Modified planar functions and their components

TL;DR

This paper explores the structure of modified planar functions over finite fields, providing a complete description of their component functions in multivariate and univariate forms using bent$_4$ functions and generalized Walsh-Hadamard transforms.

Contribution

It offers a comprehensive characterization of the component functions of modified planar functions, linking them to bent$_4$ functions and flat-spectrum transforms.

Findings

Complete multivariate description of modified planar functions.

Characterization of components via generalized Walsh-Hadamard transforms.

Modified planar functions are described similarly to classical planar functions in odd characteristic.

Abstract

Zhou 2013 introduced modified planar functions to describe $(2^n,2^n,2^n,1)$ relative difference sets $R$ as a graph of a function on the finite field $\F_{2^n}$, and pointed out that projections of $R$ are difference sets that can be described by negabent or bent$_4$ functions, which are Boolean functions given in multivariate form. Objective of this paper is to contribute to the understanding of these component functions of modified planar functions. We first completely describe a multivariate version of modified planar functions in terms of their bent$_4$ components. In the second part we characterize the component functions of (univariate) modified planar functions in terms of appropriate generalizations of the Walsh-Hadamard transform, with respect to which they have a flat spectrum. We hereby obtain a description of modified planar functions by their components which is similar to that of the classical planar functions in odd characteristic as a vectorial bent function.