//--------------------------------------------------------------
// Recovering secret by some shades
// for theoretical GO TO WIKIPEDIA, reference - shamir secret sharing, lagrange interpolation
//--------------------------------------------------------------
private BigInteger deShamir(Dictionary <BigInteger, BigInteger> kv_pairs, BigInteger p)
{
int k = kv_pairs.Count;
BigInteger secret = new BigInteger(0);
BigInteger high_part = new BigInteger(1); // high part of lagrange intepolation
BigInteger low_part = new BigInteger(1); // low part of lagrange interpolation
// first_kv - shade number 1
// second_kv - shade number 2
foreach (KeyValuePair <BigInteger, BigInteger> first_kv in kv_pairs)
{
foreach (KeyValuePair <BigInteger, BigInteger> second_kv in kv_pairs)
{
if (first_kv.Key != second_kv.Key)
{
high_part = high_part * ((-1) * second_kv.Key); // -1 caused that we need x - xj, but our x = 0, that now (-xj)
low_part = low_part * (first_kv.Key - second_kv.Key); // calculate low part
}
}
// Lagrange interpolation have division, but we need only integer numbers, so
// a * a^(-1) = 1
// but we can make some fun and:
// a * a^(-1) = 1 (mod p)
// so, when we have modulo, we can change a^(-1) to another number, that give us similar result ^.^
// a * b = 1 (mod p).
// b can be calculated by Evklid algorithm in form 'ax + by = gcd(a,b)' (*1)
// so we set 2 numbers, a = low_part and b = p, and gets gcd and 2 numbers, x and y.
// We know that gcd(prime_number, any_number) = 1, so change (*1) to our rules:
// low_part * x + p*y = 1
// In this form y always be the '0' (ave MATH!), so we get
// low_part*x = 1
// where x - our (low_part)^(-1)
// MAGIIIIIIIIIIC!
iVector temp = Evklid(low_part, p);
low_part = MathMod(temp.y, p);
high_part = MathMod((high_part * low_part), p); // let high part temporary storage all lart of 'li' (see lagrange interpolation)
secret += high_part * first_kv.Value; // summ_all( y * li )
high_part = 1;
low_part = 1;
}
secret = MathMod(secret, p); // Let restrict out sercet by out square.
return(secret); // Done. You are delicious ^_^
}
public iVector Evklid(BigInteger a, BigInteger b)
{
iVector u = new iVector(a, 1, 0);
iVector v = new iVector(b, 0, 1);
iVector T = new iVector(0, 0, 0);
BigInteger q = 0;
while (v.x != 0)
{
q = u.x / v.x;
T.x = MathMod(u.x, v.x);
T.y = u.y - q * v.y;
T.z = u.z - q * v.z;
u.set(v); // u = v, but we need to CHANGE value, not CHANGE POINTER.
v.set(T); // u = v, but we need to CHANGE value, not CHANGE POINTER.
}
return(u);
}
public void set(iVector sec)
{
this.x = sec.x;
this.y = sec.y;
this.z = sec.z;
}