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

Daniele Bartoli, Mohit Pal, Pantelimon Stanica, Tommaso Toccotelli

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.

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

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 , their function is not a permutation over for any , resolving the permutation part of their conjecture for sufficiently large fields.

Paper Structure

This paper contains 9 sections, 10 theorems, 52 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and algebraic geometry tools
  3. The general trivariate family
  4. Necessary conditions for non-permutation
  5. Complete characterization for $m=3$
  6. Role of computations.
  7. Resolution of a conjecture by Beierle et al. MR4414815
  8. Conclusion and open problems
  9. Polynomial definitions

Key Result

Theorem 2.1

Let $\mathcal{V} \subseteq \mathbb{A}^n(\overline{\mathbb{F}_q})$ be an absolutely irreducible $\mathbb{F}_q$-rational variety of dimension $r > 0$ and degree $\delta$. If $q > 2(r + 1)\delta^2$, then

Theorems & Definitions (21)

  • Theorem 2.1: Strengthened Lang-Weil MR2206396
  • Lemma 2.2: MR2648536
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • ...and 11 more