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