Bent functions using Maiorana-McFarland secondary construction

Abstract

Canteaut et al. demonstrated that bent Boolean functions are balanced over one of two complementary affine spaces. In this work, we provide an alternative proof of this property and leverage this result to establish the existence of $1$-plateaued functions whose restrictions to these affine spaces remain bent. By incorporating these results into the secondary Maiorana-McFarland construction, we obtain bent functions that are inherently balanced over a specific affine subspace (the underlying vector subspace). Furthermore, we analyse the balancedness of these new functions under linear perturbations. Finally, two algorithms are developed to simplify the research findings. It is worth noting that the sets of vectors with even and odd Hamming weights constitute a particular case of such complementary affine spaces.