A Note on "Constructing Bent Functions Outside the Maiorana-McFarland Class Using a General Form of Rothaus"
Fei Guo, Zilong Wang, Guang Gong
TL;DR
This work addresses when Rothaus' bent-function construction yields bent functions and whether iterating the method can produce new bent functions outside the Maiorana-McFarland class. By leveraging Walsh-Hadamard transform composition properties, it proves that the Rothaus sufficiency conditions are also necessary, namely $f$ is bent iff $A,B,C$ and $A⊕B⊕C$ are bent, and that the relation $AB⊕AC⊕BC = A⊕B⊕C$ implies $A=B=C$, making iterative Rothaus constructions effectively a direct sum of bent components. Consequently, Zhang et al.'s iterative approach does not contribute to generating genuinely new bent functions. The paper then advocates Hodzic's generalized secondary construction to realize an infinite sequence of bent functions on $n+4k$ variables, offering a viable path for iterative bent-function generation.
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].