On the number of the closest bent functions to some Maiorana--McFarland bent functions

D. A. Bykov ; N. A. Kolomeets

Prikladnaya Diskretnaya Matematika. Supplement, no. 17 (2024), pp. 24-27

Résumé

We consider the numbers of bent functions that are closest to some bent functions from the Maiorana — McFarland class $\mathcal{M}_{2n}$, specifically, the numbers near to their lower $\mathcal{l}_{2n} = 2^{2n + 1} - 2^n$ and tight upper $\mathcal{L}_{2n}$ bounds. For a bent function $f(x, y) = \langle x, \sigma(y)\rangle \oplus \varphi(y) \in \mathcal{M}_{2n}$, where $\sigma$ is a function based on the inverse function of elements of the finite field, the number of closest bent functions is calculated for identically zero $\varphi$. Moreover, it is shown that this number is less than $\mathcal{l}_{2n} + 82(2^n - 1)$ and asymptotically equals to $\mathcal{l}_{2n} + o(\mathcal{l}_{2n})$ for some $\varphi$. An explicit formula for the number of bent functions closest to $f(x, y) = \langle x, y\rangle \oplus y_1 y_2 \dots y_m$, where $3 \leq m \leq n$, has been derived. The values for $m = 3$ and $m = n$ are equal to $o(\mathcal{L}_{2n})$ and $\dfrac{1}{3}\mathcal{L}_{2n} + o(\mathcal{L}_{2n})$ respectively as $n \to \infty$. A complete classification of $\mathcal{M}_6$ using the number of closest bent functions is obtained.

Détail

Keywords: affine subspaces, bent functions, Maiorana — McFarland class, minimal distance, the closest functions.