Design of ideal secret sharing based on new results on representable quadripartite matroids

Abstract

With the rapid development of the next generation of mobile networks and communications (5G networks), group-oriented applications show the great potential ability in 5G and multipartite group communications occur frequently in a single network, where a multipartite access structure is denied to be a collection of the subsets of users who may come from different parts of the network such that only users in an authorized subset of users can use their shares to build up a group key for a secure group communication. Most existing group key establishment schemes based on a secret sharing target on building up a group key for a threshold access structure and need to compute a t-degree interpolating polynomial in order to encrypt and decrypt the secret group key. This approach is not suitable and inefficient in terms of computational complexity for multipartite group environments which need to realize the multipartite access structures. In this paper, we study the characterization of ideal access structures related to representable multipartite matroids. By employing some existing results, such as the well-known connection between ideal secret sharing and matroids, the recent discovery of ideal multipartite access structures, and the connection between multipartite matroids and discrete polymatroids, we obtain the characterization of a family of representable multipartite matroids, including a sufficient condition for an access structure to be ideal. Based on this result, we provide a complete characterization of representable quadripartite matroids, showing that these access structures related to representable quadripartite matroids are the ideal ones. From these ideal access structures, we can then construct ideal secret sharing schemes for multipartite group communications. Our results are attractive for efficient and secure multi-group-oriented applications.