Two Constructions on Limits of Entropy Functions

2013

Des. Codes Cryptogr.

One important result in secret sharing is the Brickell-Davenport Theorem: every ideal perfect secret sharing scheme defines a matroid that is uniquely determined by the access structure. Even though a few attempts have been made, there is no satisfactory definition of ideal secret sharing scheme for the general case, in which non-perfect schemes are considered as well. Without providing another unsatisfactory definition of ideal non-perfect secret sharing scheme, we present a generalization of the Brickell-Davenport Theorem to the general case. After analyzing that result under a new point of view and identifying its combinatorial nature, we present a characterization of the (not necessarily perfect) secret sharing schemes that are associated to matroids. Some optimality properties of such schemes are discussed.

Section: Examplesmentioning

confidence: 99%

“…These techniques provided a characterization of the hierarchical access structures that admit an ideal perfect secret sharing scheme [12]. Other relevant results on secret sharing have been obtained by using matroid theory as, for instance, the ones in [2,9,13,23].…”

Section: Introductionmentioning

confidence: 99%

Extending Brickell–Davenport theorem to non-perfect secret sharing schemes

2013

Des. Codes Cryptogr.

“…It is not difficult to check that S is Π-partite with Π = ({a, b}, {c, d}). Matúš [32] pointed out that the rank function of S violates the non-Shannon information inequality given by Zhang and Yeung [43]. This implies that S is not entropic.…”

Section: Propositionmentioning

confidence: 99%

On the optimization of bipartite secret sharing schemes

Metcalf-Burton

et al. 2011

Des. Codes Cryptogr.

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.

“…At this point, the remaining open question about the characterization of ideal access structures is determining the matroids that can be defined from ideal secret sharing schemes. Some important results, ideas and techniques to solve this question have been given by Matúš [21,22].…”

Section: Introductionmentioning

confidence: 99%

Ideal Hierarchical Secret Sharing Schemes

Lecture Notes in Computer Science

2010

Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention since 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 is based on the well known connection between ideal secret sharing schemes and matroids and, more specifically, on the connection between ideal multipartite secret sharing schemes and integer polymatroids. In particular, we prove that every hierarchical matroid port admits an ideal linear secret sharing scheme over every large enough finite field. Finally, we use our results to present a new proof for the existing characterization of the ideal weighted threshold access structures.