Secret sharing schemes on graphs

Given a graph G, a perfect secret sharing scheme based on G is a method to distribute a secret data among the vertices of G, the participants, so that a subset of participants can recover the secret if they contain an edge of G, otherwise they can obtain no information regarding the key. The average information rate is the ratio of the size of the secret and the average size of the share a participant must remember. The information rate of G is the supremum of the information rates realizable by perfect secret sharing schemes. Based on the entropy-theoretical arguments due to Capocelli et al [3], and extending the results of M. van Dijk [6] we construct a graph Gn on n vertices with average information rate below < 4/ log n. We obtain this result by determining, up to a constant factor, the average information rate of the d-dimensional cube.