Weighted threshold secret sharing schemes

Morillo, Paz; Padrö, Carles; Sáez, Germán; Villar, Jorge L.

doi:10.1016/s0020-0190(99)00070-8

Cited by 46 publications

(30 citation statements)

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“…This problem has been considered as well for some families of multipartite access structures. Partial results about weighted threshold access structures were given in [29,35]. Subsequently, a complete characterization of the ideal access structures in this family was presented by Beimel, Tassa and Weinreb [1].…”

Section: Theorem 22 ([25]mentioning

confidence: 99%

Ideal Multipartite Secret Sharing Schemes

Farràs¹,

Martí-Farré²,

Padrö³

2007

Advances in Cryptology - EUROCRYPT 2007

108

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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.

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Section: Theorem 22 ([25]mentioning

confidence: 99%

Ideal Multipartite Secret Sharing Schemes

Farràs¹,

Martí-Farré²,

Padrö³

2007

Advances in Cryptology - EUROCRYPT 2007

108

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“…Access structures that may be realized by an ideal secret sharing scheme are called ideal. Ideal secret sharing schemes and ideal access structures have been studied in, e.g., [1,7,8,15,18,19,21,23,25,30,33]. Ideal access structures are known to have certain combinatorial properties.…”

Section: Introductionmentioning

confidence: 99%

Characterizing Ideal Weighted Threshold Secret Sharing

Beimel

Tassa

Weinreb

2005

Lecture Notes in Computer Science

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Abstract. Weighted threshold secret sharing was introduced by Shamir in his seminal work on secret sharing. In such settings, there is a set of users where each user is assigned a positive weight. A dealer wishes to distribute a secret among those users so that a subset of users may reconstruct the secret if and only if the sum of weights of its users exceeds a certain threshold. A secret sharing scheme is ideal if the size of the domain of shares of each user is the same as the size of the domain of possible secrets (this is the smallest possible size for the domain of shares). The family of subsets authorized to reconstruct the secret in a secret sharing scheme is called an access structure. An access structure is ideal if there exists an ideal secret sharing scheme that realizes it.It is known that some weighted threshold access structures are not ideal, while other nontrivial weighted threshold access structures do have an ideal scheme that realizes them. In this work we characterize all weighted threshold access structures that are ideal. We show that a weighted threshold access structure is ideal if and only if it is a hierarchical threshold access structure (as introduced by Simmons), or a tripartite access structure (these structures, that we introduce here, generalize the concept of bipartite access structures due to Padró and Sáez), or a composition of two ideal weighted threshold access structures that are defined on smaller sets of users. We further show that in all those cases the weighted threshold access structure may be realized by a linear ideal secret sharing scheme. The proof of our characterization relies heavily on the strong connection between ideal secret sharing schemes and matroids, as proved by Brickell and Davenport.

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“…An access structure is hierarchical if, for any two given participants, one of them is hierarchically superior to the other. According to this definition, the family of the hierarchical access structures contains the multilevel access structures [6,32], the hierarchical threshold access structures studied by Tassa [33] and by Tassa and Dyn [34], and also the weighted threshold access structures that were first considered by Shamir [31] and studied in [2,3,23,29]. Duality and minors are fundamental concepts in secret sharing, as they are in matroid theory.…”

Section: Our Resultsmentioning

confidence: 99%

“…Some of them dealt with families of multipartite access structures. Beimel, Tassa and Weinreb [2] presented a characterization of the ideal weighted threshold access structures that generalizes the partial results in [23,29]. Another important result about weighted threshold access structures have been obtained recently by Beimel and Weinreb [3].…”

Section: Introductionmentioning

confidence: 80%

Ideal Hierarchical Secret Sharing Schemes

Farràs

Padrö

2010

Lecture Notes in Computer Science

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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.

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Weighted threshold secret sharing schemes

Cited by 46 publications

References 0 publications

Ideal Multipartite Secret Sharing Schemes

Ideal Multipartite Secret Sharing Schemes

Characterizing Ideal Weighted Threshold Secret Sharing

Ideal Hierarchical Secret Sharing Schemes

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