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arXiv · 2404.16220

When does a bent concatenation not belong to the completed Maiorana-McFarland class?

Sadmir Kudin·Enes Pasalic·Alexandr Polujan·Fengrong Zhang
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Abstract

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.

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Sadmir Kudin, Enes Pasalic, Alexandr Polujan, Fengrong Zhang. 2024-04-24. When does a bent concatenation not belong to the completed Maiorana-McFarland class?. https://arxiv.org/abs/2404.16220

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