The combinatorial structure and value distributions of plateaued functions

TL;DR

This paper investigates the combinatorial and spectral properties of plateaued functions, revealing their structural characteristics, existence conditions, and implications for cryptographic primitives.

Contribution

It provides new insights into the interplay between Walsh transform, linearity, and value distributions of plateaued functions, including existence conditions and bounds.

Findings

Plateaued $d$-to-$1$ functions only exist for specific $d$ values.

Bounds on differential uniformity of plateaued functions.

Connections between Walsh transform and value distribution of plateaued APN functions.

Abstract

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.