Necessary and sufficient conditions of two classes of permutation polynomials

Abstract

In this paper, two types of permutation polynomials, (x2m+x+δ)s+cx and xr(xq−1+a), are studied. To formulate the necessary and sufficient conditions for the polynomials to be permutation polynomials over finite fields, the structures and properties of the field elements are analyzed. Meanwhile, the number of solutions to some equations over finite fields is investigated. For quadratic equations, the necessary and sufficient conditions are utilized thoroughly to study the permutation polynomials. But for cubic equations, only one direction is investigated by the existing literatures. This paper employs the properties of these equations in full strength to study (x2m+x+δ)s+cx over F23m⁎. For the binomials of the form xr(xq−1+a), two different methods are exploited to formulate the necessary and sufficient conditions over Fq3, and some partial results are obtained for Fqe.