A characterization of span program size and improved lower bounds for monotone span programs

Gál, Anna

doi:10.1145/276698.276855

Cited by 28 publications

(33 citation statements)

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“…The best known lower bounds for linear secret-sharing schemes is n Ω(log n) [1,16,17]. By modification of the claims in [17], we show that these lower bounds hold also for multi-linear secret-sharing schemes.…”

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

confidence: 78%

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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: 78%

“…In contrast to general secret-sharing schemes, super-polynomial lower bounds are known for linear secret-sharing schemes. That is, there exist explicit access structures such that the total share size of any linear secret-sharing scheme realizing them is n Ω(log n) times the size of the secret [1,16,17].…”

Section: Introductionmentioning

confidence: 99%

Multi-linear Secret-Sharing Schemes

Beimel

Ben-Efraim

Padrö

et al. 2014

Theory of Cryptography

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“…More details about them are given in Section 6. Stronger separation results are needed to prove the limitations of the polymatroid technique to find good asymptotic lower bounds for σ(Γ) in the same way as the separation results between the parameters λ and σ [1,6,23] indicate the limits of the search of asymptotic upper bounds by constructing linear schemes.…”

Section: Lower Bounds From Polymatroidsmentioning

confidence: 99%

“…Therefore, determining the values of the new parameter κ and finding out the separation between κ and σ are new open problems that have important implications. Some important results showing a separation between the parameters σ and λ have been given in [1,6,23]. They imply that linear schemes are not among the optimal schemes for every access structure.…”

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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“…Linear secret-sharing schemes are equivalent to monotone span programs, defined by [46]. Super-polynomial lower bounds for monotone span programs and, therefore, for linear secret-sharing schemes were proved in [5,2,36].…”

Section: Example 1 (Attribute Based Encryption)mentioning

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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A characterization of span program size and improved lower bounds for monotone span programs

Cited by 28 publications

References 28 publications

Multi-linear Secret-Sharing Schemes

Multi-linear Secret-Sharing Schemes

On Secret Sharing Schemes, Matroids and Polymatroids

Secret-Sharing Schemes: A Survey

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