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 \(e=0,d=2^{(n+s-2)/2}\)). Moreover, a Boolean function f is s-plateaued (of weight \(\ne 2^{(n+s-2)/2}\)) if and only if the associated Cayley graph is strongly 3-walk-regular (and also strongly \(\ell \)-walk-regular, for all odd \(\ell \ge 3\)) with some explicitly given parameters.