Value Distributions of Perfect Nonlinear Functions

TL;DR

This paper analyzes the value distributions of perfect nonlinear functions, providing new classification results, conditions, and methods to determine their properties using combinatorial and Fourier analysis tools.

Contribution

It introduces a general combinatorial framework for studying value distributions of perfect nonlinear functions and classifies their distributions in key cases, advancing understanding of their structure.

Findings

Classified value distributions of vectorial Boolean bent functions with output dimension ≤ 4

Most classical perfect nonlinear functions are nearly balanced in their value distributions

Discrete Fourier transform can identify perfect nonlinear functions and their equivalences

Abstract

In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions in the most general setting, generalizing several results that were achieved under specific constraints. For the particularly interesting elementary abelian case, we derive several new strong conditions and classification results on the value distributions. Moreover, we show that most of the classical constructions of perfect nonlinear functions have very specific value distributions, in the sense that they are almost balanced. Consequently, we completely determine the possible value distributions of vectorial Boolean bent functions with output dimension at most 4. Finally, using the discrete Fourier transform, we show that in some cases value distributions can be used to determine whether a given function is perfect nonlinear, or to decide whether given perfect nonlinear functions are equivalent.