/*
Author: Pate Williams (c) 1997
Exercise
"5.6 The field GF(2 ^ 5) can be constructed as
Z_2[x]/(x ^ 5 + x ^ 2 + 1). Perform the following
computations in this field.
(a) Compute (x ^ 4 + x ^ 2) * (x ^ 3 + x + 1).
(b) Using the Extended Euclidean algorithm
compute (x ^ 3 + x ^ 2) ^ - 1.
(c) Using the square and multiply algorithm,
compute x ^ 25." -Douglas R. Stinson-
See "Cryptography: Theory and Practice" by
Douglas R. Stinson page 201.
*/
#include <math.h>
#include <stdio.h>
#define SIZE 32l
void poly_mul(long m, long n, long *a, long *b, long *c, long *p)
{
long ai, bj, i, j, k, sum;
*p = m + n;
for (k = 0; k <= *p; k++) {
sum = 0;
for (i = 0; i <= k; i++) {
j = k - i;
if (i > m) ai = 0; else ai = a[i];
if (j > n) bj = 0; else bj = b[j];
sum += ai * bj;
}
c[k] = sum;
}
}
void poly_div(long m, long n, long *u, long *v,
long *q, long *r, long *p, long *s)
{
long j, jk, k, nk, vn = v[n];
for (j = 0; j <= m; j++) r[j] = u[j];
if (m < n) {
*p = 0, *s = m;
q[0] = 0;
}
else {
*p = m - n, *s = n - 1;
for (k = *p; k >= 0; k--) {
nk = n + k;
q[k] = r[nk] * pow(vn, k);
for (j = nk - 1; j >= 0; j--) {
jk = j - k;
if (jk >= 0)
r[j] = vn * r[j] - r[nk] * v[j - k];
else
r[j] = vn * r[j];
}
}
while (*p > 0 && q[*p] == 0) *p = *p - 1;
while (*s > 0 && r[*s] == 0) *s = *s - 1;
}
}
void poly_exp_mod(long degreeA, long degreem, long n,
long modulus, long *A, long *m, long *s,
long *ds)
{
int zero;
long dp, dq, dx = degreeA, i;
long p[SIZE], q[SIZE], x[SIZE];
*ds = 0, s[0] = 1;
for (i = 0; i <= dx; i++) x[i] = A[i];
while (n > 0) {
if ((n & 1) == 1) {
/* s = (s * x) % m; */
poly_mul(*ds, dx, s, x, p, &dp);
poly_div(dp, degreem, p, m, q, s, &dq, ds);
for (i = 0; i <= *ds; i++) s[i] %= modulus;
zero = s[*ds] == 0, i = *ds;
while (i > 0 && zero) {
if (zero) *ds = *ds - 1;
zero = s[--i] == 0;
}
}
n >>= 1;
/* x = (x * x) % m; */
poly_mul(dx, dx, x, x, p, &dp);
poly_div(dp, degreem, p, m, q, x, &dq, &dx);
for (i = 0; i <= dx; i++) x[i] %= modulus;
zero = x[dx] == 0, i = dx;
while (i > 0 && zero) {
if (zero) dx--;
zero = x[--i] == 0;
}
}
}
void poly_copy(long db, long *a, long *b, long *da)
/* a = b */
{
long i;
*da = db;
for (i = 0; i <= db; i++) a[i] = b[i];
}
int poly_Extended_Euclidean(long db, long dn,
long *b, long *n,
long *t, long *dt)
{
int nonzero;
long db0, dn0, dq, dr, dt0 = 0, dtemp, du, i;
long b0[SIZE], n0[SIZE], q[SIZE];
long r[SIZE], t0[SIZE], temp[SIZE], u[SIZE];
*dt = 0;
poly_copy(dn, n0, n, &dn0);
poly_copy(db, b0, b, &db0);
t0[0] = 0;
t[0] = 1;
poly_div(dn0, db0, n0, b0, q, r, &dq, &dr);
nonzero = r[0] != 0;
for (i = 1; !nonzero && i <= dr; i++)
nonzero = r[i] != 0;
while (nonzero) {
poly_mul(dq, *dt, q, t, u, &du);
if (dt0 < du)
for (i = dt0 + 1; i <= du; i++) t0[i] = 0;
for (i = 0; i <= du; i++)
temp[i] = t0[i] - u[i];
dtemp = du;
poly_copy(*dt, t0, t, &dt0);
poly_copy(dtemp, t, temp, dt);
poly_copy(db0, n0, b0, &dn0);
poly_copy(dr, b0, r, &db0);
poly_div(dn0, db0, n0, b0, q, r, &dq, &dr);
nonzero = r[0] != 0;
for (i = 1; !nonzero && i <= dr; i++)
nonzero = r[i] != 0;
}
if (db0 != 0 && b0[0] != 1) return 0;
return 1;
}
void poly_mod(long da, long p, long *a, long *new_da)
{
int zero;
long i;
for (i = 0; i <= da; i++) {
a[i] %= p;
if (a[i] < 0) a[i] += p;
}
zero = a[da] == 0;
for (i = da - 1; zero && i >= 0; i--) {
da--;
zero = a[i] == 0;
}
*new_da = da;
}
void poly_write(char *label, long da, long *a)
{
long i;
printf("%s", label);
for (i = da; i >= 0; i--)
printf("%ld ", a[i]);
printf("\n");
}
int main(void)
{
long da = 4, db = 3, dc = 3, dd = 5;
long de, dq, dr, ds, p = 2;
long a[5] = { 0, 0, 1, 0, 1 };
long b[4] = { 1, 1, 0, 1 };
long c[4] = { 0, 0, 1, 1 };
long d[6] = { 1, 0, 1, 0, 0, 1 };
long e[SIZE], q[SIZE], r[SIZE], s[SIZE];
poly_write("A = ", da, a);
poly_write("B = ", db, b);
poly_write("C = ", dc, c);
poly_write("D = ", dd, d);
poly_mul(da, db, a, b, e, &de);
poly_mod(de, p, e, &de);
poly_div(de, dd, e, d, q, r, &dq, &dr);
poly_mod(dq, p, q, &dq);
poly_mod(dr, p, r, &dr);
poly_write("A * B mod 2 = ", de, e);
poly_write("A * B / D mod 2 = ", dq, q);
poly_write("A * B mod D, 2 = ", dr, r);
if (!poly_Extended_Euclidean(dc, dd, c, d, e, &de))
printf("*error*\in poly_Extended_Euclidean\n");
poly_mod(de, p, e, &de);
poly_write("C ^ - 1 mod D = ", de, e);
poly_mul(dc, de, c, e, s, &ds);
poly_mod(ds, p, s, &ds);
poly_div(ds, dd, s, d, q, r, &dq, &dr);
poly_mod(dr, p, r, &dr);
poly_write("C * C ^ - 1 mod D, 2 = ", dr, r);
dq = 1;
q[0] = 0;
q[1] = 1;
poly_exp_mod(dq, dd, 7, p, q, d, s, &ds);
poly_mod(ds, p, s, &ds);
poly_write("x ^ 7 mod D, 2 = ", ds, s);
poly_exp_mod(dq, dd, 9, p, q, d, s, &ds);
poly_mod(ds, p, s, &ds);
poly_write("x ^ 9 mod D, 2 = ", ds, s);
poly_exp_mod(dq, dd, 10, p, q, d, s, &ds);
poly_mod(ds, p, s, &ds);
poly_write("x ^ 10 mod D, 2 = ", ds, s);
poly_exp_mod(dq, dd, 25, p, q, d, s, &ds);
poly_mod(ds, p, s, &ds);
poly_write("x ^ 25 mod D, 2 = ", ds, s);
return 0;
}
Numerical Explorations 