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/*
Author: Pate Williams (c) 1997
Polynomial extended Euclidean algorithm. See
"Handbook of Applied Cryptography" by Alfred
J. Menezes et al 2.6.2 Section 2.2.1 Algorithm
page 82.
*/
#include <stdio.h>
#include "lip.h"
#define DEBUG
#define POLY_SIZE 8192l
void zpoly_mul(long m, long n, verylong *za, verylong *zb,
verylong *zc, long *p)
{
long i, j, k;
verylong zd = 0, zai = 0, zbk = 0, zsum = 0, zterm = 0;
*p = m + n;
for (k = 0; k <= *p; k++) {
zzero(&zsum);
for (i = 0; i <= k; i++) {
j = k - i;
if (i > m) zzero(&zai);
else zcopy(za[i], &zai);
if (j > n) zzero(&zbk);
else zcopy(zb[j], &zbk);
zmul(zai, zbk, &zterm);
zcopy(zsum, &zd);
zadd(zterm, zd, &zsum);
}
zcopy(zsum, &zc[k]);
}
zfree(&zd);
zfree(&zai);
zfree(&zbk);
zfree(&zsum);
zfree(&zterm);
}
void zpoly_div(long m, long n, verylong *zu, verylong *zv,
verylong *zq, verylong *zr, long *p, long *s)
{
long j, jk, k, nk;
verylong za = 0, zb = 0, zvn = 0;
zcopy(zv[n], &zvn);
for (j = 0; j <= m; j++)
zcopy(zu[j], &zr[j]);
if (m < n) {
*p = 0, *s = m;
zzero(&zq[0]);
}
else {
*p = m - n, *s = n - 1;
for (k = *p; k >= 0; k--) {
nk = n + k;
zsexp(zvn, k, &za);
zmul(zr[nk], za, &zq[k]);
for (j = nk - 1; j >= 0; j--) {
jk = j - k;
if (jk >= 0) {
zmul(zvn, zr[j], &za);
zmul(zr[nk], zv[jk], &zb);
zsub(za, zb, &zr[j]);
}
else {
zcopy(zr[j], &za);
zmul(zvn, za, &zr[j]);
}
}
}
while (*p > 0 && zscompare(zq[*p], 0l) == 0) *p = *p - 1;
while (*s > 0 && zscompare(zr[*s], 0l) == 0) *s = *s - 1;
}
zfree(&za);
zfree(&zb);
zfree(&zvn);
}
void zpoly_pow(long degreeA, long degreem, verylong zn,
verylong *zA, verylong *zm, verylong *zs,
long *ds)
{
long dp, dq, dx = degreeA, i;
verylong za = 0, zb = 0, zp[POLY_SIZE], zq[POLY_SIZE],
zx[POLY_SIZE], zy[POLY_SIZE];
for (i = 0; i < POLY_SIZE; i++)
zp[i] = zq[i] = zx[i] = zy[i] = 0;
*ds = 0;
zcopy(zn, &za);
zone(&zs[0]);
for (i = 0; i <= dx; i++) zcopy(zA[i], &zx[i]);
while (zscompare(za, 0l) > 0) {
if (zodd(za)) {
/* s = (s * x) % m; */
zpoly_mul(*ds, dx, zs, zx, zp, &dp);
zpoly_div(dp, degreem, zp, zm, zq, zs, &dq, ds);
}
zcopy(za, &zb);
zrshift(zb, 1l, &za);
if (zscompare(za, 0l) > 0) {
/* x = (x * x) % m; */
for (i = 0; i <= dx; i++) zcopy(zx[i], &zy[i]);
zpoly_mul(dx, dx, zx, zy, zp, &dp);
zpoly_div(dp, degreem, zp, zm, zq, zx, &dq, &dx);
}
}
zfree(&za);
zfree(&zb);
for (i = 0; i < POLY_SIZE; i++) {
zfree(&zp[i]);
zfree(&zq[i]);
zfree(&zx[i]);
zfree(&zy[i]);
}
}
void zpoly_sub(long da, long db, verylong *za, verylong *zb,
verylong *zc, long *dc)
{
long i;
verylong zz = 0;
zzero(&zz);
if (da >= db) {
for (i = 0; i <= db; i++)
zsub(za[i], zb[i], &zc[i]);
for (i = db + 1; i <= da; i++)
zcopy(za[i], &zc[i]);
*dc = da;
}
else {
for (i = 0; i <= da; i++)
zsub(za[i], zb[i], &zc[i]);
for (i = da + 1; i <= db; i++)
zsub(zz, zb[i], &zc[i]);
*dc = db;
}
zfree(&zz);
}
void zpoly_gcd(long degreeA, long degreeB, long p,
verylong *zA, verylong *zB, verylong *za,
long *da)
{
int nonzero = 0, zero;
long db, dq, dr, i;
verylong zc = 0, zp = 0;
verylong zb[POLY_SIZE], zq[POLY_SIZE], zr[POLY_SIZE];
for (i = 0; i < POLY_SIZE; i++)
zb[i] = zq[i] = zr[i] = 0;
if (degreeA > degreeB) {
*da = degreeA;
db = degreeB;
for (i = 0; i <= *da; i++) zcopy(zA[i], &za[i]);
for (i = 0; i <= db; i++) zcopy(zB[i], &zb[i]);
}
else {
*da = degreeB;
db = degreeA;
for (i = 0; i <= *da; i++) zcopy(zB[i], &za[i]);
for (i = 0; i <= db; i++) zcopy(zA[i], &zb[i]);
}
for (i = 0; i <= db && !nonzero; i++)
nonzero = zscompare(zb[i], 0l) != 0;
while (nonzero) {
zpoly_div(*da, db, za, zb, zq, zr, &dq, &dr);
zintoz(p, &zp);
for (i = 0; i <= dr; i++) {
zcopy(zr[i], &zc);
zmod(zc, zp, &zr[i]);
}
zero = 1;
for (i = dr; i >= 0 && zero; i--) {
zero = zscompare(zr[i], 0l) == 0;
if (zero && dr > 0) dr--;
}
for (i = 0; i <= db; i++) zcopy(zb[i], &za[i]);
*da = db;
for (i = 0; i <= dr; i++) zcopy(zr[i], &zb[i]);
db = dr;
nonzero = 0;
for (i = 0; i <= db && !nonzero; i++)
nonzero = zscompare(zb[i], 0l) != 0;
}
zfree(&zc);
zfree(&zp);
for (i = 0; i < POLY_SIZE; i++) {
zfree(&zb[i]);
zfree(&zq[i]);
zfree(&zr[i]);
}
}
void zpoly_copy(long da, verylong *za, verylong *zb, long *db)
{
long i;
*db = da;
for (i = 0; i <= da; i++) zcopy(za[i], &zb[i]);
}
void zpoly_print(long da, verylong *za)
{
long i;
for (i = da; i >= 0; i--) {
zwrite(za[i]);
printf(" ");
}
printf("\n");
}
void zpoly_mod(long p, verylong *za, long *da)
{
long i;
for (i = 0; i <= *da; i++)
zintoz(zsmod(za[i], p), &za[i]);
while (*da > 0 && zscompare(za[*da], 0l) == 0) *da = *da - 1;
}
void zpoly_ext_euclid(long dg, long dh, long p, verylong *zg,
verylong *zh, verylong *zs,
verylong *zt, verylong *zd,
long *ds, long *dt, long *dd)
{
long da, dq, dr, ds1 = 0, ds2 = 0, dt1 = 0, dt2 = 0, i;
verylong za[POLY_SIZE], zb[POLY_SIZE];
verylong zq[POLY_SIZE], zr[POLY_SIZE];
verylong zs1[POLY_SIZE], zs2[POLY_SIZE];
verylong zt1[POLY_SIZE], zt2[POLY_SIZE];
if (dh == 0 && zscompare(zh[0], 0l) == 0) {
zpoly_copy(dg, zg, zd, dd);
*ds = *dt = 0;
zone(&zs[0]);
zzero(&zt[0]);
}
for (i = 0; i < POLY_SIZE; i++) {
za[i] = zb[i] = zq[i] = zr[i] = 0;
zs1[i] = zs2[i] = zt1[i] = zt2[i] = 0;
}
zone(&zs2[0]);
zzero(&zs1[0]);
zzero(&zt2[0]);
zone(&zt1[0]);
while (dh != 0 || zscompare(zh[0], 0l) != 0) {
zpoly_div(dg, dh, zg, zh, zq, zr, &dq, &dr);
zpoly_mod(p, zq, &dq);
zpoly_mod(p, zr, &dr);
zpoly_mul(dq, ds1, zq, zs1, za, &da);
zpoly_sub(ds2, da, zs2, za, zs, ds);
zpoly_mul(dq, dt1, zq, zt1, za, &da);
zpoly_sub(dt2, da, zt2, za, zt, dt);
zpoly_mod(p, zs, ds);
zpoly_mod(p, zt, dt);
zpoly_copy(dh, zh, zg, &dg);
zpoly_copy(dr, zr, zh, &dh);
zpoly_copy(ds1, zs1, zs2, &ds2);
zpoly_copy(*ds, zs, zs1, &ds1);
zpoly_copy(dt1, zt1, zt2, &dt2);
zpoly_copy(*dt, zt, zt1, &dt1);
#ifdef DEBUG
printf("q = "); zpoly_print(dq, zq);
printf("r = "); zpoly_print(dr, zr);
printf("s = "); zpoly_print(*ds, zs);
printf("t = "); zpoly_print(*dt, zt);
printf("g = "); zpoly_print(dg, zg);
printf("h = "); zpoly_print(dh, zh);
printf("s2 = "); zpoly_print(ds2, zs2);
printf("s1 = "); zpoly_print(ds1, zs1);
printf("t2 = "); zpoly_print(dt2, zt2);
printf("t1 = "); zpoly_print(dt1, zt1);
#endif
}
zpoly_copy(dg, zg, zd, dd);
zpoly_copy(ds2, zs2, zs, ds);
zpoly_copy(dt2, zt2, zt, dt);
for (i = 0; i < POLY_SIZE; i++) {
zfree(&za[i]);
zfree(&zb[i]);
zfree(&zq[i]);
zfree(&zr[i]);
zfree(&zs1[i]);
zfree(&zs2[i]);
zfree(&zt1[i]);
zfree(&zt2[i]);
}
}
int main(void)
{
long dd, dg, dh, ds, dt, i, p = 2;
verylong zg[POLY_SIZE], zh[POLY_SIZE];
verylong zd[POLY_SIZE], zs[POLY_SIZE];
verylong zt[POLY_SIZE];
for (i = 0; i < POLY_SIZE; i++)
zd[i] = zg[i] = zh[i] = zs[i] = zt[i] = 0;
dg = 10;
for (i = 0; i <= dg; i++) zzero(&zg[i]);
zintoz(1l, &zg[10]);
zintoz(1l, &zg[9]);
zintoz(1l, &zg[8]);
zintoz(1l, &zg[6]);
zintoz(1l, &zg[5]);
zintoz(1l, &zg[4]);
zintoz(1l, &zg[0]);
dh = 9;
for (i = 0; i <= dh; i++) zzero(&zh[i]);
zintoz(1l, &zh[9]);
zintoz(1l, &zh[6]);
zintoz(1l, &zh[5]);
zintoz(1l, &zh[3]);
zintoz(1l, &zh[2]);
zintoz(1l, &zh[0]);
zpoly_ext_euclid(dg, dh, p, zg, zh, zs, zt, zd, &ds, &dt, &dd);
printf("s = "); zpoly_print(ds, zs);
printf("t = "); zpoly_print(dt, zt);
printf("d = "); zpoly_print(dd, zd);
for (i = 0; i < POLY_SIZE; i++) {
zfree(&zd[i]);
zfree(&zg[i]);
zfree(&zh[i]);
zfree(&zs[i]);
zfree(&zt[i]);
}
return 0;
}
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