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Bent functions and strongly regular graphs

Valentino Smaldore

TL;DR

The parameters of such Cayley graphs are listed, given a condition on ( n, m)-bent functions F = ( f 1, . . . , f m ), involving the support of their components f i , and their n -ary symmetric differences.

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 $srg(v,kλ,μ)$, with $λ=μ$. 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.

Bent functions and strongly regular graphs

TL;DR

The parameters of such Cayley graphs are listed, given a condition on ( n, m)-bent functions F = ( f 1, . . . , f m ), involving the support of their components f i , and their n -ary symmetric differences.

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 by the support of a bent function is a strongly regular graph , with . In this note we list the parameters of such Cayley graphs. Moreover, it is given a condition on -bent functions , involving the support of their components , and their -ary symmetric differences.
Paper Structure (6 sections, 4 theorems, 16 equations)

This paper contains 6 sections, 4 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Cayley graph $Cay(\mathbb{Z}_{2}^{n},\Omega_{f})$
  4. Strongly regular graphs
  5. Vectorial bent function
  6. Conclusion

Key Result

Corollary 4.3

Consider a $srg(v,k,\lambda,\mu)$, with spectrum $k,\theta_{1}^{m_{1}},\theta_{2}^{m_{2}}$. Then $\lambda=\mu$ if and only if $\theta_{1}=-\theta_{2}$.

Theorems & Definitions (12)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Corollary 4.3
  • Proposition 4.7
  • proof
  • Example 4.8
  • Definition 5.1
  • Definition 5.2
  • Proposition 5.3
  • ...and 2 more