On Non-representable Secret Sharing Matroids

Abstract

The characterization of the access structures of ideal secret sharing schemes is one of the main open problems in secret sharing and has important connections with matroid theory. Because of its difficulty, it has been studied for several particular families of access structures. Multipartite access structures, in which the set of participants is divided into several parts and all participants in the same part play an equivalent role, have been studied in seminal works on secret sharing by Shamir, Simmons, and Brickell, and also recently by several authors.. In the EUROCRYPT'07, Farras made a important contribution to this work: By using discrete polymatroids, they obtained a necessary condition and a sufficient condition for a multipartite access structure to be ideal respectively. In particular, they further gave a very difficult open problem, that is, characterizing the representable discrete polymatroids, i.e., which discrete polymatroids are representable and which ones are non-representable. In this paper, by dealing with a family of matroids derived from the Vamos matroid, which was the first matroid that was proved to be non-representable, we obtain a family of non-representable matroids. As a consequence, we extend it to the general case and obtain a sufficient condition for a discrete polymatroid to be non-representable, which is a new contribution to the open problem given by Farras.