A Family of Non-Representable Secret Sharing Matroids

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. In the EUROCRYPT´07, Farras made a important contribution to this work by using discrete polymatroids. 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 (it was the first matroid that was proved to be non-representable), we obtain a family of non-representable matroids, which is a new contribution to the open problem given by Farras.