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47 * ====================================================================
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).
55 #include "internal/cryptlib.h"
58 /* least significant word */
59 #define BN_lsw(n) (((n)->top == 0) ? (BN_ULONG) 0 : (n)->d[0])
61 /* Returns -2 for errors because both -1 and 0 are valid results. */
62 int BN_kronecker(const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
65 int ret = -2; /* avoid 'uninitialized' warning */
69 * In 'tab', only odd-indexed entries are relevant:
70 * For any odd BIGNUM n,
72 * is $(-1)^{(n^2-1)/8}$ (using TeX notation).
73 * Note that the sign of n does not matter.
75 static const int tab[8] = { 0, 1, 0, -1, 0, -1, 0, 1 };
94 * Kronecker symbol, implemented according to Henri Cohen,
95 * "A Course in Computational Algebraic Number Theory"
102 ret = BN_abs_is_word(A, 1);
106 /* Cohen's step 2: */
108 if (!BN_is_odd(A) && !BN_is_odd(B)) {
113 /* now B is non-zero */
115 while (!BN_is_bit_set(B, i))
117 err = !BN_rshift(B, B, i);
122 /* (thus B was even, thus A must be odd!) */
124 /* set 'ret' to $(-1)^{(A^2-1)/8}$ */
125 ret = tab[BN_lsw(A) & 7];
138 * now B is positive and odd, so what remains to be done is to compute
139 * the Jacobi symbol (A/B) and multiply it by 'ret'
143 /* Cohen's step 3: */
145 /* B is positive and odd */
148 ret = BN_is_one(B) ? ret : 0;
152 /* now A is non-zero */
154 while (!BN_is_bit_set(A, i))
156 err = !BN_rshift(A, A, i);
161 /* multiply 'ret' by $(-1)^{(B^2-1)/8}$ */
162 ret = ret * tab[BN_lsw(B) & 7];
165 /* Cohen's step 4: */
166 /* multiply 'ret' by $(-1)^{(A-1)(B-1)/4}$ */
167 if ((A->neg ? ~BN_lsw(A) : BN_lsw(A)) & BN_lsw(B) & 2)
170 /* (A, B) := (B mod |A|, |A|) */
171 err = !BN_nnmod(B, B, A, ctx);