A complete characterization of plateaued Boolean functions in terms of their Cayley graphs

TL;DR

This paper provides a complete characterization of plateaued Boolean functions through their Cayley graphs, linking function properties to specific graph structures like complete bipartite and strongly regular graphs.

Contribution

It establishes a precise correspondence between s-plateaued Boolean functions and particular classes of Cayley graphs, including complete bipartite and strongly walk-regular graphs.

Findings

s-plateaued functions correspond to complete bipartite Cayley graphs

Non-weight-2^{(n+s-2)/2} s-plateaued functions relate to strongly 3-walk-regular graphs

Characterization includes explicit graph parameters for different plateaued functions

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 $\neq 2^{(n+s-2)/2}$) if and only if the associated Cayley graph is strongly $3$-walk-regular (and also strongly $\ell$-walk-regular, for all odd $\ell\geq 3$) with some explicitly given parameters.