A Novel Approach for Bent Functions with Dillon-like Exponents and Characterizing Three Classes of Bent Functions via Kloosterman Sums

TL;DR

This paper introduces new classes of Dillon-like bent functions using rational trace functions, characterizes three classes via Kloosterman sums, and identifies functions not EA-equivalent to known monomials.

Contribution

It generalizes a criterion for bentness of Dillon-like functions and explicitly characterizes three new classes using Kloosterman sums, including novel functions.

Findings

Characterization of three classes of bent functions via Kloosterman sums

Introduction of a special family of trace rational functions as building blocks

Discovery of new bent functions not EA-equivalent to known monomials

Abstract

Dillon-like Boolean functions are known, in the literature, to be those trace polynomial functions from $\mathbb{F}_{2^{2n}}$ to $\mathbb{F}_{2}$, with all the exponents being multiples of $2^n-1$ often called Dillon-like exponents. This paper is devoted to bent functions in which we study the bentness of some classes of Dillon-like Boolean functions connected with rational trace functions. Specifically, we introduce a special infinite family of trace rational functions. We shall use these functions as building blocks and generalise notably a criterion due to Li et al. published in [IEEE Trans. Inf. Theory 59(3), pp. 1818-1831, 2013] on the bentness of Dillon-like functions in the binary case, we explicitly characterize three classes of bent functions. These characterizations are expressed in terms of the well-known binary Kloosterman sums. Furthermore, analysis and experiments indicate that new functions not EA-equivalent to all known classes of monomial functions are included in our classes.