Ideal Multipartite Secret Sharing Schemes

Multipartite secret sharing schemes are those having a multipartite access structure, in which the set of participants is divided into several parts and all participants in the same part play an equivalent role. Several particular families of multipartite schemes, such as the weighted threshold schemes, the hierarchical and the compartmented schemes, and the ones with bipartite or tripartite access structure have been considered in the literature. The characterization of the access structures of ideal secret sharing schemes is one of the main open problems in secret sharing. In this work, the characterization of ideal multipartite access structures is studied with all generality. Our results are based on the well-known connections between ideal secret sharing schemes and matroids. One of the main contributions of this paper is the application of discrete polymatroids to secret sharing. They are proved to be a powerful tool to study the properties of multipartite matroids. In this way, we obtain some necessary conditions and some sufficient conditions for a multipartite access structure to be ideal. Our results can be summarized as follows. First, we present a characterization of matroid-related multipartite access structures in terms of discrete polymatroids. As a consequence of this characterization, a necessary condition for a multipartite access structure to be ideal is obtained. Second, we use linear representations of discrete polymatroids to characterize the linearly representable multipartite matroids. In this way we obtain a sufficient condition for a multipartite access structure to be ideal. Finally, we apply our general results to obtain a complete characterization of ideal tripartite access structures, which was until now an open problem.