On Generalizations of Maiorana-McFarland and $\mathcal{PS}_{ap}$ Functions

TL;DR

This paper introduces new classes of Boolean bent functions extending classical constructions, analyzes their decomposability, and presents a secondary construction method using concatenation, employing algebraic curve theory.

Contribution

It constructs generalized Maiorana--McFarland bent functions outside classical classes and investigates the non-decomposability of generalized $ ext{PS}_{ap}$ functions, offering new secondary construction techniques.

Findings

Generalized Maiorana--McFarland bent functions are not equivalent to classical classes.

Small-degree generalized $ ext{PS}_{ap}$ functions generally do not admit decomposition.

A new secondary construction method based on concatenation of vectorial components.

Abstract

We study generalizations of two classical primary constructions of Boolean bent functions, namely the Maiorana-McFarland ($MM$) class and the (Desarguesian) partial spread ($\mathcal{PS}_{ap}$) class. The construction of bent functions lying outside the completed $MM$ class has attracted considerable attention in recent years. In this direction, we construct families of generalized Maiorana--McFarland bent functions that are not equivalent to any function in the classical $MM$ or $\mathcal{PS}_{ap}$ classes, and hence lie outside their completed classes. As a second contribution, we investigate the decomposition of generalized $\mathcal{PS}_{ap}$ functions. We prove that when the degree is sufficiently small relative to the size of the underlying finite field, such functions do not, in general, admit a decomposition into bent or semibent functions. Consequently, they cannot be obtained from known secondary constructions based on concatenation. Finally, we present a secondary construction of Boolean bent functions arising from the concatenation of components of vectorial generalized $\mathcal{PS}_{ap}$ functions. Our constructions and proofs rely on classical results concerning second-order derivatives of bent functions and their duals. In addition, we employ methods from the theory of algebraic curves and their function fields.