On the Maiorana-McFarland Class Extensions

• URL: http://arxiv.org/abs/2503.21440v1
• Date: Thu, 27 Mar 2025 12:34:05 GMT
• Title: On the Maiorana-McFarland Class Extensions
• Authors: Nikolay Kolomeec, Denis Bykov,
• Abstract summary: The closure $mathcalM_m#$ and the extension $widehatmathcalM_m$ of the Maiorana--McFarland class $mathcalM_m$ are investigated.<n>We find the number of all closest bent functions to the set $mathcalM_m$ and provide an upper bound of the same number for $mathcalM_m#$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The closure $\mathcal{M}_{m}^{\#}$ and the extension $\widehat{\mathcal{M}}_{m}$ of the Maiorana--McFarland class $\mathcal{M}_{m}$ in $m = 2n$ variables relative to the extended-affine equivalence and the bent function construction $f \oplus \mathrm{Ind}_{U}$ are considered, where $U$ is an affine subspace of $\mathbb{F}_{2}^{m}$ of dimension $m/2$. We obtain an explicit formula for $|\widehat{\mathcal{M}}_{m}|$ and an upper bound for $|\widehat{\mathcal{M}}_{m}^{\#}|$. Asymptotically tight bounds for $|\mathcal{M}_{m}^{\#}|$ are proved as well, for instance, $|\mathcal{M}_{8}^{\#}| \approx 2^{77.865}$. Metric properties of $\mathcal{M}_{m}$ and $\mathcal{M}_{m}^{\#}$ are also investigated. We find the number of all closest bent functions to the set $\mathcal{M}_{m}$ and provide an upper bound of the same number for $\mathcal{M}_{m}^{\#}$. The average number $E(\mathcal{M}_{m})$ of $m/2$-dimensional affine subspaces of $\mathbb{F}_{2}^{m}$ such that a function from $\mathcal{M}_{m}$ is affine on each of them is calculated. We obtain that similarly defined $E(\mathcal{M}_{m}^{\#})$ satisfies $E(\mathcal{M}_{m}^{\#}) < E(\mathcal{M}_{m})$ and $E(\mathcal{M}_{m}^{\#}) = E(\mathcal{M}_{m}) - o(1)$.

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