Natural Generalizations of Threshold Secret Sharing

Farràs, Oriol; Padrö, Carles; Xing, Chaoping; An, Yang

doi:10.1007/978-3-642-25385-0_33

Cited by 12 publications

(10 citation statements)

References 37 publications

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“…[13] [14] [15] [16] are recent scheme supporting general access structure. Farras et al [17] suggested natural generalizations of threshold secret sharing.…”

Section: Literature Surveymentioning

confidence: 99%

General Access Structure Secret Sharing in Matrix Projection

Patil¹,

Deshmukh²

2014

IJCA

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Image Secret Sharing is a technique which provides security to confidential images by dispersed storages. General access structure image secret sharing provides flexibility for deciding qualified subsets of participants which can reconstruct the original secret image. Non qualified subsets of participants cannot get any information about original secret image. This paper proposes general access structure image secret sharing based on matrix projection. The experimental results show high accuracy, high security and less overhead in the network due to highly reduced size of image shares.

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“…[13] [14] [15] [16] are recent scheme supporting general access structure. Farras et al [17] suggested natural generalizations of threshold secret sharing.…”

Section: Literature Surveymentioning

confidence: 99%

General Access Structure Secret Sharing in Matrix Projection

Patil¹,

Deshmukh²

2014

IJCA

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“…These techniques provided a characterization of the hierarchical access structures that admit an ideal perfect secret sharing scheme [12]. Other relevant results on secret sharing have been obtained by using matroid theory as, for instance, the ones in [2,9,13,23].…”

Section: Introductionmentioning

confidence: 99%

Extending Brickell–Davenport theorem to non-perfect secret sharing schemes

Farràs

Padrö

2013

Des. Codes Cryptogr.

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One important result in secret sharing is the Brickell-Davenport Theorem: every ideal perfect secret sharing scheme defines a matroid that is uniquely determined by the access structure. Even though a few attempts have been made, there is no satisfactory definition of ideal secret sharing scheme for the general case, in which non-perfect schemes are considered as well. Without providing another unsatisfactory definition of ideal non-perfect secret sharing scheme, we present a generalization of the Brickell-Davenport Theorem to the general case. After analyzing that result under a new point of view and identifying its combinatorial nature, we present a characterization of the (not necessarily perfect) secret sharing schemes that are associated to matroids. Some optimality properties of such schemes are discussed.

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“…Because of that, the representability of certain classes of matroids is closely connected to the search for efficient constructions of secret sharing schemes for certain classes of access structures. The reader is referred to [13] for more information about secret sharing and its connections to matroid theory.Several constructions of ideal linear secret sharing schemes for families of relatively simple access structures with interesting properties for the applications have been proposed [2,4,8,9,11,15,18,21,22,23]. They are basic and natural generalizations of Shamir's [19] threshold secret sharing scheme.…”

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confidence: 99%

“…Several constructions of ideal linear secret sharing schemes for families of relatively simple access structures with interesting properties for the applications have been proposed [2,4,8,9,11,15,18,21,22,23]. They are basic and natural generalizations of Shamir's [19] threshold secret sharing scheme.…”

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confidence: 99%

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On the Representability of the Biuniform Matroid

Ball¹,

Padrö²,

Weiner³

et al. 2013

SIAM J. Discrete Math.

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Abstract. Every bi-uniform matroid is representable over all sufficiently large fields. But it is not known exactly over which finite fields they are representable, and the existence of efficient methods to find a representation for every given bi-uniform matroid has not been proved. The interest of these problems is due to their implications to secret sharing. The existence of efficient methods to find representations for all bi-uniform matroids is proved here for the first time. The previously known efficient constructions apply only to a particular class of bi-uniform matroids, while the known general constructions were not proved to be efficient. In addition, our constructions provide in many cases representations over smaller finite fields.Key words. Matroid theory, representable matroid, bi-uniform matroid, secret sharing AMS subject classifications. 05B35, 94A621. Introduction. Given a class of representable matroids, the following are two basic questions about the class. Over which fields are the members of the class representable? Are there efficient algorithms to construct representations for every member of the class? Here an algorithm is efficient if its running time is polynomial in the size of the ground set. For instance, every transversal matroid is representable over all sufficiently large fields [17, Corollary 12.2.17], but it is not known exactly over which fields they are representable, and the existence of efficient algorithms to construct representations is an open problem too.The interest for these problems has been mainly motivated by their connections to coding theory and cryptology, mainly to secret sharing. Determining over which fields the uniform matroids are representable is equivalent to solving the Main Conjecture for Maximum Distance Separable Codes. For more details, and a proof of this conjecture in the prime case, see [1], and for further information on when the conjecture is known to hold, see [12, Section 3]. As a consequence of the results by Brickell [4], every representation of a matroid M over a finite field provides ideal linear secret sharing schemes for the access structures that are ports of the matroid M . Because of that, the representability of certain classes of matroids is closely connected to the search for efficient constructions of secret sharing schemes for certain classes of access structures. The reader is referred to [13] for more information about secret sharing and its connections to matroid theory.Several constructions of ideal linear secret sharing schemes for families of relatively simple access structures with interesting properties for the applications have been proposed [2,4,8,9,11,15,18,21,22,23]. They are basic and natural generalizations of Shamir's [19] threshold secret sharing scheme. A unified approach to all those proposals was presented in [6]. As a consequence, the open questions about the existence of such secret sharing schemes for some sizes of the secret value and the

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Natural Generalizations of Threshold Secret Sharing

Cited by 12 publications

References 37 publications

General Access Structure Secret Sharing in Matrix Projection

General Access Structure Secret Sharing in Matrix Projection

Extending Brickell–Davenport theorem to non-perfect secret sharing schemes

On the Representability of the Biuniform Matroid

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