Research On Permutation Polynomials Based On The AGW Criterion

Permutation polynomials over finite fields usually have a wide application in combinatorial theory,cryptography,coding theory and so on.For example,the permutation function with low difference uniformity and high non-linearity is often used in cryptographic algorithm design.Constructing new permutation polynomials has always been a hot spot in cryptography.Hermite Criterion and Dickson polynomial have ever been the main tools of construction of permutation polynomials.Nowadays,trace functions,linearized polynomials,APN functions and Kloosterman sums are important tools to construct permutation polynomials.Especially,the AGW criterion,proposed by Akbary,Ghioca and Wang,has become a powerful tool for constructing permutation polynomials.In the paper we discuss the constructions of several classes of permutation polynomials based on the AGW criterion.Many scholars pay attention to permutation polynomials of the form(xpk-x +δ)s + L(x)over Fq,which are connected with trace functions when the characteristic of Fq is 2.Based on the AGW criterion,we get a necessary and sufficient condition for a polynomial of the form f(x)= x +(Trmn(x)k+δ)sover F22m to be permutation polynomial,which transforms the question to judge whether a polynomial f(x)is a permutation polynomial over F22m.or not,to the question to judge whether another polynomial over F2m is bijective or not.Next,based on the necessary and sufficient condition,we characterize the equivalent conditions for two classes of polynomials to be permutation polynomials over F22m.Finally,by using the necessary and sufficient condition,we construct three classes of new permutation polynomials when m is odd,and we generalize the permutation polynomials of the form f(x)= x +(Trmn(x)k+δ)s to these of the form f(x)= x+(Trmn(x)k+δ)S1 +(Trmn(x)+ δ)s2.