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.
55 * We guarantee that the square root returned is the smallest
56 * one (i.e., the `positive' square root).
59 mp
*mp_modsqrt(mp
*d
, mp
*a
, mp
*p
)
70 /* --- Cope if %$a \not\in Q_p$% --- */
72 if (mp_jacobi(a
, p
) != 1) {
77 /* --- Choose some quadratic non-residue --- */
80 grand
*g
= fibrand_create(0);
84 b
= mprand_range(b
, p
, g
, 0);
85 while (mp_jacobi(b
, p
) != -1);
89 /* --- Find the inverse of %$a$% --- */
91 ainv
= mp_modinv(MP_NEW
, a
, p
);
93 /* --- Split %$p - 1$% into a power of two and an odd number --- */
95 t
= mp_sub(MP_NEW
, p
, MP_ONE
);
98 /* --- Now to really get going --- */
100 mpmont_create(&mm
, p
);
101 b
= mpmont_mul(&mm
, b
, b
, mm
.r2
);
102 c
= mpmont_expr(&mm
, b
, b
, t
);
103 t
= mp_add(t
, t
, MP_ONE
);
105 dd
= mpmont_mul(&mm
, MP_NEW
, a
, mm
.r2
);
106 r
= mpmont_expr(&mm
, t
, dd
, t
);
108 ainv
= mpmont_mul(&mm
, ainv
, ainv
, mm
.r2
);
110 mone
= mp_sub(MP_NEW
, p
, mm
.r
);
114 for (i
= 1; i
< s
; i
++) {
116 /* --- Compute %$d_0 = r^2a^{-1}$% --- */
119 dd
= mpmont_reduce(&mm
, dd
, dd
);
120 dd
= mpmont_mul(&mm
, dd
, dd
, ainv
);
122 /* --- Now %$d = d_0^{2^{s - i - 1}}$% --- */
124 for (j
= i
; j
< s
- 1; j
++) {
126 dd
= mpmont_reduce(&mm
, dd
, dd
);
129 /* --- Fiddle at the end --- */
132 r
= mpmont_mul(&mm
, r
, r
, c
);
134 c
= mpmont_reduce(&mm
, c
, c
);
137 /* --- Done, so tidy up --- *
139 * Canonify the answer.
142 d
= mpmont_reduce(&mm
, d
, r
);
144 if (MP_CMP(r
, <, d
)) { mp
*tt
= r
; r
= d
; d
= tt
; }
146 mp_drop(r
); mp_drop(c
);
154 /*----- Test rig ----------------------------------------------------------*/
158 #include <mLib/testrig.h>
160 static int verify(dstr
*v
)
162 mp
*a
= *(mp
**)v
[0].buf
;
163 mp
*p
= *(mp
**)v
[1].buf
;
164 mp
*rr
= *(mp
**)v
[2].buf
;
165 mp
*r
= mp_modsqrt(MP_NEW
, a
, p
);
170 else if (MP_EQ(r
, rr
))
174 fputs("\n*** fail\n", stderr
);
175 fputs("a = ", stderr
); mp_writefile(a
, stderr
, 10); fputc('\n', stderr
);
176 fputs("p = ", stderr
); mp_writefile(p
, stderr
, 10); fputc('\n', stderr
);
178 fputs("r = ", stderr
);
179 mp_writefile(r
, stderr
, 10);
182 fputs("r = <undef>\n", stderr
);
183 fputs("rr = ", stderr
); mp_writefile(rr
, stderr
, 10); fputc('\n', stderr
);
191 assert(mparena_count(MPARENA_GLOBAL
) == 0);
195 static test_chunk tests
[] = {
196 { "modsqrt", verify
, { &type_mp
, &type_mp
, &type_mp
, 0 } },
200 int main(int argc
, char *argv
[])
203 test_run(argc
, argv
, tests
, SRCDIR
"/tests/mp");
209 /*----- That's all, folks -------------------------------------------------*/