Four new classes of permutation trinomials and their compositional inverses

TL;DR

The paper addresses the problem of finding permutation polynomials with few terms over finite fields, focusing on the cubic extension ${\mathbb F}_{q^3}$ with even characteristic ($q=2^m$). It constructs four classes of permutation trinomials of the form $f_t(X)=X^d+L(X^s)$ over ${\mathbb F}_{q^3}$ and derives explicit compositional inverses using Dickson-matrix techniques for $q$-linearized components, along with a complete inverse for the XLXZT trinomial with $\alpha=\beta=1$ via the local method, revealing a new permutation quadrinomial inverse. The results include precise permutation conditions for each class (e.g., dependencies on $a,b\in{\mathbb F}_q^*$ and $m\bmod 3$) and explicit inverse formulas, enriching the catalog of invertible sparse PPs and their inverses in characteristic two. The QM-inequivalence analysis shows the new families are genuinely distinct from known classes, strengthening their novelty and potential utility in cryptography and coding theory.

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 $αX^{q(q^2 - q + 1)} + βX^{q^2 - q + 1} + 2X$ for $α= 1$ and $β= 1$, originally proposed by Xie, Li, Xu, Zeng, and Tang (2023).

Theorems & Definitions (31)

• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• Theorem 3.1
• proof
• Proposition 3.2
• proof
• Theorem 3.3