Perfect secret sharing schemes on five participants

Jackson, Wen-Ai; Martin, Keith M.

doi:10.1007/bf00129769

Cited by 60 publications

(95 citation statements)

References 13 publications

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“…One of the most remarkable consequences of Theorem 4.4 is related to a repeated phenomenon was observed in several families of access structures as, for instance, the ones defined by graphs [15], the ones on at most five participants [43,26], the bipartite access structures [37], the structures with at most four minimal qualified subsets [31], or the ones with intersection number equal to one [32]. Namely, in all these families, the optimal complexity of every nonideal access structure is at least 3/2.…”

Section: Introductionmentioning

confidence: 94%

“…Obviously, λ(Γ) ≥ σ(Γ) and, of course, every construction of a linear secret sharing scheme for Γ provides an upper bound for λ(Γ), and hence for σ(Γ). Efficient constructions can be obtained by using the existing methods to combine some given linear secret sharing schemes into a new one [14,18,26,38,44,45]. In order to develop a fully formal exposition, specially when dealing with minors in Section 2.4, we consider that σ(Γ) = λ(Γ) = 0 if Γ is a trivial access structure.…”

Section: H(ementioning

confidence: 99%

“…On one hand, upper bounds have been obtained from different decomposition methods by means of which several linear secret sharing schemes are combined into a new one [11,14,18,26,38,44,45]. Upper bounds on λ(Γ) are obtained in this way.…”

Section: Introductionmentioning

confidence: 99%

“…Upper bounds on λ(Γ) are obtained in this way. On the other hand, lower bounds on the optimal complexity have been derived by combining the basic inequalities of Shannon entropy with the requirements given by the access structure [9,10,15,26]. Csirmaz [17] simplified and unified the techniques in those works by revealing the combinatorial nature of the method.…”

Section: Introductionmentioning

confidence: 99%

“…The use of this result would greatly simplify, for instance, the computation in [26] of the lower bounds for the access structures on five participants. A similar behavior was known for the parameter λ(Γ).…”

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: 94%

Section: H(ementioning

confidence: 99%

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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On Matroids and Non-ideal Secret Sharing

Beimel

Livne

2006

Theory of Cryptography

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Abstract. Secret-sharing schemes are a tool used in many cryptographic protocols. In these schemes, a dealer holding a secret string distributes shares to the parties such that only authorized subsets of participants can reconstruct the secret from their shares. The collection of authorized sets is called an access structure. An access structure is ideal if there is a secret-sharing scheme realizing it such that the shares are taken from the same domain as the secrets. Brickell and Davenport (J. of Cryptology, 1991) have shown that ideal access structures are closely related to matroids. They give a necessary condition for an access structure to be ideal -the access structure must be induced by a matroid. Seymour (J. of Combinatorial Theory B, 1992) showed that the necessary condition is not sufficient: There exists an access structure induced by a matroid that does not have an ideal scheme.In this work we continue the research on access structures induced by matroids. Our main result in this paper is strengthening the result of Seymour. We show that in any secret sharing scheme realizing the access structure induced by the Vamos matroid with domain of the secrets of size k, the size of the domain of the shares is at least k + Ω( √ k). Our second result considers non-ideal secret sharing schemes realizing access structures induced by matroids. We prove that the fact that an access structure is induced by a matroid implies lower and upper bounds on the size of the domain of shares of subsets of participants even in nonideal schemes (this generalized results of Brickell and Davenport for ideal schemes).

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