On a lower bound for the number of bent functions at the minimum distance from a bent function in the Maiorana–McfFrland class

Bent functions at the minimum distance $2^n$ from a given bent function in $2n$ variables belonging to the Maiorana–McFarland class $\mathcal{M}_{2n}$ are investigated. We provide a criterion for a function obtained using the addition of the indicator of an $n$-dimensional affine subspace to a given bent function from $\mathcal{M}_{2n}$ to be a bent function as well. In other words, all bent functions at the minimum distance from a Maiorana–McFarland bent function are characterized. It is shown that the lower bound $2^{2n+1}-2^n$ for the number of bent functions at the minimum distance from $f \in \mathcal{M}_{2n}$ is not attained if the permutation used for constructing $f$ is not an APN function. It is proven that for any prime $n\geq 5$ there are functions from $\mathcal{M}_{2n}$ for which this lower bound is accurate. Examples of such bent functions are found. It is also established that the permutations of EA-equivalent functions from $\mathcal{M}_{2n}$ are affinely equivalent if the second derivatives of at least one of the permutations are not identically zero. Bibliogr. 31.