On secret sharing and 3-uniform hypergraphs

Abstract

In a secret sharing scheme a piece of information – the secret – is distributed among a finite set of participants, such that only some predefined coalitions can recover it. This set of subsets is supposed to be monotone, hence the system can be characterized by its minimal elements. From this point of view, a secret sharing scheme can be described with a hypergraph where the hyperedges are these minimal elements. The efficiency of the scheme is measured by the amount of information the most heavily loaded participant must remember. This amount is called complexity and one of the most interesting problem of this topic is the characterization of systems of complexity 1, or ideal structures. We outline some results about 2-uniform hypergraphs (i.e. graph-based systems) and discuss the generalizations of the respective known results for 3-uniform hypergraphs, where every hyperedge contains exactly 3 participants. We present several families of ideal systems, disprove an older characterization of ideal hyperstars and demonstrate the strength of the so-called forbidden minor characterization method by describing the forbidden 3-uniform sub-hypergraphs of ideal hypergraphs.