Exact information ratios for secret sharing on small graphs with girth at least 5

In a secret-sharing scheme, a piece of information – the secret – is distributed among a finite set of participants in such a way that only some predefined coalitions can recover it. The efficiency of the scheme is measured by the amount of information the most heavily loaded participant must remember. This amount is called information ratio, and one of the most interesting problems of this topic is to calculate the exact information ratio of given structures. In this paper, the information ratios of all but one graph-based schemes on 8 or 9 vertices with a girth at least 5 and all graph-based schemes on 10 vertices and 10 edges with a girth at least 5 are determined using two polyhedral combinatoric tools: the entropy method and covering with stars. Beyond the investigation of new graphs, the paper contains a few improvements and corrections of recent results on graphs with 9 vertices. Furthermore, we determine the exact information ratio of a large class of generalized sunlet graphs consisting of some pendant paths attached to a cycle of length at least 5

Secret Sharing Schemes With Three Or Four Minimal Qualified Subsets

In this paper we study secret sharing schemes whose access structure has three or four minimal qualified subsets. The ideal case is completely characterized and for the non-ideal case we provide bounds on the optimal information rate

On Representable Matroids and Ideal Secret Sharing

In secret sharing, the exact characterization of ideal access structures is a longstanding open problem. Brickell and Davenport (J. of Cryptology, 1991) proved that ideal access structures are induced by matroids. Subsequently, ideal access structures and access structures induced by matroids have attracted a lot of attention. Due to the difficulty of finding general results, the characterization of ideal access structures has been studied for several particular families of access structures. In all these families, all the matroids that are related to access structures in the family are representable and, then, the matroid-related access structures coincide with the ideal ones. In this paper, we study the characterization of representable matroids. By using the well known connection between ideal secret sharing and matroids and, in particular, the recent results on ideal multipartite access structures and the connection between multipartite matroids and discrete polymatroids, we obtain a characterization of a family of representable multipartite matroids, which implies a sufficient condition for an access structure to be ideal. By using this result and further introducing the reduced discrete polymatroids, we provide a complete characterization of quadripartite representable matroids, which was until now an open problem, and hence, all access structures related to quadripartite representable matroids are the ideal ones. By the way, using our results, we give a new and simple proof that all access structures related to unipartite, bipartite and tripartite matroids coincide with the ideal ones

Secret sharing using artificial neural network

Secret sharing is a fundamental notion for secure cryptographic design. In a secret sharing scheme, a set of participants shares a secret among them such that only pre-specified subsets of these shares can get together to recover the secret. This dissertation introduces a neural network approach to solve the problem of secret sharing for any given access structure. Other approaches have been used to solve this problem. However, the yet known approaches result in exponential increase in the amount of data that every participant need to keep. This amount is measured by the secret sharing scheme information rate. This work is intended to solve the problem with better information rate