A complete characterization of plateaued boolean functions in terms of their cayley graphs

Abstract

In this paper we find a complete characterization of plateaued Boolean functions in terms of the associated Cayley graphs. Precisely, we show that a Boolean function f is s-plateaued (of weight = 2(n+s-2)/2) if and only if the associated Cayley graph is a complete bipartite graph between the support of f and its complement (hence the graph is strongly regular of parameters e= 0, d= 2(n+s2)/2)Moreover, a Boolean function f is s-plateaued (of weight ≠ 2 2(n+s2)/2) if and only if the associated Cayley graph is strongly 3-walk-regular (and also strongly ℓ -walk-regular, for all odd ℓ≥ 3) with some explicitly given parameters.