On the Maiorana-McFarland Class Extensions
Nikolay Kolomeec, Denis Bykov
TL;DR
This work studies two secondary constructions of the Maiorana--McFarland class $\mathcal{M}_{m}$ of bent functions: (i) the extended-affine (EA) closure $\mathcal{M}_{m}^{\#}$ and (ii) the Dillon-type extension $\widehat{\mathcal{M}}_{m}$ via $f \oplus \mathrm{Ind}_{U}$ with $U$ an $m/2$-dimensional affine subspace. It provides exact and asymptotically tight results for the cardinalities of these families, in particular explicit formulas for $|\widehat{\mathcal{M}}_{m}|$ and bounds for $|\mathcal{M}_{m}^{\#}|$ and $|\widehat{\mathcal{M}}_{m}^{\#}|$, along with asymptotics for $|\mathcal{M}_{m}^{\#}|$ and the average affinity $E(\mathcal{M}_{m})$. A central component is a criterion for the set $\mathcal{N}(f)$ of closest bent functions to a given bent $f$ in $\mathcal{M}_{m}$, expressed via subspace data and affine conditions, which enables precise counting and structural descriptions. The paper also analyzes metric properties, showing that the average number of $m/2$-dimensional affine subspaces on which a random $f\in\mathcal{M}_{m}$ is affine is $E(\mathcal{M}_{m}) = \frac{10}{3} 2^{m} - 2^{m/2} + \frac{176}{21} + o(1)$ and that $E(\mathcal{M}_{m}^{\#}) < E(\mathcal{M}_{m})$ with $E(\mathcal{M}_{m}^{\#}) = E(\mathcal{M}_{m}) - o(1)$. Consequently, explicit asymptotics for $|\widehat{\mathcal{M}}_{m}|$ and strong bounds for $|\mathcal{M}_{m}^{\#}|$ are obtained, advancing understanding of the size and structure of these bent-function families and informing searches for bent functions outside the completed Maiorana--McFarland class.
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 $E(\mathcal{M}_{m}^{\#}) = E(\mathcal{M}_{m}) - o(1)$.