A Family of Non-representable Multipartite 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.Actually,every access structure is multipartite and,hence,in the EUROCRYPT'07,Farras et al dealed with the characterization of ideal multipartite access structures.In their paper,a necessary condition and a sufficient condition for a multipartite access structure to be ideal is obtained.At the same time,they proved that a multipartite matroid is representable if and only if the corresponding discrete polymatroid is representable.In particular,they present an open problem:the characterization of the representable discrete polymatroids,that is,which discrete polymatroids are representable? In this paper,we present a species of non-representable discrete polymatroids,which implies a sufficient condition for a discrete polymatroid to be non-representable.We apply this general result to Vamos matroid and obtain a family of non-representable multipartite matroids.Furthermore,by the linearly dependent and independent vectors,we prove that the Vamos matroid is a non-representable multipartite matroid.