xref: /openssl/crypto/bn/bn_kron.c (revision 706457b7)
1 /*
2  * Copyright 2000-2016 The OpenSSL Project Authors. All Rights Reserved.
3  *
4  * Licensed under the Apache License 2.0 (the "License").  You may not use
5  * this file except in compliance with the License.  You can obtain a copy
6  * in the file LICENSE in the source distribution or at
7  * https://www.openssl.org/source/license.html
8  */
9 
10 #include "internal/cryptlib.h"
11 #include "bn_local.h"
12 
13 /* least significant word */
14 #define BN_lsw(n) (((n)->top == 0) ? (BN_ULONG) 0 : (n)->d[0])
15 
16 /* Returns -2 for errors because both -1 and 0 are valid results. */
BN_kronecker(const BIGNUM * a,const BIGNUM * b,BN_CTX * ctx)17 int BN_kronecker(const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
18 {
19     int i;
20     int ret = -2;               /* avoid 'uninitialized' warning */
21     int err = 0;
22     BIGNUM *A, *B, *tmp;
23     /*-
24      * In 'tab', only odd-indexed entries are relevant:
25      * For any odd BIGNUM n,
26      *     tab[BN_lsw(n) & 7]
27      * is $(-1)^{(n^2-1)/8}$ (using TeX notation).
28      * Note that the sign of n does not matter.
29      */
30     static const int tab[8] = { 0, 1, 0, -1, 0, -1, 0, 1 };
31 
32     bn_check_top(a);
33     bn_check_top(b);
34 
35     BN_CTX_start(ctx);
36     A = BN_CTX_get(ctx);
37     B = BN_CTX_get(ctx);
38     if (B == NULL)
39         goto end;
40 
41     err = !BN_copy(A, a);
42     if (err)
43         goto end;
44     err = !BN_copy(B, b);
45     if (err)
46         goto end;
47 
48     /*
49      * Kronecker symbol, implemented according to Henri Cohen,
50      * "A Course in Computational Algebraic Number Theory"
51      * (algorithm 1.4.10).
52      */
53 
54     /* Cohen's step 1: */
55 
56     if (BN_is_zero(B)) {
57         ret = BN_abs_is_word(A, 1);
58         goto end;
59     }
60 
61     /* Cohen's step 2: */
62 
63     if (!BN_is_odd(A) && !BN_is_odd(B)) {
64         ret = 0;
65         goto end;
66     }
67 
68     /* now  B  is non-zero */
69     i = 0;
70     while (!BN_is_bit_set(B, i))
71         i++;
72     err = !BN_rshift(B, B, i);
73     if (err)
74         goto end;
75     if (i & 1) {
76         /* i is odd */
77         /* (thus  B  was even, thus  A  must be odd!)  */
78 
79         /* set 'ret' to $(-1)^{(A^2-1)/8}$ */
80         ret = tab[BN_lsw(A) & 7];
81     } else {
82         /* i is even */
83         ret = 1;
84     }
85 
86     if (B->neg) {
87         B->neg = 0;
88         if (A->neg)
89             ret = -ret;
90     }
91 
92     /*
93      * now B is positive and odd, so what remains to be done is to compute
94      * the Jacobi symbol (A/B) and multiply it by 'ret'
95      */
96 
97     while (1) {
98         /* Cohen's step 3: */
99 
100         /*  B  is positive and odd */
101 
102         if (BN_is_zero(A)) {
103             ret = BN_is_one(B) ? ret : 0;
104             goto end;
105         }
106 
107         /* now  A  is non-zero */
108         i = 0;
109         while (!BN_is_bit_set(A, i))
110             i++;
111         err = !BN_rshift(A, A, i);
112         if (err)
113             goto end;
114         if (i & 1) {
115             /* i is odd */
116             /* multiply 'ret' by  $(-1)^{(B^2-1)/8}$ */
117             ret = ret * tab[BN_lsw(B) & 7];
118         }
119 
120         /* Cohen's step 4: */
121         /* multiply 'ret' by  $(-1)^{(A-1)(B-1)/4}$ */
122         if ((A->neg ? ~BN_lsw(A) : BN_lsw(A)) & BN_lsw(B) & 2)
123             ret = -ret;
124 
125         /* (A, B) := (B mod |A|, |A|) */
126         err = !BN_nnmod(B, B, A, ctx);
127         if (err)
128             goto end;
129         tmp = A;
130         A = B;
131         B = tmp;
132         tmp->neg = 0;
133     }
134  end:
135     BN_CTX_end(ctx);
136     if (err)
137         return -2;
138     else
139         return ret;
140 }
141