_GF2m_mod_sqrt_arr(r
, a
, arr
, ctx
);
856 if (arr
) OPENSSL_free(arr
);
860 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
861 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
863 int BN_GF2m_mod_solve_quad_arr(BIGNUM
*r
, const BIGNUM
*a_
, const int p
[], BN_CTX
*ctx
)
865 int ret
= 0, count
= 0, j
;
866 BIGNUM
*a
, *z
, *rho
, *w
, *w2
, *tmp
;
872 /* reduction mod 1 => return 0 */
881 if (w
== NULL
) goto err
;
883 if (!BN_GF2m_mod_arr(a
, a_
, p
)) goto err
;
892 if (p
[0] & 0x1) /* m is odd */
894 /* compute half-trace of a */
895 if (!BN_copy(z
, a
)) goto err
;
896 for (j
= 1; j
<= (p
[0] - 1) / 2; j
++)
898 if (!BN_GF2m_mod_sqr_arr(z
, z
, p
, ctx
)) goto err
;
899 if (!BN_GF2m_mod_sqr_arr(z
, z
, p
, ctx
)) goto err
;
900 if (!BN_GF2m_add(z
, z
, a
)) goto err
;
906 rho
= BN_CTX_get(ctx
);
907 w2
= BN_CTX_get(ctx
);
908 tmp
= BN_CTX_get(ctx
);
909 if (tmp
== NULL
) goto err
;
912 if (!BN_rand(rho
, p
[0], 0, 0)) goto err
;
913 if (!BN_GF2m_mod_arr(rho
, rho
, p
)) goto err
;
915 if (!BN_copy(w
, rho
)) goto err
;
916 for (j
= 1; j
<= p
[0] - 1; j
++)
918 if (!BN_GF2m_mod_sqr_arr(z
, z
, p
, ctx
)) goto err
;
919 if (!BN_GF2m_mod_sqr_arr(w2
, w
, p
, ctx
)) goto err
;
920 if (!BN_GF2m_mod_mul_arr(tmp
, w2
, a
, p
, ctx
)) goto err
;
921 if (!BN_GF2m_add(z
, z
, tmp
)) goto err
;
922 if (!BN_GF2m_add(w
, w2
, rho
)) goto err
;
925 } while (BN_is_zero(w
) && (count
< MAX_ITERATIONS
));
928 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR
,BN_R_TOO_MANY_ITERATIONS
);
933 if (!BN_GF2m_mod_sqr_arr(w
, z
, p
, ctx
)) goto err
;
934 if (!BN_GF2m_add(w
, z
, w
)) goto err
;
935 if (BN_GF2m_cmp(w
, a
))
937 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR
, BN_R_NO_SOLUTION
);
941 if (!BN_copy(r
, z
)) goto err
;
951 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
953 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper
954 * function is only provided for convenience; for best performance, use the
955 * BN_GF2m_mod_solve_quad_arr function.
957 int BN_GF2m_mod_solve_quad(BIGNUM
*r
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
960 const int max
= BN_num_bits(p
) + 1;
964 if ((arr
= (int *)OPENSSL_malloc(sizeof(int) *
965 max
)) == NULL
) goto err
;
966 ret
= BN_GF2m_poly2arr(p
, arr
, max
);
967 if (!ret
|| ret
> max
)
969 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD
,BN_R_INVALID_LENGTH
);
972 ret
= BN_GF2m_mod_solve_quad_arr(r
, a
, arr
, ctx
);
975 if (arr
) OPENSSL_free(arr
);
979 /* Convert the bit-string representation of a polynomial
980 * ( \sum_{i=0}^n a_i * x^i) into an array of integers corresponding
981 * to the bits with non-zero coefficient. Array is terminated with -1.
982 * Up to max elements of the array will be filled. Return value is total
983 * number of array elements that would be filled if array was large enough.
985 int BN_GF2m_poly2arr(const BIGNUM
*a
, int p
[], int max
)
993 for (i
= a
->top
- 1; i
>= 0; i
--)
996 /* skip word if a->d[i] == 0 */
999 for (j
= BN_BITS2
- 1; j
>= 0; j
--)
1003 if (k
< max
) p
[k
] = BN_BITS2
* i
+ j
;
1018 /* Convert the coefficient array representation of a polynomial to a
1019 * bit-string. The array must be terminated by -1.
1021 int BN_GF2m_arr2poly(const int p
[], BIGNUM
*a
)
1027 for (i
= 0; p
[i
] != -1; i
++)
1029 if (BN_set_bit(a
, p
[i
]) == 0)