Mathematical preliminaries of crypto More Concepts of Number Theory

Theorem 1 (Chinese Remainder Theorem). If ni ⊥ nj for each 1 ≤ i < j ≤ k, then the system of congruences (1) has a unique solution modulo N = ∏k i=1 ni. The proof of the existence of the solution will also give us an algorithm for finding that solution. Proof. For each i ∈ {1, . . . , k} let mi = N/ni. The numbers mi are natural numbers, because ni was one of the factors of N . We have ni ⊥ mi, because (1) mi is the product of all nj, where i 6= j, and (2) ni ⊥ nj for all these nj . Let ri be the inverse of mi modulo ni. In other words, ri is such that rimi ≡ 1 (mod ni). Such an ri exists because mi is invertible in the ring Zni . We can use the extended Euclid’s algorithm to compute ri. Define x0 = k