APN functions play a central role as building blocks in the design of many block ciphers, serving as optimal functions to resist differential attacks. One of the most important properties of APN functions is their linearity, which is directly related to the Walsh spectrum of the function. In this paper, we establish two novel connections that allow us to derive strong conditions on the Walsh spectra of quadratic APN functions. We prove that the Walsh transform of a quadratic APN function $F$ operating on $n=2k$ bits is uniquely associated with a vector space partition of $\mathbb{F}_2^n$ and a specific blocking set in the corresponding projective space $PG(n-1,2)$. These connections allow us to prove a variety of results on the Walsh spectrum of $F$. We prove for instance that $F$ can have at most one component function of amplitude larger than $2^{\frac{3}{4}n}$. We also find the first nontrivial upper bound on the number of bent component functions of a quadratic APN function, and and provide conditions for a function to be CCZ-equivalent to a permutation, based on its number of bent components.

翻译：APN函数作为构建许多分组密码的核心组件，在抵抗差分攻击方面发挥着最优函数的作用。APN函数最重要的性质之一是其线性度，这与函数的Walsh谱直接相关。本文建立了两个新颖的联系，使我们能够推导出关于二次APN函数Walsh谱的强约束条件。我们证明，在$n=2k$比特上运算的二次APN函数$F$的Walsh变换，唯一对应于$\mathbb{F}_2^n$的一个向量空间划分以及相应射影空间$PG(n-1,2)$中的一个特定阻塞集。这些联系使我们能够证明关于$F$的Walsh谱的多种结果。例如，我们证明$F$至多只能有一个振幅大于$2^{\frac{3}{4}n}$的分量函数。我们还首次给出了二次APN函数的弯曲分量函数数量的非平凡上界，并基于其弯曲分量数量，给出了函数CCZ等价于置换的条件。