Bent Functions

Chapter 11

Distances Between Bent Functions

Abstract

Hamming distances between bent functions are studied. In general, we follow the PhD thesis of N.A. Kolomeec published in 2014 and totally devoted to the topic of this chapter. It is shown that the minimal possible distance between two distinct bent functions in n variables is equal to 2^n/2. Moreover, bent functions are at this distance if and only if they differ in all elements of some affine subspace of dimension n/2 and both functions are affine on it. It was proven by Kolomeec that if f is a bent function in n variables, then the number of bent functions at distance 2^n/2 from it is not more than $2^{n/2}{\prod }_{i=1}^{n/2}(2^i+1)$; this bound is achieved if and only if f is a quadratic bent function. Complete ...