Non-permutation phenomena in trivariate families over $\F_{2^m}$ and resolution of a conjecture

TL;DR

This paper investigates when certain trivariate functions over finite fields fail to be permutations, providing algebraic conditions, computational classifications, and resolving a conjecture for large fields.

Contribution

It establishes necessary conditions for non-permutation, classifies small cases computationally, and proves the permutation failure for large fields, resolving a key conjecture.

Findings

Necessary conditions for non-permutation functions

Complete classification for small field extensions

Proved non-permutation for all large odd characteristic-2 fields

Abstract

Constructing permutation polynomials over finite fields, particularly those with simple algebraic structure in multiple variables, is a fundamental problem with applications in cryptography and coding theory. Recently, Li and Kaleyski (IEEE Trans. Inf. Theory, 2024) generalized two sporadic quadratic APN permutations into infinite families of trivariate functions. Motivated by their work, we investigate conditions under which generalized trivariate functions fail to be permutations. We establish necessary conditions on coefficient parameters that prevent the permutation property, provide a complete computational classification for small field extensions, and prove general non-permutation results. As a key application of our algebraic geometry approach, we resolve the permutation part of a conjecture by Beierle, Carlet, Leander, and Perrin (Finite Fields Appl., 2022) regarding a related trivariate form. Specifically, we prove that for all odd characteristic-2 extension degrees $m \geq 23$, their function $C_u$ is not a permutation over $\mathbb{F}_{2^m}^3$ for any $u \in \mathbb{F}_{2^m}^*$, resolving the permutation part of their conjecture for sufficiently large fields.