A general approach to permutation polynomials from quadratic forms

TL;DR

This paper presents a unified method for characterizing permutation polynomials over finite fields of characteristic 2 by linking them to quadratic forms and analyzing related character sums.

Contribution

It introduces a general framework connecting permutation polynomials and quadratic forms, enabling explicit descriptions of new classes of permutation polynomials.

Findings

Characterization of permutation polynomials via quadratic forms

Explicit descriptions of several classes of permutation polynomials

Connection between permutation polynomials and character sums

Abstract

We investigate a family of permutation polynomials of finite fields of characteristic 2. Through a connection between permutation polynomials and quadratic forms, a general treatment is presented to characterize these permutation polynomials. By determining some character sums associated with quadratic forms, we explicitly describe several classes of permutation polynomials.