2 * Copyright 2000-2016 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the OpenSSL license (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
10 #include "internal/cryptlib.h"
13 /* least significant word */
14 #define BN_lsw(n) (((n)->top == 0) ? (BN_ULONG) 0 : (n)->d[0])
16 /* Returns -2 for errors because both -1 and 0 are valid results. */
17 int BN_kronecker(const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
20 int ret = -2; /* avoid 'uninitialized' warning */
24 * In 'tab', only odd-indexed entries are relevant:
25 * For any odd BIGNUM n,
27 * is $(-1)^{(n^2-1)/8}$ (using TeX notation).
28 * Note that the sign of n does not matter.
30 static const int tab[8] = { 0, 1, 0, -1, 0, -1, 0, 1 };
49 * Kronecker symbol, implemented according to Henri Cohen,
50 * "A Course in Computational Algebraic Number Theory"
57 ret = BN_abs_is_word(A, 1);
63 if (!BN_is_odd(A) && !BN_is_odd(B)) {
68 /* now B is non-zero */
70 while (!BN_is_bit_set(B, i))
72 err = !BN_rshift(B, B, i);
77 /* (thus B was even, thus A must be odd!) */
79 /* set 'ret' to $(-1)^{(A^2-1)/8}$ */
80 ret = tab[BN_lsw(A) & 7];
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'
100 /* B is positive and odd */
103 ret = BN_is_one(B) ? ret : 0;
107 /* now A is non-zero */
109 while (!BN_is_bit_set(A, i))
111 err = !BN_rshift(A, A, i);
116 /* multiply 'ret' by $(-1)^{(B^2-1)/8}$ */
117 ret = ret * tab[BN_lsw(B) & 7];
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)
125 /* (A, B) := (B mod |A|, |A|) */
126 err = !BN_nnmod(B, B, A, ctx);