The compositional inverses of the permutation polynomials of the form $x+γ\operatorname{Tr}_{q}^{q^n}(H(x))$ over $\mathbb{F}_{q^n}$

Abstract

This paper focuses on computing the compositional inverses of permutation polynomials of the form $x+γ\operatorname{Tr}_{q}^{q^{n}}(H(x))$ over finite fields via the local method. We explicitly construct compositional inverses for four families of permutation polynomials of this type over $\mathbb{F}_{q^2}$, another four families over $\mathbb{F}_{q^3}$, and one general family over $\mathbb{F}_{q^n}$. The closed-form inverse expressions derived in this work supplement the theory of trace permutation polynomials.