3 * $Id: mp-modsqrt.c,v 1.1 2000/06/22 19:01:31 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 /*----- Revision history --------------------------------------------------*
32 * $Log: mp-modsqrt.c,v $
33 * Revision 1.1 2000/06/22 19:01:31 mdw
34 * Compute square roots in a prime field.
38 /*----- Header files ------------------------------------------------------*/
46 /*----- Main code ---------------------------------------------------------*/
48 /* --- @mp_modsqrt@ --- *
50 * Arguments: @mp *d@ = destination integer
51 * @mp *a@ = source integer
52 * @mp *p@ = modulus (must be prime)
54 * Returns: If %$a$% is a quadratic residue, a square root of %$a$%; else
57 * Use: Returns an integer %$x$% such that %$x^2 \equiv a \pmod{p}$%,
58 * if one exists; else a null pointer. This function will not
59 * work if %$p$% is composite: you must factor the modulus, take
60 * a square root mod each factor, and recombine the results
61 * using the Chinese Remainder Theorem.
64 mp
*mp_modsqrt(mp
*d
, mp
*a
, mp
*p
)
75 /* --- Cope if %$a \not\in Q_p$% --- */
77 if (mp_jacobi(a
, p
) != 1) {
83 /* --- Choose some quadratic non-residue --- */
86 grand
*g
= fibrand_create(0);
90 b
= mprand_range(b
, p
, g
, 0);
91 while (mp_jacobi(b
, p
) != -1);
95 /* --- Find the inverse of %$a$% --- */
98 mp_gcd(0, &ainv
, 0, a
, p
);
100 /* --- Split %$p - 1$% into a power of two and an odd number --- */
102 t
= mp_sub(MP_NEW
, p
, MP_ONE
);
103 t
= mp_odd(t
, t
, &s
);
105 /* --- Now to really get going --- */
107 mpmont_create(&mm
, p
);
108 c
= mpmont_expr(&mm
, b
, b
, t
);
109 t
= mp_add(t
, t
, MP_ONE
);
111 r
= mpmont_expr(&mm
, t
, a
, t
);
112 ainv
= mpmont_mul(&mm
, ainv
, ainv
, mm
.r2
);
114 mone
= mp_sub(MP_NEW
, p
, mm
.r
);
118 for (i
= 1; i
< s
; i
++) {
120 /* --- Compute %$d_0 = r^2a^{-1}$% --- */
123 dd
= mpmont_reduce(&mm
, dd
, dd
);
124 dd
= mpmont_mul(&mm
, dd
, dd
, ainv
);
126 /* --- Now %$d = d_0^{s - i - 1}$% --- */
128 for (j
= i
; j
< s
- 1; j
++) {
130 dd
= mpmont_reduce(&mm
, dd
, dd
);
133 /* --- Fiddle at the end --- */
135 if (MP_CMP(dd
, ==, mone
))
136 r
= mpmont_mul(&mm
, r
, r
, c
);
138 c
= mpmont_reduce(&mm
, c
, c
);
141 /* --- Done, so tidy up --- */
143 d
= mpmont_reduce(&mm
, d
, r
);
145 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_CMP(r
, ==, rr
))
174 if (MP_CMP(r
, ==, rr
))
179 fputs("\n*** fail\n", stderr
);
180 fputs("a = ", stderr
); mp_writefile(a
, stderr
, 10); fputc('\n', stderr
);
181 fputs("p = ", stderr
); mp_writefile(p
, stderr
, 10); fputc('\n', stderr
);
183 fputs("r = ", stderr
);
184 mp_writefile(r
, stderr
, 10);
187 fputs("r = <undef>\n", stderr
);
188 fputs("rr = ", stderr
); mp_writefile(rr
, stderr
, 10); fputc('\n', stderr
);
197 assert(mparena_count(MPARENA_GLOBAL
) == 0);
201 static test_chunk tests
[] = {
202 { "modsqrt", verify
, { &type_mp
, &type_mp
, &type_mp
, 0 } },
206 int main(int argc
, char *argv
[])
209 test_run(argc
, argv
, tests
, SRCDIR
"/tests/mp");
215 /*----- That's all, folks -------------------------------------------------*/