Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Sezel Alkan, Nurdagül Anbar, Athina Avrantini, Erroxe Etxabarri-Alberdi, Tekgül Kalaycı, Beatrice Toesca

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.

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

Abstract

We study generalizations of two classical primary constructions of Boolean bent functions, namely the Maiorana-McFarland () class and the (Desarguesian) partial spread () class. The construction of bent functions lying outside the completed 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 or classes, and hence lie outside their completed classes. As a second contribution, we investigate the decomposition of generalized 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 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.

Paper Structure

This paper contains 9 sections, 20 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Generalized Maiorana--McFarland and generalized $\mathcal{PS}_{ap}$ functions
  3. Maiorana--McFarland ($MM$) class.
  4. Generalized Maiorana--McFarland ($GMM$) class.
  5. Desarguesian partial spread ($\mathcal{PS}_{ap}$) class.
  6. Generalized $\mathcal{PS}_{ap}$ class.
  7. Construction of $GMM$ functions outside the $MM^{\#}$ class
  8. Decomposition of the generalized $\mathcal{PS}_{ap}$ functions
  9. Decomposition and concatenation of bent functions

Key Result

Lemma 1

dillon Let $n = 2m$. A Boolean bent function $f : \mathbb{V}_n^{(2)} \rightarrow \mathbb{F}_2$ belongs to the $MM^{\#}$ class if and only if there exists an $m$-dimensional vector subspace $\mathcal{U} \subseteq \mathbb{V}_n^{(2)}$ such that, for all $a,b \in \mathcal{U}$, the second-order derivativ vanish identically.

Theorems & Definitions (38)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • proof
  • Remark 2
  • Remark 3
  • Proposition 3
  • proof
  • Corollary 4
  • Remark 4
  • ...and 28 more