Four new classes of permutation trinomials and their compositional inverses

Abstract: We construct four new classes of permutation trinomials over the cubic extension of a finite field with even characteristic. Additionally, we explicitly provide the compositional inverse of each class of permutation trinomials in polynomial form. Furthermore, we derive the compositional inverse of the permutation trinomial $\alpha X^{q(q^2 - q + 1)} + \beta X^{q^2 - q + 1} + 2X$ for $\alpha = 1$ and $\beta = 1$, originally proposed by Xie, Li, Xu, Zeng, and Tang (2023).