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：本文研究了平坦函数 $F \colon \mathbb{F}_p^n \rightarrow \mathbb{F}_p^m$ 的组合性质。所有二次函数、bent函数及大多数已知的APN函数均为平坦函数，因此许多密码学原语以平坦函数作为基本构件。研究重点在于平坦函数的Walsh变换与线性性质、微分性质及其值分布（即像集和原像集的大小）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——该类函数仅存在两种非零原像集大小，例如泛化了所有单项式函数。针对此类函数，我们获得了若干直接关联及存在/不存在条件，例如证明了平坦的 $d$-to-$1$ 函数（进而包括平坦单项式）仅对特定的 $d$ 选择存在，并推导了所有这些函数的线性度及微分均匀度的界。此外，我们还专门研究了平坦APN函数的Walsh变换及其与值分布的关系。