Work

Almost Maiorana-McFarland Bent Functions

Year: 2025

Type: article

Abstract: In this article, we study bent functions on F2m2of the formf(x,y) =x· ϕ(y) +h(y), wherex∈ Fm−12andy∈ Fm+12, which form the generalized Maiorana-McFarland class (denoted byGMMm+1) and are referred to as almost Maiorana-McFarland bent functions. We provide a complete characterization of the bent property for such functions and determine their duals. Specifically, we show thatfis bent if and only if the mapping ϕ partitions Fm+12into 2-dimensional affine subspaces, on each of which the functionhhas odd weight. While the partition of Fm+12into 2-dimensional affine subspaces is crucial for the bentness, we demonstrate that the algebraic structure of these subspaces plays an even greater role in ensuring that the constructed bent functionsfare excluded from the completed Maiorana-McFarland classM#(the set of bent functions that are extended-affine equivalent to bent functions from the Maiorana-McFarland classM). Consequently, we investigate which properties of mappings ϕ: Fm+12→ Fm−12lead to bent functions of the formf(x,y) =x· ϕ(y)+h(y) both inside and outsideM#and provide construction methods for suitable Boolean functionshon Fm+12. As part of this framework, we present a simple algorithm for constructing partitions of the vector space Fm+12together with appropriate Boolean functionshthat generate bent functions outsideM#. When 2m= 8, we explicitly identify many such partitions that produce at least 278distinct bent functions on F82that do not belong toM#, thereby generating more bent functions outsideM#than the total number of 8-variable bent functions inM#(whose cardinality is approximately 277). Additionally, we demonstrate that concatenating four almost Maiorana-McFarland bent functions outsideM#, i.e., definingf=f1||f2||f3||f4wherefi/∈M#, can result in a bent functionf∈M#. This finding essentially answers an open problem posed recently in Kudin et al. (IEEE Trans. Inf. Theory 71(5): 3999-4011, 2025). Conversely, using a similar approach to concatenate four functionsf1||f2||f3||f4, where eachfi∈M#, we generate bent functions that are provably outsideM#.

Source: IEEE Transactions on Information Theory

DOI: 10.1109/tit.2025.3614379

Authors: Sadmir Kudin, Enes Pasalic, Alexandr Polujan, Fengrong Zhang, Haixia Zhao

Institutions: University of Primorska, Otto-von-Guericke-Universität Magdeburg, Xidian University, Guilin University of Electronic Technology

Language: English

Topic: Coding theory and cryptography

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,645

Funders: National Natural Science Foundation of China, Javna Agencija za Raziskovalno Dejavnost RS