text: /*
   * 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>
  
  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");
      if (power > n) n = power;
    }
  
    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);
  }
  