Binomial Permutations Over Finite Fields With Even Characteristic

Permutation polynomial over the finite field play an important role not only in the design of the S-box applied in the advanced encryption standard AES,but also in the key exchange protocol and the construction of cryptographic functions with low-differential uniformity.Moreover,they are widely used in the research field of number theory combinatorial design,such as the orthogonal Latin square and the new type of skew Hadamard difference.In recent years,a variety of ways to constructing permu-tation polynomials over finite fields have been proposed,su ch as AGW criterion and switch construction over finite fields,which makes great progress in the research of permutation polynomials over finite fields.Permutation polynomials with a small number of terms on finite fields have very simple algebraic expressions,but how to construct such permutation polynomials is still a very difficult problem.The permutation characteristics of monomials are easy to characterize,but for some binomials,the permutation characteristics have not been fully explored.Based on this,this article firstly discusses the relationship between the zero roots and the permutation properties of the binomial f(x)over a finite field IFq2.Secondly,we consider the binomial f(x)?Fq2 of the form xr(a+x3(q-1))with g=2m.By employing the Hermite criterion,all the exponents r and the coefficients a such that f(x)permute Fq2 are determined.Finally,when m is even or m is odd,the necessary and sufficient conditions for r and a for which f(x)is a permutation binomial on the finite field Fq 2 are given,respectively.