Ideal secret sharing schemes on graph-based $3$-homogeneous access structures

Abstract

‎The characterization of the ideal access structures is one of the main open problems in secret sharing and is important from both practical and theoretical points of views‎. ‎A graph-based $3-$homogeneous access structure is an access structure in which the participants are the vertices of a connected graph and every subset of the vertices is a minimal qualified subset if it has three vertices and induces a connected graph‎. ‎In this paper‎, ‎we introduce the graph-based $3-$homogeneous access structures and characterize the ideal graph-based $3$-homogeneous access structures‎. ‎We prove that for every non-ideal graph-based $3$-homogeneous access structure over the graph $G$ with the maximum degree $d$ there exists a secret sharing scheme with an information rate $frac{1}{d+1}$‎. ‎Furthermore‎, ‎we mention three forbidden configurations that are useful in characterizing other families of ideal access structures‎.