1 /* mpz_legendre (op1, op2).
2 Contributed by Bennet Yee (bsy) at Carnegie-Mellon University
4 Copyright (C) 1992, 1996 Free Software Foundation, Inc.
6 This file is part of the GNU MP Library.
8 The GNU MP Library is free software; you can redistribute it and/or modify
9 it under the terms of the GNU Lesser General Public License as published by
10 the Free Software Foundation; either version 2.1 of the License, or (at your
11 option) any later version.
13 The GNU MP Library is distributed in the hope that it will be useful, but
14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
16 License for more details.
18 You should have received a copy of the GNU Lesser General Public License
19 along with the GNU MP Library; see the file COPYING.LIB. If not, write to
20 the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
21 MA 02111-1307, USA. */
29 /* Precondition: both p and q are positive */
33 mpz_legendre (mpz_srcptr pi, mpz_srcptr qi)
44 register mpz_ptr pptr, qptr;
45 register int retval = 1;
46 register unsigned long int s;
49 mpz_init_set (pptr, pi);
51 mpz_init_set (qptr, qi);
60 printf ("tail_recurse2: p=");
61 mpz_out_str (stdout, 10, pptr);
63 mpz_out_str (stdout, 10, qptr);
66 s = mpz_scan1 (qptr, 0);
67 if (s) mpz_tdiv_q_2exp (qptr, qptr, s); /* J(a,2) = 1 */
69 printf ("2 factor decomposition: p=");
70 mpz_out_str (stdout, 10, pptr);
72 mpz_out_str (stdout, 10, qptr);
75 /* postcondition q odd */
76 if (!mpz_cmp_ui (qptr, 1L)) /* J(a,1) = 1 */
78 mpz_mod (pptr, pptr, qptr); /* J(a,q) = J(b,q) when a == b mod q */
80 printf ("mod out by q: p=");
81 mpz_out_str (stdout, 10, pptr);
83 mpz_out_str (stdout, 10, qptr);
86 /* quick calculation to get approximate size first */
87 /* precondition: p < q */
88 if ((mpz_sizeinbase (pptr, 2) + 1 >= mpz_sizeinbase (qptr,2))
89 && (mpz_tdiv_q_2exp (qdiv2, qptr, 1L), mpz_cmp (pptr, qdiv2) > 0))
92 mpz_sub (pptr, qptr, pptr);
93 /* J(-1,q) = (-1)^((q-1)/2), q odd */
94 if (mpz_get_ui (qptr) & 2)
99 mpz_sub_ui (q_minus_q, qptr, 1L);
101 tail_recurse: /* we use tail_recurse only if q has not changed */
103 printf ("tail_recurse1: p=");
104 mpz_out_str (stdout, 10, pptr);
106 mpz_out_str (stdout, 10, qptr);
111 * this occurs only if gcd(p,q) != 1 which is never true for
114 if (!mpz_cmp_ui (pptr, 0L))
120 if (!mpz_cmp_ui (pptr, 1L))
127 if (!mpz_cmp (pptr, q_minus_1))
129 /* J(-1,q) = (-1)^((q-1)/2) */
130 if (mpz_get_ui (qptr) & 2)
132 /* else retval *= 1; */
137 * we do not handle J(xy,q) except for x==2
138 * since we do not want to factor
140 if ((s = mpz_scan1 (pptr, 0)) != 0)
143 * J(2,q) = (-1)^((q^2-1)/8)
145 * Note that q odd guarantees that q^2-1 is divisible by 8:
146 * Let a: q=2a+1. q^2 = 4a^2+4a+1, (q^2-1)/8 = a(a+1)/2, qed
148 * Now, note that this means that the low two bits of _a_
149 * (or the low bits of q shifted over by 1 determines
152 mpz_tdiv_q_2exp (pptr, pptr, s);
154 /* even powers of 2 gives J(2,q)^{2n} = 1 */
157 s = mpz_get_ui (qptr) >> 1;
165 * we know p is odd since we have cast out 2s
166 * precondition that q is odd guarantees both odd.
168 * quadratic reciprocity
169 * J(p,q) = (-1)^((p-1)(q-1)/4) * J(q,p)
171 if ((s = mpz_scan1 (pptr, 1)) <= 2 && (s + mpz_scan1 (qptr, 1)) <= 2)
174 mtmp = pptr; pptr = qptr; qptr = mtmp;
181 mpz_clear (q_minus_1);