Simple and fast Algorithm for Finding Roots of Error-Locator Polynomials: Modulus Search

Abstract

A novel very simple method for finding roots of polynomials over finite fields has been proposed. The essence of the proposed method is to search the roots via nested cycles over the subgroups of the multiplicative group of the Galois field. The modified Chien search is actually used in the inner cycles, but the internal polynomials are small. The word "modulus" was used because the search is doing on subsets like $α^{a+bi}$, where a,b=const. In addition, modulo division of polynomials is actively used. The algorithm is applicable not for all Galois fields, but for selective ones, starting from GF($2^8$). The algorithm has an advantage for large polynomials. The number of operations is significant for small polynomials, but it grows very slowly with the degree of the polynomial. When the polynomial is large or very large, the proposed method can be 10-100 times faster than Chien search.