The compositional inverses of linearized permutation binomials over finite fields

Abstract

Let $q$ be a prime power and $n$ and $r$ be positive integers. It is well known that the linearized binomial $L_r(x)=x^{q^r}+ax\in\mathbb{F}_{q^n}[x]$ is a permutation polynomial if and only if $(-1)^{n/d}a^{{(q^n-1)}/{(q^{d}-1)}}\neq 1$ where $d=(n,r)$. In this paper, the compositional inverse of $L_r(x)$ is explicitly determined when this condition holds.