On a Lower Bound for the Number of Bent Functions at the Minimum Distance from a Bent Function in the Maiorana–McFarland Class

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.