Bent functions and line ovals

TL;DR

This paper explores the properties of bent functions linear on spread elements, their geometric connections with ovals and line ovals, and provides explicit formulas and characterizations for duals and Niho bent functions.

Contribution

It offers a geometric characterization of Niho bent functions, explicit formulas for their duals, and insights into their equivalence under EA-transformations.

Findings

Geometric characterization of Niho bent functions

Explicit formulas for dual bent functions

Connections between bent functions and ovals/line ovals

Abstract

In this paper we study those bent functions which are linear on elements of spreads, their connections with ovals and line ovals, and we give descriptions of their dual bent functions. In particular, we give a geometric characterization of Niho bent functions and of their duals, we give explicit formula for the dual bent function and present direct connections with ovals and line ovals. We also show that bent functions which are linear on elements of inequivalent spreads can be EA-equivalent.