Permutation Polynomials Over Finite Fields

Let Fqn be a finite field of order qn, where q is a power of a prime number p and n is a positive integer. As well known, any mapping from a finite field into itself is given by a polynomial. A polynomia1f(x) Fq[x] is called a permutation polynomial over Fq if it induces a bijective map from Fq to itself.The paper is dedicated to the constructions of permutation polynomials over finite fields.Firstly, we give an introduction about the development and applications of permutation polynomials, and give a summary of the judgement methods for permutation polynomials including the following criterion(see Lidl’ Finite Fields):Theorem1(Hermite’ criterion) Let Fq be of characteristic p. Then f(x)∈Fq[x] is a permutation polynomial of Fq if and only if the following two conditions hold:(1) f(x) has exactly one root in Fq;(2) For each integer t, with1≤t≤q—2and t(?)0(mod p), the reduction of (f(x))t mod x9—x has degree≤q-2.Secondly, in this paper, we construct several specific types of permutation polynomials over finite fields, for example, we get the following result:Theorem2Let m, e be positive integers, p is a prime and q=pe. Let where T(x) is a surjective mapping from Fqm to with T(ax+y)=aT(x)+T(y) Then F(x)is a permutation polynomial of Fqm if and only if the following condi-tions hold:(1)f(x)=L(x)+xh(x)is a permutation polynomial of Fq;(2)For any y∈Fq,x∈Fqm satisfies L(x)+xh(y)=0and T(x)=0if and only if x=0.