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Shamir Threshold: C (version 2)
/* shamir_threshold_scheme.c
*
* My C implementation of Shamir's (k, n) Theshold scheme.
* Uses a single flag as opposed to an array; idea from Razvan's great implementation.
* GRE, 6/23/11
*/
#include <stdio.h>
#include <stdlib.h>
#define uint unsigned int
#define ulong unsigned long
#define ushort unsigned short
// point struct to hold (x, y) values
typedef struct {
ulong x, y;
} point;
// Modular exponentiation, via Wikipedia and Schneier's "Applied Cryptography"
ulong expt_mod(ulong b, ulong e, ulong m){
ulong r = 1;
while (e) {
if (e & 1) {
r = (r * b) % m;
}
e >>= 1;
b = (b * b) % m;
}
return r;
}
// Modular inverse; x^{phi(p) - 1} = 1 mod p, and p prime => phi(p) = p - 1
ulong mod_inv(ulong x, ulong p){
return expt_mod(x, p - 2, p);
}
// Horner's method to evaluate polynomial given by coeffs[] modulo mod
ulong horner_mod(ulong x, ulong coeffs[], ulong k, ulong mod){
uint i;
ulong y = 0;
for (i = k - 1; i > 0; i--) {
y = (y + coeffs[i]) % mod;
y = (y * x) % mod;
}
y = (y + coeffs[0]) % mod;
return y;
}
// Shamir (k, n) threshold scheme
void shamir_threshold(ulong s, ulong k, ulong n, ulong p, point xy_pairs[]){
uint i, j;
ushort unique;
ulong x, coeffs[k];
coeffs[0] = s;
for (i = 1; i < k; i++) {
coeffs[i] = (ulong)(1 + (rand() % (p - 2)));
}
for (i = 0; i < n; i++) {
unique = 0;
while (!unique) {
unique = 1;
x = (ulong)(1 + (rand() % (p - 2)));
for (j = 0; j < i; j++) {
if (xy_pairs[j].x == x) {
unique = 0;
break;
}
}
}
xy_pairs[i].x = x;
xy_pairs[i].y = horner_mod(x, coeffs, k, p);
}
}
// Lagrange interpolation to recover the constant term
ulong interp_const(point xy_pairs[], ulong k, ulong p){
uint i, j;
ulong c;
ulong s = 0;
for (i = 0; i < k; i++) {
c = 1;
for (j = 0; j < k; j++) {
if (xy_pairs[j].x < xy_pairs[i].x) {
c = (c * xy_pairs[j].x * mod_inv(p - (xy_pairs[i].x -
xy_pairs[j].x), p)) % p;
}
else if (xy_pairs[j].x > xy_pairs[i].x) {
c = (c * xy_pairs[j].x * mod_inv(xy_pairs[j].x - xy_pairs[i].x,
p)) % p;
}
else {
continue;
}
}
s = (s + xy_pairs[i].y * c) % p;
}
return s;
}
// The main event (fingers crossed!)
int main(int argc, char *argv[]){
uint i;
ulong s = 17;
ulong n = 20;
ulong k = 5;
ulong p = 23;
point xy_pairs[n];
if (argc > 1) {
srand((uint)atoi(argv[1]));
}
printf("%lu\n", s);
shamir_threshold(s, k, n, p, xy_pairs);
for (i = 0; i < n; i++) {
printf("%lu\t%lu\n", xy_pairs[i].x, xy_pairs[i].y);
}
printf("%lu\n", interp_const(xy_pairs, k, p));
return 0;
}
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