On the Classification of Dillon's APN Hexanomials

2025-11-02

AI Summary

This paper investigates the APN (Almost Perfect Nonlinear) property of six-term polynomial functions over finite fields of characteristic two, as introduced by Dillon (2006). Method: We systematically derive algebraic necessary conditions for APNness in terms of the coefficients (A,B,C,D,E), and construct a classification framework based on the zero-set distribution and irreducible component structure of associated polynomials. Key algebraic obstructions—such as degree mismatches—are identified to drastically reduce the search space. Our approach integrates tools from algebraic number theory, the theory of algebraic varieties over finite fields, and polynomial factorization, complemented by exhaustive enumeration over small fields and randomized sampling over larger ones. Contribution/Results: We discover thousands of new APN hexanomials across multiple fields; many are provably CCZ-inequivalent to the known Budaghyan–Carlet family. This work provides the first systematic algebraic criteria and constructive methodology for this class of hexanomials, advancing the theoretical understanding and construction of APN functions.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

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).

Problem

• Analyzing Dillon's hexanomial functions for APN properties over finite fields
• Establishing necessary coefficient conditions for optimal differential uniformity
• Identifying algebraic obstructions that prevent achieving APN characteristics

Innovation

• Used algebraic number theory for APN analysis
• Applied algebraic-geometric techniques on varieties
• Conducted computational verification via exhaustive searches