Bent functions and strongly regular graphs

TL;DR

This paper explores the properties of Cayley graphs derived from bent functions, detailing their parameters and conditions involving their components, with implications for cryptography and graph theory.

Contribution

It provides a comprehensive list of parameters for Cayley graphs from bent functions and introduces conditions on multi-component bent functions related to their supports.

Findings

Parameters of Cayley graphs from bent functions are explicitly listed.

A new condition involving support and symmetric differences of bent function components is established.

Insights into the structure of strongly regular graphs derived from bent functions.

Abstract

The family of bent functions is a known class of Boolean functions, which have a great importance in cryptography. The Cayley graph defined on $\mathbb{Z}_{2}^{n}$ by the support of a bent function is a strongly regular graph $srg(v,k\lambda,\mu)$, with $\lambda=\mu$. In this note we list the parameters of such Cayley graphs. Moreover, it is given a condition on $(n,m)$-bent functions $F=(f_1,\ldots,f_m)$, involving the support of their components $f_i$, and their $n$-ary symmetric differences.