Some observations on bent and planar functions

TL;DR

This paper investigates bent and planar functions over finite fields, focusing on Fourier-analytic properties of bent graphs, redundancy in the associated difference operators, and minimum distance within planar function classes. It proves that the graph of a bent function is a Salem set with constant $C=1$, yielding extremal Fourier decay. It further shows that all difference operators are controlled by a fixed $d\ell$-dimensional basis, revealing strong redundancy for PN and bent functions in odd characteristic. Finally, it provides an elementary argument that distinct planar functions over ${\mathbb F}_q$ with $q=p^\ell$, $p>3$, must differ on at least two inputs, informing distance properties in this function class.

Abstract

We show that the graph of a bent function is a Salem set in an appropriate sense. We also establish a simple result that quantifies redundancies in the difference operators of a function, which applies to bent functions over fields of odd characteristic via their equivalence to perfect non-linear functions in that setting. We end by demonstrating, by entirely elementary means, that the distance between two distinct planar functions must be at least two.

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof