A Study on the Average Information Ratio of Perfect Secret-Sharing Schemes for Access Structures Based On Bipartite Graphs

A perfect secret-sharing scheme is a method to distribute a secret among a set of participants in such a way that only qualified subsets of participants can recover the secret and the joint share of participants in any unqualified subset is statistically independent of the secret. The collection of all qualified subsets is called the access structure of the perfect secret-sharing scheme. In a graph-based access structure, each vertex of a graph G represents a participant and each edge of G represents a minimal qualified subset. The average information ratio of a perfect secret-sharing scheme Σ realizing the access structure based on G is defined as ARΣ = ( ∑ v∈V (G) H(ζv))/(|V (G)|H(ζs)), where ζs is the secret and ζv is the share of v, both are random variables from Σ and H is the Shannon entropy. The infimum of the average information ratio of all possible perfect secret-sharing schemes realizing a given access structure is called the optimal average information ratio of that access structure. Most known results about the optimal average information ratio give upper bounds or lower bounds on it. In this present paper, we study the access structures based on bipartite graphs and determine the exact values of the optimal average information ratio of some infinite classes of them. Keywords—secret-sharing scheme, average information ratio, star covering, core sequence.