Ideal Secret Sharing Schemes for Useful Multipartite Access Structures

Farràs, Oriol; Padrö, Carles

doi:10.1007/978-3-642-20901-7_6

Cited by 4 publications

(1 citation statement)

References 54 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As a consequence, the open questions about the existence of such secret sharing schemes for some sizes of the secret value and the possibility of constructing them efficiently are equivalent to determining the representability of some classes of multi-uniform matroids. See [7,9] for more information on this line of work.…”

mentioning

confidence: 99%

On the Representability of the Biuniform Matroid

Ball¹,

Padrö²,

Weiner³

et al. 2013

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

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

show abstract

mentioning

confidence: 99%