On the Classification of Dillon's APN Hexanomials
Daniele Bartoli, Giovanni Giuseppe Grimaldi, Pantelimon Stanica
TL;DR
This work analyzes Dillon's hexanomials over $\\mathbb{F}_{q^2}$, recasting the APN condition as a problem about algebraic varieties and Frobenius-fixed irreducible components. A central geometric theorem links the existence of a φ-fixed, absolutely irreducible component not lying in forbidden hyperplanes to non-APN for large $q$, while cases lying entirely in forbidden hyperplanes can preserve APN in specific degeneracies. The authors provide a thorough case split ( $B=0$ and $B\neq 0$ ) with precise algebraic obstructions, complemented by extensive computational searches that uncover both BC-equivalent and novel CCZ-inequivalent APN classes, though none are permutations. The results sharply narrow the APN search space in Dillon's class, reveal new APN instances at larger field sizes, and suggest a general, geometry-guided path for classifying APN candidates in other polynomial families.
Abstract
In this paper, we undertake a systematic analysis of a class of hexanomial functions over finite fields of characteristic 2 proposed by Dillon in 2006 as potential candidates for almost perfect nonlinear (APN) functions, pushing the analysis a lot further than what has been done via the partial APN concept in (Budaghyan et al., DCC 2020). These functions, defined over $\mathbb{F}_{q^2}$ where $q=2^n$, have the form $F(x) = x(Ax^2 + Bx^q + Cx^{2q}) + x^2(Dx^q + Ex^{2q}) + x^{3q}.$ Using algebraic number theory and methods on algebraic varieties over finite fields, we establish necessary conditions on the coefficients $A, B, C, D, E$ that must hold for the corresponding function to be APN. Our main contribution is a comprehensive case-by-case analysis that systematically excludes large classes of Dillon's hexanomials from being APN based on the vanishing patterns of certain key polynomials in the coefficients. Through a combination of number theory, algebraic-geometric techniques and computational verification, we identify specific algebraic obstructions-including the existence of absolutely irreducible components in associated varieties and degree incompatibilities in polynomial factorizations-that prevent these functions from achieving optimal differential uniformity. Our results significantly narrow the search space for new APN functions within this family and provide a theoretical roadmap applicable to other classes of potential APN functions. We complement our theoretical work with extensive computations. Through exhaustive searches on $\mathbb{F}_{2^2}$ and $\mathbb{F}_{2^4}$ and random sampling on $\mathbb{F}_{2^6}$ and $\mathbb{F}_{2^8}$, we identified thousands of APN hexanomials, many of which are not CCZ-equivalent to the known Budaghyan-Carlet family (Budaghyan-Carlet, IEEE Trans. Inf. Th., 2008).