We systematically analyze a class of hexanomial functions over finite fields of characteristic $2$ proposed by Dillon (2006) as candidates for almost perfect nonlinear (APN) functions, significantly extending earlier partial-APN results. For functions over $\mathbb{F}_{q^2}$, where $q=2^n$, of the form \[ F(x)=x(Ax^2+Bx^q+Cx^{2q})+x^2(Dx^q+Ex^{2q})+x^{3q}, \] we derive necessary conditions on the coefficients $A,B,C,D,E$ for APNness using algebraic number theory and algebraic-geometry methods over finite fields. Our main contribution is a comprehensive case-by-case analysis that excludes large classes of Dillon hexanomials via vanishing patterns of key coefficient polynomials. We identify algebraic obstructions -- including absolutely irreducible components of associated varieties and degree incompatibilities in polynomial factorizations -- that prevent these functions from attaining optimal differential uniformity. These results substantially narrow the search space for new APN functions in this family and provide a framework applicable to other APN candidates. We complement the theory with extensive computations: exhaustive searches over $\mathbb{F}_{2^2}$ and $\mathbb{F}_{2^4}$, and random sampling over $\mathbb{F}_{2^6}$ and $\mathbb{F}_{2^8}$, yielding hundreds of APN hexanomials. Complete CCZ-equivalence testing shows that, although many examples occur, they fall into few distinct classes. For $q\in\{2,4\}$, all examples are CCZ-equivalent to the Budaghyan--Carlet family, while in larger dimensions none appear equivalent to that family.

翻译：我们系统分析了Dillon（2006）提出的特征为$2$的有限域上一类六项式函数作为几乎完美非线性（APN）函数的候选，显著扩展了早期部分APN的结果。对于$\mathbb{F}_{q^2}$（其中$q=2^n$）上形如\[ F(x)=x(Ax^2+Bx^q+Cx^{2q})+x^2(Dx^q+Ex^{2q})+x^{3q} \]的函数，我们利用有限域上的代数数论与代数几何方法，推导出系数$A,B,C,D,E$满足APN性质的必要条件。我们的主要贡献是通过关键系数多项式的零点模式进行全面的案例分析，排除了Dillon六项式中的大类函数。我们识别了阻碍这些函数达到最优差分均匀性的代数障碍——包括相关簇的绝对不可约分量和多项式分解中的次数不兼容性。这些结果显著缩小了该族中新APN函数的搜索空间，并提供了适用于其他APN候选函数的分析框架。我们通过大量计算补充理论结果：在$\mathbb{F}_{2^2}$和$\mathbb{F}_{2^4}$上的穷举搜索，以及在$\mathbb{F}_{2^6}$和$\mathbb{F}_{2^8}$上的随机抽样，获得了数百个APN六项式实例。完整的CCZ等价性测试表明，尽管出现大量实例，它们仅属于少数不同等价类。对于$q\in\{2,4\}$，所有实例均与Budaghyan--Carlet族CCZ等价，而在更高维情形中未发现与该族等价的实例。