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Simonis, Juriaan; Ashikhmin, Alexei

doi:10.1023/a:1008244215660

Cited by 78 publications

(29 citation statements)

References 9 publications

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“…On the other hand, the columns labelled by 1, 2, g 1 are dependent, thus, they lie on a line. Figure 2 is not linearly representable over any field [28, Proposition 6.1.10], but has a 2-linear representation over F 3 [35]. Therefore, the Non-Pappus matroid is 2-minimally representable.…”

Section: The Rank Of the Matroid Is Defined R(m ) := R(e) A Base Of mentioning

confidence: 99%

“…However, not all access structures induced by matroids are ideal [32] [25]. The class of matroids inducing ideal access structures are called secret-sharing matroids and also almost affinely representable, and discussed in [35]. Every multilinearly representable matroid is a secret-sharing matroid.…”

Section: Ideal Secret-sharing Schemes and Matroidsmentioning

confidence: 99%

“…[32,35,25,24]. In particular, if an access structure is induced by a multi-linear representable matroid, then it is ideal [35].…”

Section: Introductionmentioning

confidence: 99%

“…Simonis and Ashikhmin [35] considered the access structure induced by the Non-Pappus matroid. They construct an ideal multi-linear secret-sharing scheme realizing this access structure, where the secret contains two field elements, and they prove (using known results about matroids) that there is no ideal linear secret-sharing realizing this access structure (that is, in any linear secret-sharing realizing this access structure at least one share must contain more than one field element).…”

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: The Rank Of the Matroid Is Defined R(m ) := R(e) A Base Of mentioning

confidence: 99%

Section: Ideal Secret-sharing Schemes and Matroidsmentioning

confidence: 99%

“…[32,35,25,24]. In particular, if an access structure is induced by a multi-linear representable matroid, then it is ideal [35].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multi-linear Secret-Sharing Schemes

Beimel

Ben-Efraim

Padrö

et al. 2014

Theory of Cryptography

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“…The access structure of the scheme is then determined by the matroid associated to the code. The connection between ideal secret sharing schemes and matroids, which applies to non-linear schemes as well, was discovered by Brickell and Davenport [4] and has been studied afterwards in may other works, being [17,16,12, 2] some of them. It plays a key role in one of the main open problems in secret sharing: the characterization of the access structures of ideal secret sharing schemes.…”

Section: Ideal Multiplicative Linear Secret Sharing Schemesmentioning

confidence: 99%

Representing Small Identically Self‐Dual Matroids by Self‐Dual Codes

Padrö¹,

Gracia²

2006

SIAM J. Discrete Math.

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The matroid associated to a linear code is the representable matroid that is defined by the columns of any generator matrix. The matroid associated to a self-dual code is identically self-dual, but it is not known whether every identically self-dual representable matroid can be represented by a self-dual code.This open problem was proposed in [8], where it was proved to be equivalent to an open problem on the complexity of multiplicative linear secret sharing schemes.Some contributions to its solution are given in this paper. A new family of identically self-dual matroids that can be represented by self-dual codes is presented. Besides, we prove that every identically self-dual matroid on at most eight points is representable by a self-dual code.

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On Codes, Matroids and Secure Multi-party Computation from Linear Secret Sharing Schemes

Cramer

Daza²,

Gracia³

et al. 2005

Advances in Cryptology – CRYPTO 2005

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Abstract-Error-correcting codes and matroids have been widely used in the study of ordinary secret sharing schemes. In this paper, the connections between codes, matroids, and a special class of secret sharing schemes, namely, multiplicative linear secret sharing schemes (LSSSs), are studied. Such schemes are known to enable multiparty computation protocols secure against general (nonthreshold) adversaries.Two open problems related to the complexity of multiplicative LSSSs are considered in this paper. The first one deals with strongly multiplicative LSSSs. As opposed to the case of multiplicative LSSSs, it is not known whether there is an efficient method to transform an LSSS into a strongly multiplicative LSSS for the same access structure with a polynomial increase of the complexity. A property of strongly multiplicative LSSSs that could be useful in solving this problem is proved. Namely, using a suitable generalization of the well-known Berlekamp-Welch decoder, it is shown that all strongly multiplicative LSSSs enable efficient reconstruction of a shared secret in the presence of malicious faults. The second one is to characterize the access structures of ideal multiplicative LSSSs. Specifically, the considered open problem is to determine whether all self-dual vector space access structures are in this situation. By the aforementioned connection, this in fact constitutes an open problem about matroid theory, since it can be restated in terms of representability of identically self-dual matroids by self-dual codes. A new concept is introduced, the flat-partition, that provides a useful classification of identically self-dual matroids. Uniform identically self-dual matroids, which are known to be representable by self-dual codes, form one of the classes. It is proved that this property also holds for the family of matroids that, in a natural way, is the next class in the above classification: the identically self-dual bipartite matroids.

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Cited by 78 publications

References 9 publications

Multi-linear Secret-Sharing Schemes

Multi-linear Secret-Sharing Schemes

Representing Small Identically Self‐Dual Matroids by Self‐Dual Codes

On Codes, Matroids and Secure Multi-party Computation from Linear Secret Sharing Schemes

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