/* -*-c-*-
*
- * $Id: mp-gcd.c,v 1.1 1999/11/17 18:02:16 mdw Exp $
+ * $Id$
*
* Extended GCD calculation
*
* (c) 1999 Straylight/Edgeware
*/
-/*----- Licensing notice --------------------------------------------------*
+/*----- Licensing notice --------------------------------------------------*
*
* This file is part of Catacomb.
*
* it under the terms of the GNU Library General Public License as
* published by the Free Software Foundation; either version 2 of the
* License, or (at your option) any later version.
- *
+ *
* Catacomb is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU Library General Public License for more details.
- *
+ *
* You should have received a copy of the GNU Library General Public
* License along with Catacomb; if not, write to the Free
* Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
* MA 02111-1307, USA.
*/
-/*----- Revision history --------------------------------------------------*
- *
- * $Log: mp-gcd.c,v $
- * Revision 1.1 1999/11/17 18:02:16 mdw
- * New multiprecision integer arithmetic suite.
- *
- */
-
/*----- Header files ------------------------------------------------------*/
#include "mp.h"
*
* Use: Calculates @gcd(a, b)@, and two numbers @x@ and @y@ such that
* @ax + by = gcd(a, b)@. This is useful for computing modular
- * inverses. Neither @a@ nor @b@ may be zero. Note that,
- * unlike @mp_div@ for example, it is not possible to specify
- * explicit destinations -- new MPs are always allocated.
+ * inverses.
*/
void mp_gcd(mp **gcd, mp **xx, mp **yy, mp *a, mp *b)
{
- mp *X = MP_ONE, *Y = MP_ZERO;
- mp *x = MP_ZERO, *y = MP_ONE;
+ mp *x = MP_ONE, *X = MP_ZERO;
+ mp *y = MP_ZERO, *Y = MP_ONE;
mp *u, *v;
- size_t shift = 0;
- int ext = xx || yy;
- int swap = 0;
+ mp *q = MP_NEW, *t, *spare = MP_NEW;
+ unsigned f = 0;
- /* --- Ensure that @a@ is larger than @b@ --- */
+#define f_swap 1u
+#define f_aneg 2u
+#define f_bneg 4u
+#define f_ext 8u
- if (MP_CMP(a, <, b)) {
- { mp *t = a; a = b; b = t; }
- swap = 1;
- }
+ /* --- Sort out some initial flags --- */
- /* --- Take a reference to the arguments --- */
+ if (xx || yy)
+ f |= f_ext;
- a = MP_COPY(a);
- b = MP_COPY(b);
+ if (MP_NEGP(a))
+ f |= f_aneg;
+ if (MP_NEGP(b))
+ f |= f_bneg;
- /* --- Make sure @a@ and @b@ are not both even --- */
+ /* --- Ensure that @a@ is larger than @b@ --- *
+ *
+ * Use absolute values here!
+ */
- if (((a->v[0] | b->v[0]) & 1) == 0) {
- mpscan asc, bsc;
+ if (MPX_UCMP(a->v, a->vl, <, b->v, b->vl)) {
+ t = a; a = b; b = t;
+ f |= f_swap;
+ }
- /* --- Break off my copies --- */
+ /* --- Check for zeroness --- */
- MP_SPLIT(a);
- MP_SPLIT(b);
- MP_SCAN(&asc, a);
- MP_SCAN(&bsc, b);
+ if (MP_ZEROP(b)) {
- /* --- Start scanning --- */
+ /* --- Store %$|a|$% as the GCD --- */
- for (;;) {
- if (!MP_STEP(&asc) || !MP_STEP(&bsc))
- assert(((void)"zero argument passed to mp_gcd", 0));
- if (MP_BIT(&asc) || MP_BIT(&bsc))
- break;
- shift++;
+ if (gcd) {
+ if (*gcd) MP_DROP(*gcd);
+ a = MP_COPY(a);
+ if (MP_NEGP(a)) {
+ MP_SPLIT(a);
+ a->f &= ~MP_NEG;
+ f |= f_aneg;
+ }
+ *gcd = a;
}
- /* --- Shift @a@ and @b@ down --- */
+ /* --- Store %$1$% and %$0$% in the appropriate bins --- */
- a = mp_lsr(a, a, shift);
- b = mp_lsr(b, b, shift);
+ if (f & f_ext) {
+ if (f & f_swap) {
+ mp **tt = xx; xx = yy; yy = tt;
+ }
+ if (xx) {
+ if (*xx) MP_DROP(*xx);
+ if (MP_EQ(a, MP_ZERO))
+ *xx = MP_ZERO;
+ else if (f & f_aneg)
+ *xx = MP_MONE;
+ else
+ *xx = MP_ONE;
+ }
+ if (yy) {
+ if (*yy) MP_DROP(*yy);
+ *yy = MP_ZERO;
+ }
+ }
+ return;
}
- /* --- Set up @u@ and @v@ --- */
+ /* --- Force the signs on the arguments and take copies --- */
+
+ a = MP_COPY(a);
+ b = MP_COPY(b);
+
+ MP_SPLIT(a); a->f &= ~MP_NEG;
+ MP_SPLIT(b); b->f &= ~MP_NEG;
u = MP_COPY(a);
v = MP_COPY(b);
- /* --- Start the main loop --- */
-
- for (;;) {
-
- /* --- While @u@ is even --- */
-
- {
- mpscan sc, xsc, ysc;
- size_t n = 0, nn = 0;
-
- MP_SCAN(&sc, u);
- MP_SCAN(&xsc, X); MP_SCAN(&ysc, Y);
- for (;;) {
- MP_STEP(&sc);
- MP_STEP(&xsc); MP_STEP(&ysc);
- if (MP_BIT(&sc))
- break;
- if (ext && (MP_BIT(&xsc) | MP_BIT(&ysc))) {
- if (n) {
- X = mp_lsr(X, X, n);
- Y = mp_lsr(Y, Y, n);
- n = 0;
- }
- X = mp_add(X, X, b);
- Y = mp_sub(Y, Y, a);
- MP_SCAN(&xsc, X);
- MP_SCAN(&ysc, Y);
- MP_STEP(&xsc); MP_STEP(&ysc);
- }
- n++; nn++;
- }
-
- if (nn) {
- u = mp_lsr(u, u, nn);
- if (ext && n) {
- X = mp_lsr(X, X, n);
- Y = mp_lsr(Y, Y, n);
- }
- }
+ /* --- Main extended Euclidean algorithm --- */
+
+ while (!MP_ZEROP(v)) {
+ mp_div(&q, &u, u, v);
+ if (f & f_ext) {
+ t = mp_mul(spare, X, q);
+ t = mp_sub(t, x, t);
+ spare = x; x = X; X = t;
+ t = mp_mul(spare, Y, q);
+ t = mp_sub(t, y, t);
+ spare = y; y = Y; Y = t;
}
+ t = u; u = v; v = t;
+ }
- /* --- While @v@ is even --- */
-
- {
- mpscan sc, xsc, ysc;
- size_t n = 0, nn = 0;
-
- MP_SCAN(&sc, v);
- MP_SCAN(&xsc, x); MP_SCAN(&ysc, y);
- for (;;) {
- MP_STEP(&sc);
- MP_STEP(&xsc); MP_STEP(&ysc);
- if (MP_BIT(&sc))
- break;
- if (ext && (MP_BIT(&xsc) | MP_BIT(&ysc))) {
- if (n) {
- x = mp_lsr(x, x, n);
- y = mp_lsr(y, y, n);
- n = 0;
- }
- x = mp_add(x, x, b);
- y = mp_sub(y, y, a);
- MP_SCAN(&xsc, x);
- MP_SCAN(&ysc, y);
- MP_STEP(&xsc); MP_STEP(&ysc);
- }
- n++; nn++;
- }
+ MP_DROP(q); if (spare) MP_DROP(spare);
+ if (!gcd)
+ MP_DROP(u);
+ else {
+ if (*gcd) MP_DROP(*gcd);
+ u->f &= ~MP_NEG;
+ *gcd = u;
+ }
- if (nn) {
- v = mp_lsr(v, v, nn);
- if (ext && n) {
- x = mp_lsr(x, x, n);
- y = mp_lsr(y, y, n);
- }
- }
+ /* --- Perform a little normalization --- *
+ *
+ * Ensure that the coefficient returned is positive, if there is only one.
+ * If there are two, favour @y@. Of course, if the original arguments were
+ * negative then I'll need to twiddle their signs as well.
+ */
+
+ if (f & f_ext) {
+
+ /* --- If @a@ and @b@ got swapped, swap the coefficients back --- */
+
+ if (f & f_swap) {
+ t = x; x = y; y = t;
+ t = a; a = b; b = t;
}
- /* --- End-of-loop fiddling --- */
+ /* --- Sort out the signs --- *
+ *
+ * Note that %$ax + by = a(x - b) + b(y + a)$%.
+ *
+ * This is currently bodgy. It needs sorting out at some time.
+ */
- if (MP_CMP(u, >=, v)) {
- u = mp_sub(u, u, v);
- if (ext) {
- X = mp_sub(X, X, x);
- Y = mp_sub(Y, Y, y);
+ if (yy) {
+ if (MP_NEGP(y)) {
+ do {
+ y = mp_add(y, y, a);
+ x = mp_sub(x, x, b);
+ } while (MP_NEGP(y));
+ } else {
+ while (MP_CMP(y, >=, a)) {
+ y = mp_sub(y, y, a);
+ x = mp_add(x, x, b);
+ }
}
} else {
- v = mp_sub(v, v, u);
- if (ext) {
- x = mp_sub(x, x, X);
- y = mp_sub(y, y, Y);
+ if (MP_NEGP(x)) {
+ do
+ x = mp_add(x, x, b);
+ while (MP_NEGP(x));
+ } else {
+ while (MP_CMP(x, >=, b))
+ x = mp_sub(x, x, b);
}
}
- if (MP_CMP(u, ==, MP_ZERO))
- break;
- }
-
- /* --- Write the results out --- */
+ /* --- Twiddle the signs --- */
- if (gcd)
- *gcd = mp_lsl(v, v, shift);
- else
- MP_DROP(v);
+ if (f & f_aneg)
+ x->f ^= MP_NEG;
+ if (f & f_bneg)
+ y->f ^= MP_NEG;
- /* --- Perform a little normalization --- *
- *
- * Ensure that the coefficient returned is positive, if there is only one.
- * If there are two, favour @y@.
- */
+ /* --- Store the results --- */
- if (ext) {
- if (swap) {
- mp *t = x; x = y; y = t;
+ if (!xx)
+ MP_DROP(x);
+ else {
+ if (*xx) MP_DROP(*xx);
+ *xx = x;
}
- if (yy) {
- if (y->f & MP_NEG) {
- y = mp_add(y, y, a);
- x = mp_sub(x, x, b);
- }
- } else if (x->f & MP_NEG)
- x = mp_add(x, x, b);
- if (xx) *xx = x; else MP_DROP(x);
- if (yy) *yy = y; else MP_DROP(y);
+ if (!yy)
+ MP_DROP(y);
+ else {
+ if (*yy) MP_DROP(*yy);
+ *yy = y;
+ }
}
- MP_DROP(u);
+ MP_DROP(v);
MP_DROP(X); MP_DROP(Y);
MP_DROP(a); MP_DROP(b);
}
+/* -- @mp_modinv@ --- *
+ *
+ * Arguments: @mp *d@ = destination
+ * @mp *x@ = argument
+ * @mp *p@ = modulus
+ *
+ * Returns: The inverse %$x^{-1} \bmod p$%.
+ *
+ * Use: Computes a modular inverse. An assertion fails if %$p$%
+ * has no inverse.
+ */
+
+mp *mp_modinv(mp *d, mp *x, mp *p)
+{
+ mp *g = MP_NEW;
+ mp_gcd(&g, 0, &d, p, x);
+ assert(MP_EQ(g, MP_ONE));
+ mp_drop(g);
+ return (d);
+}
+
/*----- Test rig ----------------------------------------------------------*/
#ifdef TEST_RIG
+static int modinv(dstr *v)
+{
+ int ok = 1;
+ mp *x = *(mp **)v[0].buf;
+ mp *m = *(mp **)v[1].buf;
+ mp *r = *(mp **)v[2].buf;
+
+ mp *y = mp_modinv(MP_NEW, x, m);
+ if (!MP_EQ(y, r)) {
+ fputs("\n*** mp_modinv failed", stderr);
+ fputs("\nx = ", stderr); mp_writefile(x, stderr, 10);
+ fputs("\nm = ", stderr); mp_writefile(m, stderr, 10);
+ fputs("\nexpect = ", stderr); mp_writefile(r, stderr, 10);
+ fputs("\nresult = ", stderr); mp_writefile(y, stderr, 10);
+ ok = 0;
+ }
+ MP_DROP(x); MP_DROP(m); MP_DROP(r); MP_DROP(y);
+ assert(mparena_count(MPARENA_GLOBAL) == 0);
+ return (ok);
+}
+
static int gcd(dstr *v)
{
int ok = 1;
mp *x = *(mp **)v[3].buf;
mp *y = *(mp **)v[4].buf;
- mp *gg, *xx, *yy;
+ mp *gg = MP_NEW, *xx = MP_NEW, *yy = MP_NEW;
mp_gcd(&gg, &xx, &yy, a, b);
- if (MP_CMP(x, !=, xx)) {
+ if (!MP_EQ(x, xx)) {
fputs("\n*** mp_gcd(x) failed", stderr);
- fputs("\na = ", stderr); mp_writefile(a, stderr, 10);
- fputs("\nb = ", stderr); mp_writefile(b, stderr, 10);
+ fputs("\na = ", stderr); mp_writefile(a, stderr, 10);
+ fputs("\nb = ", stderr); mp_writefile(b, stderr, 10);
fputs("\nexpect = ", stderr); mp_writefile(x, stderr, 10);
fputs("\nresult = ", stderr); mp_writefile(xx, stderr, 10);
fputc('\n', stderr);
ok = 0;
}
- if (MP_CMP(y, !=, yy)) {
+ if (!MP_EQ(y, yy)) {
fputs("\n*** mp_gcd(y) failed", stderr);
- fputs("\na = ", stderr); mp_writefile(a, stderr, 10);
- fputs("\nb = ", stderr); mp_writefile(b, stderr, 10);
+ fputs("\na = ", stderr); mp_writefile(a, stderr, 10);
+ fputs("\nb = ", stderr); mp_writefile(b, stderr, 10);
fputs("\nexpect = ", stderr); mp_writefile(y, stderr, 10);
fputs("\nresult = ", stderr); mp_writefile(yy, stderr, 10);
fputc('\n', stderr);
mp *ax = mp_mul(MP_NEW, a, xx);
mp *by = mp_mul(MP_NEW, b, yy);
ax = mp_add(ax, ax, by);
- if (MP_CMP(ax, ==, gg))
+ if (MP_EQ(ax, gg))
fputs("\n*** (Alternative result found.)\n", stderr);
MP_DROP(ax);
MP_DROP(by);
}
- if (MP_CMP(g, !=, gg)) {
+ if (!MP_EQ(g, gg)) {
fputs("\n*** mp_gcd(gcd) failed", stderr);
- fputs("\na = ", stderr); mp_writefile(a, stderr, 10);
- fputs("\nb = ", stderr); mp_writefile(b, stderr, 10);
+ fputs("\na = ", stderr); mp_writefile(a, stderr, 10);
+ fputs("\nb = ", stderr); mp_writefile(b, stderr, 10);
fputs("\nexpect = ", stderr); mp_writefile(g, stderr, 10);
fputs("\nresult = ", stderr); mp_writefile(gg, stderr, 10);
fputc('\n', stderr);
}
MP_DROP(a); MP_DROP(b); MP_DROP(g); MP_DROP(x); MP_DROP(y);
MP_DROP(gg); MP_DROP(xx); MP_DROP(yy);
+ assert(mparena_count(MPARENA_GLOBAL) == 0);
return (ok);
}
static test_chunk tests[] = {
{ "gcd", gcd, { &type_mp, &type_mp, &type_mp, &type_mp, &type_mp, 0 } },
+ { "modinv", modinv, { &type_mp, &type_mp, &type_mp, 0 } },
{ 0, 0, { 0 } }
};
projects / u / mdw / catacomb / blobdiff