Chapter 12 - Automorphisms of the Set of Bent Functions

Abstract

The theory of bent functions contains many unsolved problems; among them there is a question about the automorphism group of the set of all bent functions in n variables. In this chapter, we give a solution to this problem proposed by the author in 2010. First, we prove that for any nonaffine Boolean function f in n variables there is a bent function g in n variables such that the function f ⊕ g is not bent. This fact implies that affine Boolean functions are precisely all Boolean functions which are at the maximal possible distance from the class of bent functions. In other words, there is a duality, in some sense, between the definitions of bent functions and affine functions. As a corollary, we obtain that the set of bent functions and the set of affine functions have the same groups of automorphisms. This common group is a semidirect product of the general affine group GA(n) and Z2n+1.