On the dual of (non)-weakly regular bent functions and self-dual bent functions

Ayça Çeşmelioǧlu,Alexander Pott,Wilfried Meidl

doi:10.3934/amc.2013.7.425

Abstract

For weakly regular bent functions in odd characteristic the dual function is also bent.We analyse a recently introduced construction of non-weakly regular bent functions andshow conditions under which their dual is bent as well. This leads to the definition ofthe class of dual-bent functions containing the class of weakly regular bent functions as a proper subclass.We analyse self-duality for bent functions in odd characteristic, and characterize quadraticself-dual bent functions. We construct non-weakly regular bent functions with and without abent dual, and bent functions with a dual bent function of a different algebraic degree.

Highlights

For a prime p, let f be a function from Fnp to Fp
Until now the dual of a bent function has only been defined for weakly regular bent functions
In order to be able to use quadratic bent functions as a starting point to construct bent functions and their duals in higher algebraic degree, we describe the duals of quadratic bent functions in the following

Summary

Introduction

Let f be a function from Fnp to Fp. The Fourier transform of f is defined to be the complex valued function f on Fnp f (b) =. A function f : Fnp → Fp is called near-bent if |f (b)|2 = pn+1 or 0 for all b ∈ Fnp. The support supp(f ) of the Fourier transform of f is defined by supp(f ) = {b ∈ Fnp | f (b) = 0}. The normalized non-zero Fourier coefficients of a near-bent function resemble those of a bent function. A weakly regular bent function f is called self-dual if f ∗ = f. A weakly regular bent function f is called anti-self-dual if f ∗ = f + e for a constant e ∈ F∗p. As we will see this construction yields non-weakly regular bent functions for which the dual is Advances in Mathematics of Communications. Until now the dual of a bent function has only been defined for weakly regular bent functions.

Bent functions and their duals

Quadratic bent functions and their duals

Self-dual bent functions in odd characteristic

Examples