crt.c

/*  Author: Joe Reagle */
/*  Purpose: Finds the inverse of an equation using Denning's Chinese */
/*	    Remainder Theorem */
/* The following code is provided as is.
 *        It may be copied and modified freely provided that
 *Joseph M. Reagle Jr. is acknowledged within the code as the
 *originator.
*/


#include <stdio.h>
#define TRUE 1
#define FALSE 0
#define MAX_PRIME 36

int factor(int n, int d[])
   /* factor a number n, and put it's factors in an array d[] */
{
   int is_prime = FALSE;
   int counter = 0;
   int i = 0;
   int p_counter = 0;

   /* first set of primes my program attempts to use as factors 
      Kind of a cheat, but this makes it a little faster, if you
      must go ahead and compute them within the program ad infinitum*/

   int p_array[] = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,
 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137
, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 
227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311,
 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401,
 409,419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 
499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 
601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 
691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 
809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 
907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 
1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069,1087, 1091, 
1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 
1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 
1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373,1381, 
1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 
1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 
1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657,
1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 
1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871,
1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979,
1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 
2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 
2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 
2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 
2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 
2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 
2593, 2609, 2617,2621,  2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 
2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 
2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 
2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 
2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 
3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 
3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 
3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 
3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 
3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 
3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 
3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 
3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 
3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 
4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 
4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 
4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 
4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 
4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 
4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 
4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 
4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 
4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 
5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 
5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 
5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 
5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 
5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 
5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737,
5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 
5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939,
5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079,
6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 
6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 
6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 
6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 
6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 
6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763,
6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869,
6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977,
6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 
7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 
7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 
7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 
7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 
7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 
7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 
7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 
7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 
8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 
8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 
8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 
8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 
8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 
8677, 8681, 8689, 8693, 8697, 8893, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 
8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 
8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 
9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 
9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 
9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 
9343, 9349, 9371, 9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 
9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 
9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679, 
9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 
9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 
9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973 };

   while(!is_prime) {

      while (n%(p_array[p_counter]) != 0 ) {  /* get to a factor */
	 printf("%d | ", n);
	 printf("%d\n", p_array[p_counter]);
	 p_counter++;
      }
      if (p_array[p_counter]==n) {            /* if the factor is = n end */
	 is_prime = TRUE;
	 printf("End case\n");
	 d[i] = p_array[p_counter];
	 i;
      }
      else {                                  /* otherwise, store in array */
	 if (d[i-1] == p_array[p_counter]) {
	    printf("%d == %d\n", d[i-1], p_array[p_counter]);
	    d[i-1] += p_array[p_counter];     
	    printf("d[%d] is now = %d\n", i-1, d[i-1]); 
	 }  /* if it's the sam as a prev factor, add it to that value */
	 else {
	    d[i] = p_array[p_counter];
	    i++;
	 }
	 n = n/p_array[p_counter];            /* decrease n by the factor */
	 printf("n = %d\n", n);
      }
   }
   d[++i] = 0;           
   printf("Factors of are ");
   while (d[counter] != 0) {
      printf(" %d", d[counter]);
      counter++;
   }
   printf("\n");
   return i;
}

int inv(int a, int n)
   /* the function inv out of Denning's book */
{
   int x,y;
   int gnext; int glast = n; int g = a;
   int vnext; int vlast = 0; int v = 1;
   int i = 1;

   while (g != 0) {
      y = glast/g;
      gnext = glast - (y*g);
      vnext = vlast - (y*v);
      glast = g;
      vlast = v;
      g = gnext;
      v = vnext;
   }
   x = vlast;
   if (x >= 0)
      return x;
   else 
      return x + n;
}

void find_x(int a, int d[], int x[], int t)
   /* finds the appropriate x value for eac d[i] */
{
   int i = 0;

   while (i < t) {              /* while there is an x to find, find it */
      x[i] = (inv(a,d[i]));
      i++;
   }
}


int crt(int n, int b, int d[], int x[], int t)
   /* crt out of Denning's book */
{
   int i;
   int y[36];
   int solution = 0;

   for(i=0; i<t; i++) {
      y[i] = inv(n/d[i]%d[i], d[i]);
   }
    for(i=0; i<t; i++) {
       printf("solution = %d, n/d[i] = %d, y[i] = %d, x[i] = %d\n", 
	      solution, n/d[i], y[i], x[i]);
       solution = (solution + (n/d[i]*y[i]*x[i]))%n;
   }
   printf("internal solution = %d, b = %d, n = %d\n", solution, b, n);
   solution = (b*solution)%n;  /* added to Denning's method for b */
   return solution;
}
   


main ()
{
   int a;                /* _A_xmodn = b */
   int b;                /* axmodn = _B_ */
   int n = 1755317;      /* the modulus */
   int d[36];            /* the pairwise relative primes */
   int x[36];            /* the solutions */
   int t;                /* number of d's */
   int counter = 0;

   while (1) {
      t = counter = a = b = n = 0;
      while (counter < 36) {
	 d[counter] = 0;       /* iniatilize the counters */
	 x[counter] = 0;
	 counter++;
      }
   
      printf("Enter for axmodn=b: a n b\n");
      scanf("%d %d %d", &a, &n, &b);
      printf("b = %d\n", b);
      t = factor(n,d);   /* fills d[] with the primes */
      find_x(a,d,x,t);  /* finds solutions for x */
      printf("The Solution is %d\n", crt(n,b,d,x,t));
   }
}