ON THE CAYLEY GRAPHS OF BOOLEAN FUNCTIONS

Lotfallah Pourfaray, Modjtaba Ghorbani

DOI Number
https://doi.org/10.22190/FUMI2003873P
First page
873
Last page
885

Abstract


A Boolean function is a function $f:\Bbb{Z}_n^2 \rightarrow \{0,1\}$ and we denote the set of all $n$-variable Boolean functions by $BF_n$. For $f\in BF_n$ the vector $[{\rm W}_f(a_0),\ldots,{\rm W}_f(a_{2n-1})]$ is called the Walsh spectrum of $f$, where ${\rm W}_f(a)= \sum_{x\in V} (-1)^{f(x) \oplus ax}$, where $V_n$ is the vector space of dimension $n$ over the two-element field $F_2$. In this paper, we shall consider the Cayley graph $\Gamma_f$ associated with a Boolean function $f$. We shall also find a complete characterization of the bent Boolean functions of order $16$ and determine the spectrum of related Cayley graphs.In addition, we shall enumerate all orbits of the action of automorphism group on the set $BF_n$. 

Keywords

Boolean function; Walsh spectrum; Cayley graph; automorphism group.

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References

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DOI: https://doi.org/10.22190/FUMI2003873P

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