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

07/01/2018

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by   Constanza Riera, et al.

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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 ≠ 2^(n+s-2)/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.