1 /* ====================================================================
2 * Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved.
3 *
4 * Redistribution and use in source and binary forms, with or without
5 * modification, are permitted provided that the following conditions
6 * are met:
7 *
8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer.
10 *
11 * 2. Redistributions in binary form must reproduce the above copyright
12 * notice, this list of conditions and the following disclaimer in
13 * the documentation and/or other materials provided with the
14 * distribution.
15 *
16 * 3. All advertising materials mentioning features or use of this
17 * software must display the following acknowledgment:
18 * "This product includes software developed by the OpenSSL Project
19 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
20 *
21 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
22 * endorse or promote products derived from this software without
23 * prior written permission. For written permission, please contact
24 * openssl-core@openssl.org.
25 *
26 * 5. Products derived from this software may not be called "OpenSSL"
27 * nor may "OpenSSL" appear in their names without prior written
28 * permission of the OpenSSL Project.
29 *
30 * 6. Redistributions of any form whatsoever must retain the following
31 * acknowledgment:
32 * "This product includes software developed by the OpenSSL Project
33 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
34 *
35 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
36 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
37 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
38 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
39 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
40 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
41 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
42 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
43 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
44 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
45 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
46 * OF THE POSSIBILITY OF SUCH DAMAGE.
47 * ====================================================================
48 *
49 * This product includes cryptographic software written by Eric Young
50 * (eay@cryptsoft.com). This product includes software written by Tim
51 * Hudson (tjh@cryptsoft.com). */
52
53 #include <openssl/bn.h>
54
55 #include "internal.h"
56
57
58 /* least significant word */
59 #define BN_lsw(n) (((n)->top == 0) ? (BN_ULONG) 0 : (n)->d[0])
60
61 /* Returns -2 for errors because both -1 and 0 are valid results. */
BN_kronecker(const BIGNUM * a,const BIGNUM * b,BN_CTX * ctx)62 int BN_kronecker(const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) {
63 int i;
64 int ret = -2;
65 BIGNUM *A, *B, *tmp;
66 /* In 'tab', only odd-indexed entries are relevant:
67 * For any odd BIGNUM n,
68 * tab[BN_lsw(n) & 7]
69 * is $(-1)^{(n^2-1)/8}$ (using TeX notation).
70 * Note that the sign of n does not matter. */
71 static const int tab[8] = {0, 1, 0, -1, 0, -1, 0, 1};
72
73 BN_CTX_start(ctx);
74 A = BN_CTX_get(ctx);
75 B = BN_CTX_get(ctx);
76 if (B == NULL) {
77 goto end;
78 }
79
80 if (!BN_copy(A, a) ||
81 !BN_copy(B, b)) {
82 goto end;
83 }
84
85 /* Kronecker symbol, imlemented according to Henri Cohen,
86 * "A Course in Computational Algebraic Number Theory"
87 * (algorithm 1.4.10). */
88
89 /* Cohen's step 1: */
90
91 if (BN_is_zero(B)) {
92 ret = BN_abs_is_word(A, 1);
93 goto end;
94 }
95
96 /* Cohen's step 2: */
97
98 if (!BN_is_odd(A) && !BN_is_odd(B)) {
99 ret = 0;
100 goto end;
101 }
102
103 /* now B is non-zero */
104 i = 0;
105 while (!BN_is_bit_set(B, i)) {
106 i++;
107 }
108 if (!BN_rshift(B, B, i)) {
109 goto end;
110 }
111 if (i & 1) {
112 /* i is odd */
113 /* (thus B was even, thus A must be odd!) */
114
115 /* set 'ret' to $(-1)^{(A^2-1)/8}$ */
116 ret = tab[BN_lsw(A) & 7];
117 } else {
118 /* i is even */
119 ret = 1;
120 }
121
122 if (B->neg) {
123 B->neg = 0;
124 if (A->neg) {
125 ret = -ret;
126 }
127 }
128
129 /* now B is positive and odd, so what remains to be done is to compute the
130 * Jacobi symbol (A/B) and multiply it by 'ret' */
131
132 while (1) {
133 /* Cohen's step 3: */
134
135 /* B is positive and odd */
136 if (BN_is_zero(A)) {
137 ret = BN_is_one(B) ? ret : 0;
138 goto end;
139 }
140
141 /* now A is non-zero */
142 i = 0;
143 while (!BN_is_bit_set(A, i)) {
144 i++;
145 }
146 if (!BN_rshift(A, A, i)) {
147 goto end;
148 }
149 if (i & 1) {
150 /* i is odd */
151 /* multiply 'ret' by $(-1)^{(B^2-1)/8}$ */
152 ret = ret * tab[BN_lsw(B) & 7];
153 }
154
155 /* Cohen's step 4: */
156 /* multiply 'ret' by $(-1)^{(A-1)(B-1)/4}$ */
157 if ((A->neg ? ~BN_lsw(A) : BN_lsw(A)) & BN_lsw(B) & 2) {
158 ret = -ret;
159 }
160
161 /* (A, B) := (B mod |A|, |A|) */
162 if (!BN_nnmod(B, B, A, ctx)) {
163 ret = -2;
164 goto end;
165 }
166 tmp = A;
167 A = B;
168 B = tmp;
169 tmp->neg = 0;
170 }
171
172 end:
173 BN_CTX_end(ctx);
174 return ret;
175 }
176