3 * $Id: mp-modsqrt.c,v 1.5 2004/04/08 01:36:15 mdw Exp $
5 * Compute square roots modulo a prime
7 * (c) 2000 Straylight/Edgeware
10 /*----- Licensing notice --------------------------------------------------*
12 * This file is part of Catacomb.
14 * Catacomb is free software; you can redistribute it and/or modify
15 * it under the terms of the GNU Library General Public License as
16 * published by the Free Software Foundation; either version 2 of the
17 * License, or (at your option) any later version.
19 * Catacomb is distributed in the hope that it will be useful,
20 * but WITHOUT ANY WARRANTY; without even the implied warranty of
21 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
22 * GNU Library General Public License for more details.
24 * You should have received a copy of the GNU Library General Public
25 * License along with Catacomb; if not, write to the Free
26 * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
30 /*----- Header files ------------------------------------------------------*/
38 /*----- Main code ---------------------------------------------------------*/
40 /* --- @mp_modsqrt@ --- *
42 * Arguments: @mp *d@ = destination integer
43 * @mp *a@ = source integer
44 * @mp *p@ = modulus (must be prime)
46 * Returns: If %$a$% is a quadratic residue, a square root of %$a$%; else
49 * Use: Returns an integer %$x$% such that %$x^2 \equiv a \pmod{p}$%,
50 * if one exists; else a null pointer. This function will not
51 * work if %$p$% is composite: you must factor the modulus, take
52 * a square root mod each factor, and recombine the results
53 * using the Chinese Remainder Theorem.
56 mp
*mp_modsqrt(mp
*d
, mp
*a
, mp
*p
)
67 /* --- Cope if %$a \not\in Q_p$% --- */
69 if (mp_jacobi(a
, p
) != 1) {
74 /* --- Choose some quadratic non-residue --- */
77 grand
*g
= fibrand_create(0);
81 b
= mprand_range(b
, p
, g
, 0);
82 while (mp_jacobi(b
, p
) != -1);
86 /* --- Find the inverse of %$a$% --- */
88 ainv
= mp_modinv(MP_NEW
, a
, p
);
90 /* --- Split %$p - 1$% into a power of two and an odd number --- */
92 t
= mp_sub(MP_NEW
, p
, MP_ONE
);
95 /* --- Now to really get going --- */
97 mpmont_create(&mm
, p
);
98 b
= mpmont_mul(&mm
, b
, b
, mm
.r2
);
99 c
= mpmont_expr(&mm
, b
, b
, t
);
100 t
= mp_add(t
, t
, MP_ONE
);
102 dd
= mpmont_mul(&mm
, MP_NEW
, a
, mm
.r2
);
103 r
= mpmont_expr(&mm
, t
, dd
, t
);
105 ainv
= mpmont_mul(&mm
, ainv
, ainv
, mm
.r2
);
107 mone
= mp_sub(MP_NEW
, p
, mm
.r
);
111 for (i
= 1; i
< s
; i
++) {
113 /* --- Compute %$d_0 = r^2a^{-1}$% --- */
116 dd
= mpmont_reduce(&mm
, dd
, dd
);
117 dd
= mpmont_mul(&mm
, dd
, dd
, ainv
);
119 /* --- Now %$d = d_0^{s - i - 1}$% --- */
121 for (j
= i
; j
< s
- 1; j
++) {
123 dd
= mpmont_reduce(&mm
, dd
, dd
);
126 /* --- Fiddle at the end --- */
129 r
= mpmont_mul(&mm
, r
, r
, c
);
131 c
= mpmont_reduce(&mm
, c
, c
);
134 /* --- Done, so tidy up --- */
136 d
= mpmont_reduce(&mm
, d
, r
);
138 mp_drop(r
); mp_drop(c
);
146 /*----- Test rig ----------------------------------------------------------*/
150 #include <mLib/testrig.h>
152 static int verify(dstr
*v
)
154 mp
*a
= *(mp
**)v
[0].buf
;
155 mp
*p
= *(mp
**)v
[1].buf
;
156 mp
*rr
= *(mp
**)v
[2].buf
;
157 mp
*r
= mp_modsqrt(MP_NEW
, a
, p
);
162 else if (MP_EQ(r
, rr
))
171 fputs("\n*** fail\n", stderr
);
172 fputs("a = ", stderr
); mp_writefile(a
, stderr
, 10); fputc('\n', stderr
);
173 fputs("p = ", stderr
); mp_writefile(p
, stderr
, 10); fputc('\n', stderr
);
175 fputs("r = ", stderr
);
176 mp_writefile(r
, stderr
, 10);
179 fputs("r = <undef>\n", stderr
);
180 fputs("rr = ", stderr
); mp_writefile(rr
, stderr
, 10); fputc('\n', stderr
);
188 assert(mparena_count(MPARENA_GLOBAL
) == 0);
192 static test_chunk tests
[] = {
193 { "modsqrt", verify
, { &type_mp
, &type_mp
, &type_mp
, 0 } },
197 int main(int argc
, char *argv
[])
200 test_run(argc
, argv
, tests
, SRCDIR
"/tests/mp");
206 /*----- That's all, folks -------------------------------------------------*/