# Mignotte's scheme

From CryptoWiki

Chinese remainder theorem provides us with a method to uniquely determine a number x modulo t-many relatively prime integers given that

A *(t,n)*-Mignotte sequence is a strictly increasing sequence of positive prime integers , such that .
We build a *(t,n)*-threshold secret sharing scheme as follows: We choose the secret *x* as a random integer in the range . We compute a share for every participant *i*: .

Now, for any *t* different shares, we consider the system of congruences:

By the Chinese remainder theorem, since are pairwise coprime, the system (1) has a unique solution. By the construction of our shares, this solution is nothing but the secret *x* to recover. [Mi83]