The exact values of the optimal average information ratio of perfect secret-sharing schemes for tree-based access structures

A perfect secret-sharing scheme is a method of distributing a secret among a set of participants such that only qualified subsets of participants can recover the secret and the joint shares of the participants in any unqualified subset is statistically independent of the secret. The set of all qualified subsets is called the access structure of the 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 information ratio of a perfect secret-sharing scheme is defined as the ratio between the maximum length of the share given to a participant and the length of the secret. The average information ratio is the ratio between the average length of the shares given to the participants and the length of the secret. The infimum of the (average) information ratios of all possible perfect secret-sharing schemes realizing a given access structure is called the (average) information ratio of the access structure. Very few exact values of the (average) information ratio of infinite families of access structures are known. Csirmaz and Tardos have found the information ratio of all trees. Based on their method, we develop our approach to determining the exact values of the average information ratio of access structures based on trees.