Efficient Explicit Constructions of 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. Secret sharing schemes for multipartite access structures have received considerable attention due to the fact that multipartite secret sharing can be seen as a natural and useful generalization of threshold secret sharing. This work deals with efficient and explicit constructions of ideal multipartite secret sharing schemes, while most of the known constructions are either inefficient or randomized. Most ideal multipartite secret sharing schemes in the literature can be classified as either hierarchical or compartmented. The main results are the constructions for ideal hierarchical access structures, a family that contains every ideal hierarchical access structure as a particular case such as the disjunctive hierarchical threshold access structure and the conjunctive hierarchical threshold access structure, the constructions for three families of compartmented access structures, and the constructions for two families compartmented access structures with compartments. On the basis of the relationship between multipartite secret sharing schemes, polymatroids, and matroids, the problem of how to construct a scheme realizing a multipartite access structure can be transformed to the problem of how to find a representation of a matroid from a presentation of its associated polymatroid. In this paper, we give efficient algorithms to find representations of the matroids associated to several families of multipartite access structures. More precisely, based on know results about integer polymatroids, for each of those families of access structures above, we give an efficient method to find a representation of the integer polymatroid over some finite field, and then over some finite extension of that field, we give an efficient method to find a presentation of the matroid associated to the integer polymatroid. Finally, we construct ideal linear schemes realizing those families of multipartite access structures by efficient methods

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

On the optimization of bipartite secret sharing schemes

Optimizing the ratio between the maximum length of the shares and the length of the secret value in secret sharing schemes for general access structures is an extremely difficult and long-standing open problem. In this paper, we study it for bipartite access structures, in which the set of participants is divided in two parts, and all participants in each part play an equivalent role. We focus on the search of lower bounds by using a special class of polymatroids that is introduced here, the tripartite ones. We present a method based on linear programming to compute, for every given bipartite access structure, the best lower bound that can be obtained by this combinatorial method. In addition, we obtain some general lower bounds that improve the previously known ones, and we construct optimal secret sharing schemes for a family of bipartite access structures.Peer ReviewedPostprint (author's final draft

Secret sharing schemes for ports of matroids of rank 3

summary:A secret sharing scheme is ideal if the size of each share is equal to the size of the secret. Brickell and Davenport showed that the access structure of an ideal secret sharing scheme is determined by a matroid. Namely, the minimal authorized subsets of an ideal secret sharing scheme are in correspondence with the circuits of a matroid containing a fixed point. In this case, we say that the access structure is a matroid port. It is known that, for an access structure, being a matroid port is not a sufficient condition to admit an ideal secret sharing scheme. In this work we present a linear secret sharing scheme construction for ports of matroids of rank 3 in which the size of each share is at most $n$ times the size of the secret. Using the previously known secret sharing constructions, the size of each share was $O(n^2/\log n)$ the size of the secret. Our construction is extended to ports of matroids of any rank $k\geq 2$, obtaining secret sharing schemes in which the size of each share is at most $n^{k-2}$ times the size of the secret. This work is complemented by presenting lower bounds: There exist matroid ports that require $(\mathbb{F}_q,\ell)$-linear secret schemes with total information ratio $\Omega(2^{n/2}/\ell n^{3/4}\sqrt{\log q})$

On the optimization of bipartite secret sharing schemes

Optimizing the ratio between the maximum length of the shares and the length of the secret value in secret sharing schemes for general access structures is an extremely difficult and long-standing open problem. In this paper, we study it for bipartite access structures, in which the set of participants is divided in two parts, and all participants in each part play an equivalent role. We focus on the search of lower bounds by using a special class of polymatroids that is introduced here, the bipartite ones. We present a method based on linear programming to compute, for every given bipartite access structure, the best lower bound that can be obtained by this combinatorial method. In addition, we obtain some general lower bounds that improve the previously known ones, and we construct optimal secret sharing schemes for a family of bipartite access structures.Postprint (author’s final draft

Secret Sharing Based on a Hard-on-Average Problem

The main goal of this work is to propose the design of secret sharing schemes based on hard-on-average problems. It includes the description of a new multiparty protocol whose main application is key management in networks. Its unconditionally perfect security relies on a discrete mathematics problem classiffied as DistNP-Complete under the average-case analysis, the so-called Distributional Matrix Representability Problem. Thanks to the use of the search version of the mentioned decision problem, the security of the proposed scheme is guaranteed. Although several secret sharing schemes connected with combinatorial structures may be found in the bibliography, the main contribution of this work is the proposal of a new secret sharing scheme based on a hard-on-average problem, which allows to enlarge the set of tools for designing more secure cryptographic applications

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