We study combinatorial properties of plateaued functions $F \colon \mathbb{F}_p^n \rightarrow \mathbb{F}_p^m$. All quadratic functions, bent functions and most known APN functions are plateaued, so many cryptographic primitives rely on plateaued functions as building blocks. The main focus of our study is the interplay of the Walsh transform and linearity of a plateaued function, its differential properties, and their value distributions, i.e., the sizes of image and preimage sets. In particular, we study the special case of ''almost balanced'' plateaued functions, which only have two nonzero preimage set sizes, generalizing for instance all monomial functions. We achieve several direct connections and (non)existence conditions for these functions, showing for instance that plateaued $d$-to-$1$ functions (and thus plateaued monomials) only exist for a very select choice of $d$, and we derive for all these functions their linearity as well as bounds on their differential uniformity. We also specifically study the Walsh transform of plateaued APN functions and their relation to their value distribution.

翻译：我们研究了平坦函数 $F \colon \mathbb{F}_p^n \rightarrow \mathbb{F}_p^m$ 的组合性质。所有二次函数、Bent函数以及大多数已知的APN函数都属于平坦函数，因此许多密码学原语都将平坦函数作为基础构建模块。本研究的核心在于探讨平坦函数的沃尔什变换与线性性质、其差分特性以及值分布（即像集与逆像集的大小）之间的相互作用。特别地，我们研究了"几乎平衡"平坦函数的特殊情况，这类函数仅具有两个非零的逆像集大小，例如所有单项式函数均属于此类推广。我们获得了关于这些函数的若干直接关联条件与（非）存在性判定，例如证明了平坦的$d$-对-$1$函数（进而包括平坦单项式函数）仅存在于特定选择的$d$值，并推导了所有此类函数的线性性质及其差分均匀性的界。我们还专门研究了平坦APN函数的沃尔什变换及其与值分布的关联。