On the Information Rate of Secret Sharing Schemes

Blundo, Carlo; Santis, Alfredo De; Gargano, Luisa; Vaccaro, Ugo

doi:10.1007/3-540-48071-4_11

Cited by 82 publications

(87 citation statements)

References 13 publications

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“…Most of the known lower bounds on the optimal complexity were obtained by informationtheoretical arguments [9,10,15,26]. Specifically, by using basic properties of the Shannon entropy function in combination with the requirements that must be satisfied by the random variables involved in a secret sharing scheme.…”

Section: Lower Bounds From Polymatroidsmentioning

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%

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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: Lower Bounds From Polymatroidsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Secret Sharing Schemes, Matroids and Polymatroids

Martí-Farré

Padrö

Theory of Cryptography

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“…As discussed in the introduction, we conjecture that this is the best possible. Lower bounds for secret-sharing schemes have been proved in, e.g., [47,23,19,33,29,30,18]. However, these lower bounds are far from the exponential upper bounds.…”

Section: Lower Bounds On the Size Of The Sharesmentioning

confidence: 99%

“…Starting from the works of Karnin et al [47] and Capocelli et al [23], the entropy was used to prove lower bounds on the share size in secret-sharing schemes [19,33,29,30]. In other words, to prove lower bounds on the information ratio of secret-sharing schemes, we use the alternative definition of secret sharing via the entropy function, Definition 3.…”

Section: Stronger Lower Boundsmentioning

confidence: 99%

Secret-Sharing Schemes: A Survey

Beimel

2011

Lecture Notes in Computer Science

505

305

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Abstract. A secret-sharing scheme is a method by which a dealer distributes shares to parties such that only authorized subsets of parties can reconstruct the secret. Secret-sharing schemes are important tools in cryptography and they are used as a building box in many secure protocols, e.g., general protocol for multiparty computation, Byzantine agreement, threshold cryptography, access control, attribute-based encryption, and generalized oblivious transfer. In this survey, we will describe the most important constructions of secret-sharing schemes, explaining the connections between secret-sharing schemes and monotone formulae and monotone span programs. The main problem with known secret-sharing schemes is the large share size: it is exponential in the number of parties. We conjecture that this is unavoidable. We will discuss the known lower bounds on the share size. These lower bounds are fairly weak and there is a big gap between the lower and upper bounds. For linear secret-sharing schemes, which is a class of schemes based on linear algebra that contains most known schemes, super-polynomial lower bounds on the share size are known. We will describe the proofs of these lower bounds. We will also present two results connecting secret-sharing schemes for a Hamiltonian access structure to the NP vs. coNP problem and to a major open problem in cryptography -constructing oblivious-transfer protocols from one-way functions.

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“…The above a* satisfies equation (2). Moreover, the information dispersal algorithm £3 obtained taking xt = a*, for i = 1,..., n, maximizes the average information rate, that is Q{A, S-i) = T^sy-Proof.…”

Section: How To Optimize the Average Information Ratementioning

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 Information Rate of Secret Sharing Schemes

Cited by 82 publications

References 13 publications

On Secret Sharing Schemes, Matroids and Polymatroids

On Secret Sharing Schemes, Matroids and Polymatroids

Secret-Sharing Schemes: A Survey

General Short Computational Secret Sharing Schemes

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