When does a bent concatenation not belong to the completed Maiorana-McFarland class?
<|reference_start|>When does a bent concatenation not belong to the completed Maiorana-McFarland class?: Every Boolean bent function $f$ can be written either as a concatenation $f=f_1||f_2$ of two complementary semi-bent functions $f_1,f_2$; or as a concatenation $f=f_1||f_2||f_3||f_4$ of four Boolean functions $f_1,f_2,f_3,f_4$, all of which are simultaneously bent, semi-bent, or 5-valued spectra-functions. In this context, it is essential to ask: When does a bent concatenation $f$ (not) belong to the completed Maiorana-McFarland class $\mathcal{M}^\#$? In this article, we answer this question completely by providing a full characterization of the structure of $\mathcal{M}$-subspaces for the concatenation of the form $f=f_1||f_2$ and $f=f_1||f_2||f_3||f_4$, which allows us to specify the necessary and sufficient conditions so that $f$ is outside $\mathcal{M}^\#$. Based on these conditions, we propose several explicit design methods of specifying bent functions outside $\mathcal{M}^\#$ in the special case when $f=g||h||g||(h+1)$, where $g$ and $h$ are bent functions.<|reference_end|>
@article{kudin2024when,
    title={When does a bent concatenation not belong to the completed
  Maiorana-McFarland class?},
    author={Sadmir Kudin and Enes Pasalic and Alexandr Polujan and Fengrong Zhang},
    journal={arXiv preprint arXiv:2404.16220},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.16220},
    primaryClass={cs.IT cs.CR cs.DM math.CO math.IT}
}