On Polynomials of the Formxrf(x(q−1)/l)

Amir Akbary,Qiang Wang

doi:10.1155/2007/23408

Abstract

We give a general criterion for permutation polynomials of the formxrf(x(q−1)/l), wherer≥1,l≥1andl∣(q−1). We employ this criterion to characterize several classes of permutation polynomials.

Highlights

We give a general criterion for permutation polynomials of the form xr f (x(q−1)/l), where r ≥ 1, l ≥ 1 and l | (q − 1)
A polynomial is a permutation polynomial (PP) of Fq if it induces a bijective map from Fq to Fq
There has been a lot of interests in studying permutation polynomials, partly due to their applications in coding theory, combinatorics, and cryptography

Summary

Introduction

Due to the importance of the polynomials of the form xr f (x(q−1)/l), it is interesting to give criteria for PPs of this type. One such criterion was given by Wan and Lidl [5, Theorem 1.2]. We give another general criterion (Theorem 2.2) for PPs of the form P(x) = xr f (x(q−1)/l). By applying our theorem, we construct some new classes of permutation polynomials, and give simplified proofs for some known classes of permutation polynomials.

A general criterion

First application

Second application