Algebraic Methods in Difference Sets and Bent Functions

TL;DR

This paper explores algebraic techniques for analyzing difference sets and bent functions, offering polynomial criteria for counting and characterizing these combinatorial objects, including specific Boolean functions.

Contribution

It introduces a polynomial criterion for difference sets, enabling enumeration and analysis of bent functions, and examines the bentness of certain Boolean functions beyond existing conditions.

Findings

Polynomial criterion effectively counts difference sets with given parameters.

New insights into the bentness of Boolean functions not satisfying the -condition.

Application of algebraic methods to characterize and analyze difference sets and bent functions.

Abstract

We provide some applications of a polynomial criterion for difference sets. These include counting the difference sets with specified parameters in terms of Hilbert functions, in particular a count of bent functions. We also consider the question about the bentness of certain Boolean functions introduced by Carlet when the $\mathcal{C}$-condition introduced by him doesn't hold.