Efficient evaluation of polynomials over finite fields

TL;DR

This paper introduces an efficient method for evaluating polynomials over finite fields, significantly reducing computational complexity for high-degree polynomials, with applications in decoding cyclic and Reed-Solomon codes.

Contribution

It presents a novel evaluation technique that outperforms standard methods for large-degree polynomials over finite fields.

Findings

Reduced complexity compared to standard techniques

Improved decoding efficiency for cyclic and Reed-Solomon codes

Applicable to polynomial evaluation in extended finite fields

Abstract

A method is described which allows to evaluate efficiently a polynomial in a (possibly trivial) extension of the finite field of its coefficients. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large with respect to the base field. Applications to the syndrome computation in the decoding of cyclic codes, Reed-Solomon codes in particular, are highlighted.