On a Lower Bound for the Number of Bent Functions at the Minimum Distance from a Bent Function in the Maiorana–McFarland Class
D. A. Bykov, Nikolay A. Kolomeec · Journal of Applied and Industrial Mathematics · 2023
Bent functions at the minimum distance $$ 2^n $$ from a given bent function of $$ 2n $$ 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 proved that for any prime $$ n\geq 5 $$ there exist functions in $$ \mathcal {M}_{2n} $$ 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 in $$ \mathcal {M}_{2n} $$ are affinely equivalent if the second derivatives of at least one of the permutations are not identically zero.