Some observations on bent and planar functions

TL;DR

This paper explores properties of bent and planar functions, showing their graph structures, redundancies in difference operators, and minimum distances between distinct planar functions, using elementary and algebraic methods.

Contribution

It establishes new insights into the structure of bent functions as Salem sets and provides elementary bounds on distances between planar functions.

Findings

Graph of a bent function is a Salem set.

Redundancies in difference operators for bent functions are quantified.

Distance between two distinct planar functions is at least two.

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.