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-/*
- * 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");
- fprintf(stderr, " See the user's manual for more information.\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);
- gf_general_add(gf, &p, a+(i+j-db), a+(i+j-db));
- }
- }
- }
- for (i = 0; i < n; i++) {
- gf_general_add(gf, a+i, &zero, &p);
- gf_general_add(gf, b+i, &zero, a+i);
- gf_general_add(gf, &p, &zero, b+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);
- gf_general_add(gf, product+(j+k), &p, product+(j+k));
- }
- }
- for (j = n*2-1; j >= n; j--) {
- if (!gf_general_is_zero(product+j, w)) {
- gf_general_add(gf, product+j, &zero, &factor);
- for (k = 0; k <= n; k++) {
- gf_general_multiply(gf, poly+k, &factor, &p);
- gf_general_add(gf, product+(j-n+k), &p, product+(j-n+k));
- }
- }
- }
- 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);
- gf_general_add(gf, product+(j+k), &p, product+(j+k));
- }
- }
- for (j = n*2-1; j >= n; j--) {
- if (!gf_general_is_zero(product+j, w)) {
- gf_general_add(gf, product+j, &zero, &factor);
- for (k = 0; k <= n; k++) {
- gf_general_multiply(gf, poly+k, &factor, &p);
- gf_general_add(gf, product+(j-n+k), &p, product+(j-n+k));
- }
- }
- }
- for (j = 0; j < n; j++) gf_general_add(gf, product+j, &zero, retval+j);
- i--;
- }
-
- gf_general_set_one(&x, w);
- gf_general_add(gf, &x, retval+1, retval+1);
-
- 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);
-}
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