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]