Some Ideal Secret Sharing Schemes

Brickell, Ernest F.

doi:10.1007/3-540-46885-4_45

Cited by 273 publications

(334 citation statements)

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“…Thus, even using multi-linear schemes one cannot construct efficient schemes for general access structures. We will use the following alternative definition of multi-linear secret sharing schemes, proven to be equivalent in [13] (following [7,20]). Note that not every labeled matrix is a multi-target span program.…”

Section: Lower Bounds For Multi-linear Secret-sharing Schemesmentioning

confidence: 99%

“…Better constructions were introduced by Benaloh and Leichter [3]. Linear secret-sharing schemes were presented by Brickel [7] for the case that each share is one field element and by Krachmer and Wigderson [20] for the case that each share can contain more than one field element. Karchmer and Wigderson's motivation was studying a complexity model called span programs; in particular, they proved that monotone span programs are equivalent to linear secret-sharing schemes.…”

Section: Introductionmentioning

confidence: 99%

“…For example, Shamir's scheme [33] is ideal. Brickell [7] considered ideal schemes and constructed ideal schemes for some access structures, e.g., for hierarchical access structures. Brickell and Davenport [8] showed an interesting connection between ideal access structures and matroids, that is, (1) If an access structure is ideal then it is induced by a matroid, (2) If an access structure is induced by a representable matroid, then the access structure is ideal.…”

Section: Introductionmentioning

confidence: 99%

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Multi-linear Secret-Sharing Schemes

Beimel

Ben-Efraim

Padrö

et al. 2014

Theory of Cryptography

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Abstract. Multi-linear secret-sharing schemes are the most common secret-sharing schemes. In these schemes the secret is composed of some field elements and the sharing is done by applying some fixed linear mapping on the field elements of the secret and some randomly chosen field elements. If the secret contains one field element, then the scheme is called linear. The importance of multi-linear schemes is that they provide a simple non-interactive mechanism for computing shares of linear combinations of previously shared secrets. Thus, they can be easily used in cryptographic protocols.In this work we study the power of multi-linear secret-sharing schemes. On one hand, we prove that ideal multi-linear secret-sharing schemes in which the secret is composed of p field elements are more powerful than schemes in which the secret is composed of less than p field elements (for every prime p). On the other hand, we prove super-polynomial lower bounds on the share size in multi-linear secret-sharing schemes. Previously, such lower bounds were known only for linear schemes.

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Section: Lower Bounds For Multi-linear Secret-sharing Schemesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multi-linear Secret-Sharing Schemes

Beimel

Ben-Efraim

Padrö

et al. 2014

Theory of Cryptography

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“…Brickell and Davenport [13] discovered strong connections of this open problem with matroid theory. Specifically, every ideal secret sharing scheme defines a unique matroid, and hence ideal secret sharing schemes can be seen as representations of matroids that actually include linear representations because of the construction of ideal linear schemes proposed by Brickell [12].…”

Section: Introductionmentioning

confidence: 99%

On Secret Sharing Schemes, Matroids and Polymatroids

Martí-Farré

Padrö

Theory of Cryptography

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The complexity of a secret sharing scheme is defined as the ratio between the maximum length of the shares and the length of the secret. The optimization of this parameter for general access structures is an important and very difficult open problem in secret sharing. We explore in this paper the connections of this open problem with matroids and polymatroids.Matroid ports were introduced by Lehman in 1964. A forbidden minor characterization of matroid ports was given by Seymour in 1976. These results are previous to the invention of secret sharing by Shamir in 1979. Important connections between ideal secret sharing schemes and matroids were discovered by Brickell and Davenport in 1991. Their results can be restated as follows: every ideal secret sharing scheme defines a matroid, and its access structure is a port of that matroid. In spite of this, the results by Lehman and Seymour and other subsequent results on matroid ports have not been noticed until now by the researchers interested in secret sharing.Lower bounds on the optimal complexity of access structures can be found by taking into account that the joint Shannon entropies of a set of random variables define a polymatroid. We introduce a new parameter, which is denoted by κ, to represent the best lower bound that can be obtained by this method. We prove that every bound that is obtained by this technique for an access structure applies to its dual structure as well.By using the aforementioned result by Seymour we obtain two new characterizations of matroid ports. The first one refers to the existence of a certain combinatorial configuration in the access structure, while the second one involves the values of the parameter κ that is introduced in this paper. Both are related to bounds on the optimal complexity. As a consequence, we generalize the result by Brickell and Davenport by proving that, if the length of every share in a secret sharing scheme is less than 3/2 times the length of the secret, then its access structure is a matroid port. This generalizes and explains a phenomenon that was observed in several families of access structures.Finally, we present a construction of linear secret sharing schemes for the ports of the Vamos matroid and the non-Desargues matroid, which do not admit any ideal secret sharing scheme. We obtain in this way upper bounds on their optimal complexity. These new bounds are a contribution on the search of examples of access structures whose optimal complexity lies between 1 and 3/2.

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“…Both disjunctive and conjunctive hierarchical access structures have been proved to be ideal (Brickell, 1990;Tassa, 2007) which means they can carry the most informationally efficient secret sharing scheme and be completely secure (i.e., not giving any information about the secret to unauthorized coalitions). Classification of ideal access structures proved to be extremely difficult problem and the focus of attention has moved to classification of ideal access structures in subclasses of access structures like weighted threshold access structures introduced by Shamir (1979).…”

Section: Introductionmentioning

confidence: 99%

Weightedness and structural characterization of hierarchical simple games

Gvozdeva

Hameed

Slinko

2013

Mathematical Social Sciences

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In this paper we give structural charaterizations for disjunctive and conjunctive hierarchical simple games by characterizing them as complete games with a unique shift-maximal losing and, respectively, shift-minimal winning coalitions. We prove canonical representation theorems for both types of hierarchical games and establish duality between them. We characterize those disjunctive and conjunctive hierarchical games which are weighted majority games. This paper was inspired by Beimel et al. (2008) and Farràs and Padró (2010) characterizations of ideal weighted threshold access structures of secret sharing schemes.

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Some Ideal Secret Sharing Schemes

Cited by 273 publications

References 4 publications

Multi-linear Secret-Sharing Schemes

Multi-linear Secret-Sharing Schemes

On Secret Sharing Schemes, Matroids and Polymatroids

Weightedness and structural characterization of hierarchical simple games

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