Permutation polynomials of the form (xq−x+δ)i(q−1)+1+L(x) over Fq2

Abstract

Let q be a power of a prime number and let $F_q$ be the finite field with q elements. Let $\delta \in F_{q^2}$ be arbitrary. In this paper, we give a relationship between the permutation property of polynomials over $F_{q^2}$ of the forms $g(x^q-x+\delta )+c{x}^q+dx$ and $g{\left(x\right)}^q-g\left(x\right)+c{x}^q+dx$ where $c,d\in F_q^{⁎}$, $g\left(x\right)\in F_{q^2}\left[x\right]$. Further, we find the necessary and sufficient conditions on the coefficients c and d such that polynomials of the forms ${(x^q-x+\delta )}^{i(q-1)+1}+cx$ and ${(x^q-x+\delta )}^{i(q-1)+1}+c{x}^q+dx$ permute $F_{q^2}$. Moreover, some results of this article supersede certain results in the related literature.