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 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 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 ∈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≥ 5 there exist functions in 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 ℳ_2n are affinely equivalent if the second derivatives of at least one of the permutations are not identically zero.