A Note on "Constructing Bent Functions Outside the Maiorana-McFarland Class Using a General Form of Rothaus"

TL;DR

This paper proves the necessary and sufficient conditions for Rothaus' construction of bent functions, showing that iterative methods yield only trivial solutions, and proposes a new iterative construction approach to generate bent functions.

Contribution

It establishes the necessity of conditions in Rothaus' construction and introduces a novel iterative method for constructing bent functions.

Findings

Necessary and sufficient conditions for Rothaus' construction are identified.

Iterative construction methods only produce trivial solutions.

A new iterative construction approach using secondary construction is proposed.

Abstract

In 2017, Zhang et al. proposed a question (not open problem) and two open problems in [IEEE TIT 63 (8): 5336--5349, 2017] about constructing bent functions by using Rothaus' construction. In this note, we prove that the sufficient conditions of Rothaus' construction are also necessary, which answers their question. Besides, we demonstrate that the second open problem, which considers the iterative method of constructing bent functions by using Rothaus' construction, has only a trivial solution. It indicates that all bent functions obtained by using Rothaus' construction iteratively can be generated from the direct sum of an initial bent function and a quadratic bent function. This directly means that Zhang et al.'s construction idea makes no contribution to the construction of bent functions. To compensate the weakness of their work, we propose an iterative construction of bent functions by using a secondary construction in [DCC 88: 2007--2035, 2020].