On the Classification of Ideal Secret Sharing Schemes

Brickell, Ernest F.; Davenport, Daniel M.

doi:10.1007/0-387-34805-0_25

Cited by 100 publications

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“…The characterization of the ideal access structures has attracted the attention of many researchers. 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%

“…Surprisingly enough, the authors interested on secret sharing have been unaware until recently that the result by Brickell and Davenport [13] can be stated in terms of matroid ports, a combinatorial object that was introduced by Lehman [28] in 1964 to solve the Shannon switching game. Actually, after a small change in the definition of matroid port to adapt it to our topic, the main result in [13] can be rewritten as follows: every ideal access structure is a matroid port.…”

Section: Introductionmentioning

confidence: 99%

“…Actually, after a small change in the definition of matroid port to adapt it to our topic, the main result in [13] can be rewritten as follows: every ideal access structure is a matroid port. This necessary condition for an access structure to be ideal is not sufficient because there exist matroids that can not be represented by any ideal secret sharing scheme, and hence there exist matroid ports that are not ideal access structures.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Secret Sharing Schemes, Matroids and Polymatroids

Martí-Farré

Padrö

Theory of Cryptography

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“…All other ideal weighted threshold access structures can be obtained by combining the indecomposable ones using the operation of composition defined by Beimel et al (2008). Their proof was indirect, the cornerstone of their approach was an application of the well-known connection between ideal secret sharing schemes and matroids (Brickell and Davenport, 1990).…”

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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“…Moreover, Csirmaz [5] proved that there are access structures on n elements so that any perfect secret sharing scheme must assign a share which is of size at least -times the size of the secret k. n log n We have better upper bounds when the access structure is based on graphs. If, for example, the graph on which the access structure is based is complete multipartite, then there exists an ideal perfect secret sharing scheme for A (see [3]) and the size of the shares becomes log ISl/pmax + log IKI. Otherwise, using bounds found in [17] we can say that log IV,l 5 log IKl(A + 1)/2, where A is the maximum degree of the graph.…”

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confidence: 99%

General Short Computational Secret Sharing Schemes

Béguin

Cresti

1995

Advances in Cryptology — EUROCRYPT ’95

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Abstract. A secret sharing scheme permits a secret to be shared among participants in such a way that only qualified subsets of participants can recover the secret. If any non qualified subset has absolutely no information about the secret, then the scheme is called perfect. Unfortunately, in this case the size of the shares cannot be less than the size of the secret. Krawczyk [9] showed how to improve this bound in the case of computational threshold schemes by using Rabin's information dispersal algorithms [14], [15]. We show how to extend the information dispersal algorithm for general access structure (we call access structure, the set of all qualified subsets). We give bounds on the amount of information each participant must have. Then we apply this to construct computational schemes for general access structures. The size of shares each participant must have in our schemes is nearly minimal: it is equal to the minimal bound plus a piece of information whose length does not depend on the secret size but just on the security parameter.

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On the Classification of Ideal Secret Sharing Schemes

Cited by 100 publications

References 2 publications

On Secret Sharing Schemes, Matroids and Polymatroids

On Secret Sharing Schemes, Matroids and Polymatroids

Weightedness and structural characterization of hierarchical simple games

General Short Computational Secret Sharing Schemes

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