Ideal hierarchical secret sharing schemes

Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention from the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization deals with the properties of the hierarchically minimal sets of the access structure, which are the minimal qualified sets whose participants are in the lowest possible levels in the hierarchy. By using our characterization, it can be efficiently checked whether any given hierarchical access structure that is defined by its hierarchically minimal sets is ideal. We use the well known connection between ideal secret sharing and matroids and, in particular, the fact that every ideal access structure is a matroid port. In addition, we use recent results on ideal multipartite access structures and the connection between multipartite matroids and integer polymatroids. We prove that every ideal hierarchical access structure is the port of a representable matroid and, more specifically, we prove that every ideal structure in this family admits ideal linear secret sharing schemes over fields of all characteristics. In addition, methods to construct such ideal schemes can be derived from the results in this paper and the aforementioned ones on ideal multipartite secret sharing. Finally, we use our results to find a new proof for the characterization of the ideal weighted threshold access structures that is simpler than the existing one.Peer ReviewedPostprint (author's final draft

Finding lower bounds on the complexity of secret sharing schemes by linear programming

Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and long-standing open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that every secret sharing scheme defines a polymatroid related to the access structure. The best bounds that can be obtained by this combinatorial method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants. By applying this linear programming approach, we improve some of the known lower bounds for the access structures on five participants and the graph access structures on six participants for which these parameters were still undetermined. Nevertheless, the lower bounds that are obtained by this combinatorial method are not tight in general. For some access structures, they can be improved by adding to the linear program non-Shannon information inequalities as new constraints. We obtain in this way new separation results for some graph access structures on eight participants and for some ports of non-representable matroids. Finally, we prove that, for two access structures on five participants, the combinatorial lower bound cannot be attained by any linear secret sharing schemePeer ReviewedPostprint (author's final draft

Ideal homogeneous access structures constructed from graphs

Starting from a new relation between graphs and secret sharing schemes introduced by Xiao, Liu and Zhang, we show a method to construct more general ideal homogeneous access structures. The method has some advantages: it efficiently gives an ideal homogeneous access structure for the desired rank, and some conditions can be imposed (such as forbidden or necessary subsets of players), even if the exact composition of the resulting access structure cannot be fully controlled. The number of homogeneous access structures that can be constructed in this way is quite limited; for example, we show that (t, l)-threshold access structures can be constructed from a graph only when t = 1, t = l - 1 or t = l.Peer ReviewedPostprint (published version

On the representability of the biuniform matroid

Every biuniform matroid is representable over all sufficiently large fields. But it is not known exactly over which finite fields they are representable, and the existence of efficient methods to find a representation for every given biuniform matroid has not been proved. The interest of these problems is due to their implications to secret sharing. The existence of efficient methods to find representations for all biuniform matroids is proved here for the first time. The previously known efficient constructions apply only to a particular class of biuniform matroids, while the known general constructions were not proved to be efficient. In addition, our constructions provide in many cases representations over smaller finite fields. © 2013, Society for Industrial and Applied MathematicsPeer ReviewedPostprint (published version

On the representability of the biuniform matroid

Every biuniform matroid is representable over all sufficiently large fields. But it is not known exactly over which finite fields they are representable, and the existence of efficient methods to find a representation for every given biuniform matroid has not been proved. The interest of these problems is due to their implications to secret sharing. The existence of efficient methods to find representations for all biuniform matroids is proved here for the first time. The previously known efficient constructions apply only to a particular class of biuniform matroids, while the known general constructions were not proved to be efficient. In addition, our constructions provide in many cases representations over smaller finite fields. © 2013, Society for Industrial and Applied MathematicsPeer ReviewedPostprint (published version

On Representable Matroids and Ideal Secret Sharing

In secret sharing, the exact characterization of ideal access structures is a longstanding open problem. Brickell and Davenport (J. of Cryptology, 1991) proved that ideal access structures are induced by matroids. Subsequently, ideal access structures and access structures induced by matroids have attracted a lot of attention. Due to the difficulty of finding general results, the characterization of ideal access structures has been studied for several particular families of access structures. In all these families, all the matroids that are related to access structures in the family are representable and, then, the matroid-related access structures coincide with the ideal ones. In this paper, we study the characterization of representable matroids. By using the well known connection between ideal secret sharing and matroids and, in particular, the recent results on ideal multipartite access structures and the connection between multipartite matroids and discrete polymatroids, we obtain a characterization of a family of representable multipartite matroids, which implies a sufficient condition for an access structure to be ideal. By using this result and further introducing the reduced discrete polymatroids, we provide a complete characterization of quadripartite representable matroids, which was until now an open problem, and hence, all access structures related to quadripartite representable matroids are the ideal ones. By the way, using our results, we give a new and simple proof that all access structures related to unipartite, bipartite and tripartite matroids coincide with the ideal ones