On the Representability of the Biuniform Matroid

Abstract

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

Natural Generalizations of Threshold Secret Sharing

“…A vector (u 1 , u 2 ) ∈ P is in Π(Γ) if and only if u 1 ≥ 4 or |u| ≥ 6 and u 1 ≥ 1. Therefore, min Π(Γ) = {(1, 5), (2,4), (3,3), (4, 0)} ∩ P.…”

“…By using a simple computer program, one can check different sets of values of the parameters until a satisfactory one is found. A possible solution is the matrix over F 23 in Equation (3). Therefore, M is the generator matrix of a linear code that defines an F 23 -vector space secret sharing scheme with access structure Γ.…”

“…Some issues related to the efficiency of the constructions of ideal schemes for several particular families of multipartite access structures have been considered [3], [7], [10], [17], [38], [39]. We describe in the following a unified framework, derived from the general results in [14], in which those open problems can be more precisely stated.…”

“…The one proposed in [38,Section 3.3], which works only for prime fields, provides a polynomial time algorithm to construct a vector space secret sharing scheme for every hierarchical threshold access structure. Recently, similar efficient constructions of representations for all bipartite matroids have been presented [3]. The existence of efficient methods for other families of multipartite access structures is an open problem.…”

“…Another efficient algorithm for the same class of access structures was presented by Tassa [38,Section 3.3]. Recently, efficient methods to construct ideal secret sharing schemes for some bipartite access structures have been presented [3].…”