Ideal Multipartite Secret Sharing Schemes

Abstract

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. 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 and on the introduction of a new combinatorial tool in secret sharing, integer polymatroids.Our results can be summarized as follows. First, we present a characterization of multipartite matroid ports in terms of integer polymatroids. As a consequence of this characterization, a necessary condition for a multipartite access structure to be ideal is obtained. Second, we use representations of integer polymatroids by collections of vector subspaces to characterize the representable multipartite matroids. In this way we obtain a sufficient condition for a multipartite access structure to be ideal, and also a unified framework to study the open problems about the efficiency of the constructions of ideal multipartite secret sharing schemes. Finally, we apply our general results to obtain a complete characterization of ideal tripartite access structures, which was until now an open problem.

Ideal Hierarchical Secret Sharing Schemes

“…Our results strongly rely on the connection between matroids and ideal secret sharing schemes discovered by Brickell and Davenport [7]. Moreover, since hierarchical access structures are in particular multipartite, the results and techniques in [12] about the characterization of ideal multipartite access structures, which are recalled in Section 6, are extremely useful. In particular, integer polymatroids play a fundamental role.…”

“…In particular, integer polymatroids play a fundamental role. Another important tool is the geometric representation introduced in [12,29] for multipartite access structures, which is adapted in Section 4 to the hierarchical case by introducing the hierarchically minimal points that represent the hierarchically minimal sets. Our characterization of the ideal hierarchical access structures is given in terms of some properties of the h-minimal points that can be efficiently checked.…”

“…In every one of these families, all matroid ports are ports of representable matroids, and hence, all ideal access structures are vector space access structures, that is, they admit an ideal linear secret sharing scheme constructed by the method proposed by Brickell [6]. The characterization of the ideal tripartite access structures in [12] was obtained actually from the much more general results about ideal multipartite access structures in that paper. Specifically, by elaborating on the connection between ideal secret sharing and matroids, integer polymatroids are introduced in [12] as a new powerful combinatorial tool to study ideal multipartite secret sharing schemes.…”

“…A complete characterization of the ideal bipartite access structures was given in [29], and related results were given independently in [25,27]. Partial results on the characterization of the ideal tripartite access structures appeared in [9,14], and this question was solved in [12]. In every one of these families, all matroid ports are ports of representable matroids, and hence, all ideal access structures are vector space access structures, that is, they admit an ideal linear secret sharing scheme constructed by the method proposed by Brickell [6].…”

“…The characterization of the ideal tripartite access structures in [12] was obtained actually from the much more general results about ideal multipartite access structures in that paper. Specifically, by elaborating on the connection between ideal secret sharing and matroids, integer polymatroids are introduced in [12] as a new powerful combinatorial tool to study ideal multipartite secret sharing schemes.…”