gf_poly.c 8.54 KB
``````/*
* GF-Complete: A Comprehensive Open Source Library for Galois Field Arithmetic
* James S. Plank, Ethan L. Miller, Kevin M. Greenan,
* Benjamin A. Arnold, John A. Burnum, Adam W. Disney, Allen C. McBride.
*
* gf_poly.c - program to help find irreducible polynomials in composite fields,
* using the Ben-Or algorithm.
*
* (This one was written by Jim)
*
* Please see the following paper for a description of the Ben-Or algorithm:
*
* author    S. Gao and D. Panario
* title     Tests and Constructions of Irreducible Polynomials over Finite Fields
* booktitle Foundations of Computational Mathematics
* year      1997
* publisher Springer Verlag
* pages     346-361
*
* The basic technique is this.  You have a polynomial f(x) whose coefficients are
* in a base field GF(2^w).  The polynomial is of degree n.  You need to do the
* following for all i from 1 to n/2:
*
* Construct x^(2^w)^i modulo f.  That will be a polynomial of maximum degree n-1
* with coefficients in GF(2^w).  You construct that polynomial by starting with x
* and doubling it w times, each time taking the result modulo f.  Then you
* multiply that by itself i times, again each time taking the result modulo f.
*
* When you're done, you need to "subtract" x -- since addition = subtraction =
* XOR, that means XOR x.
*
* Now, find the GCD of that last polynomial and f, using Euclid's algorithm.  If
* the GCD is not one, then f is reducible.  If it is not reducible for each of
* those i, then it is irreducible.
*
* In this code, I am using a gf_general_t to represent elements of GF(2^w).  This
* is so that I can use base fields that are GF(2^64) or GF(2^128).
*
* I have two main procedures.  The first is x_to_q_to_i_minus_x, which calculates
* x^(2^w)^i - x, putting the result into a gf_general_t * called retval.
*
* The second is gcd_one, which takes a polynomial of degree n and a second one
* of degree n-1, and uses Euclid's algorithm to decide if their GCD == 1.
*
* These can be made faster (e.g. calculate x^(2^w) once and store it).
*/

#include "gf_complete.h"
#include "gf_method.h"
#include "gf_general.h"
#include "gf_int.h"
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <assert.h>

char *BM = "Bad Method: ";

void usage(char *s)
{
fprintf(stderr, "usage: gf_poly w(base-field) method power:coef [ power:coef .. ]\n");
fprintf(stderr, "\n");
fprintf(stderr, "       use - for the default method.\n");
fprintf(stderr, "       use 0x in front of the coefficient if it's in hex\n");
fprintf(stderr, "       \n");
fprintf(stderr, "       For example, to test whether x^2 + 2x + 1 is irreducible\n");
fprintf(stderr, "       in GF(2^16), the call is:\n");
fprintf(stderr, "       \n");
fprintf(stderr, "       gf_poly 16 - 2:1 1:2 0:1\n");
fprintf(stderr, "       \n");
if (s != NULL) {
fprintf(stderr, "\n");
if (s == BM) {
fprintf(stderr, "%s", s);
gf_error();
} else {
fprintf(stderr, "%s\n", s);
}
}
exit(1);
}

int gcd_one(gf_t *gf, int w, int n, gf_general_t *poly, gf_general_t *prod)
{
gf_general_t *a, *b, zero, factor, p;
int i, j, da, db;

gf_general_set_zero(&zero, w);

a = (gf_general_t *) malloc(sizeof(gf_general_t) * n+1);
b = (gf_general_t *) malloc(sizeof(gf_general_t) * n);
for (i = 0; i <= n; i++) gf_general_add(gf, &zero, poly+i, a+i);
for (i = 0; i < n; i++) gf_general_add(gf, &zero, prod+i, b+i);

da = n;
while (1) {
for (db = n-1; db >= 0 && gf_general_is_zero(b+db, w); db--) ;
if (db < 0) return 0;
if (db == 0) return 1;
for (j = da; j >= db; j--) {
if (!gf_general_is_zero(a+j, w)) {
gf_general_divide(gf, a+j, b+db, &factor);
for (i = 0; i <= db; i++) {
gf_general_multiply(gf, b+i, &factor, &p);
}
}
}
for (i = 0; i < n; i++) {
}
}

}

void x_to_q_to_i_minus_x(gf_t *gf, int w, int n, gf_general_t *poly, int logq, int i, gf_general_t *retval)
{
gf_general_t x;
gf_general_t *x_to_q;
gf_general_t *product;
gf_general_t p, zero, factor;
int j, k, lq;

gf_general_set_zero(&zero, w);
product = (gf_general_t *) malloc(sizeof(gf_general_t) * n*2);
x_to_q = (gf_general_t *) malloc(sizeof(gf_general_t) * n);
for (j = 0; j < n; j++) gf_general_set_zero(x_to_q+j, w);
gf_general_set_one(x_to_q+1, w);

for (lq = 0; lq < logq; lq++) {
for (j = 0; j < n*2; j++) gf_general_set_zero(product+j, w);
for (j = 0; j < n; j++) {
for (k = 0; k < n; k++) {
gf_general_multiply(gf, x_to_q+j, x_to_q+k, &p);
}
}
for (j = n*2-1; j >= n; j--) {
if (!gf_general_is_zero(product+j, w)) {
for (k = 0; k <= n; k++) {
gf_general_multiply(gf, poly+k, &factor, &p);
}
}
}
for (j = 0; j < n; j++) gf_general_add(gf, product+j, &zero, x_to_q+j);
}
for (j = 0; j < n; j++) gf_general_set_zero(retval+j, w);
gf_general_set_one(retval, w);

while (i > 0) {
for (j = 0; j < n*2; j++) gf_general_set_zero(product+j, w);
for (j = 0; j < n; j++) {
for (k = 0; k < n; k++) {
gf_general_multiply(gf, x_to_q+j, retval+k, &p);
}
}
for (j = n*2-1; j >= n; j--) {
if (!gf_general_is_zero(product+j, w)) {
for (k = 0; k <= n; k++) {
gf_general_multiply(gf, poly+k, &factor, &p);
}
}
}
for (j = 0; j < n; j++) gf_general_add(gf, product+j, &zero, retval+j);
i--;
}

gf_general_set_one(&x, w);

free(product);
free(x_to_q);
}

int main(int argc, char **argv)
{
int w, i, power, n, ap, success;
gf_t gf;
gf_general_t *poly, *prod;
char *string, *ptr;
char buf[100];

if (argc < 4) usage(NULL);

if (sscanf(argv[1], "%d", &w) != 1 || w <= 0) usage("Bad w.");
ap = create_gf_from_argv(&gf, w, argc, argv, 2);

if (ap == 0) usage(BM);

if (ap == argc) usage("No powers/coefficients given.");

n = -1;
for (i = ap; i < argc; i++) {
if (strchr(argv[i], ':') == NULL || sscanf(argv[i], "%d:", &power) != 1) {
string = (char *) malloc(sizeof(char)*(strlen(argv[i]+100)));
sprintf(string, "Argument '%s' not in proper format of power:coefficient\n", argv[i]);
usage(string);
}
if (power < 0) {
usage("Can't have negative powers\n");
} else {
n = power;
}
}
// in case the for-loop header fails
assert (n >= 0);

poly = (gf_general_t *) malloc(sizeof(gf_general_t)*(n+1));
for (i = 0; i <= n; i++) gf_general_set_zero(poly+i, w);
prod = (gf_general_t *) malloc(sizeof(gf_general_t)*n);

for (i = ap; i < argc; i++) {
sscanf(argv[i], "%d:", &power);
ptr = strchr(argv[i], ':');
ptr++;
if (strncmp(ptr, "0x", 2) == 0) {
success = gf_general_s_to_val(poly+power, w, ptr+2, 1);
} else {
success = gf_general_s_to_val(poly+power, w, ptr, 0);
}
if (success == 0) {
string = (char *) malloc(sizeof(char)*(strlen(argv[i]+100)));
sprintf(string, "Argument '%s' not in proper format of power:coefficient\n", argv[i]);
usage(string);
}
}

printf("Poly:");
for (power = n; power >= 0; power--) {
if (!gf_general_is_zero(poly+power, w)) {
printf("%s", (power == n) ? " " : " + ");
if (!gf_general_is_one(poly+power, w)) {
gf_general_val_to_s(poly+power, w, buf, 1);
if (n > 0) {
printf("(0x%s)", buf);
} else {
printf("0x%s", buf);
}
}
if (power == 0) {
if (gf_general_is_one(poly+power, w)) printf("1");
} else if (power == 1) {
printf("x");
} else {
printf("x^%d", power);
}
}
}
printf("\n");

if (!gf_general_is_one(poly+n, w)) {
printf("\n");
printf("Can't do Ben-Or, because the polynomial is not monic.\n");
exit(0);
}

for (i = 1; i <= n/2; i++) {
x_to_q_to_i_minus_x(&gf, w, n, poly, w, i, prod);
if (!gcd_one(&gf, w, n, poly, prod)) {
printf("Reducible.\n");
exit(0);
}
}

printf("Irreducible.\n");
exit(0);
}
``````