On the inverses of permutation polynomials of the form $h(\psi (x))\phi (x)+g(\psi (x))$ over finite fields

Abstract

In this paper, we investigate the compositional inverses of permutation polynomials of the form \[ F(x)=h(ψ(x))φ(x)+g(ψ(x)) \in \mathbb{F}_{q^n}[x], \] where \(ψ(x),φ(x) \in \mathbb{F}_{q^n}[x]\) are additive polynomials, \(h(x), g(x) \in \mathbb{F}_{q^n}[x]\) satisfy $h(\psi ({\mathbb{F}}_{q^n}))\subseteq {\mathbb{F}}_q^{\ast },$ and there exists a polynomial \(\barψ(x) \in \mathbb{F}_{q^n}[x]\) such that $\bar{\psi }(F(x))=\psi (x).$