Optimal linear secret sharing schemes for graph access structures on six participants

Abstract

The efficiency of a secret sharing scheme is measured via its information ratio. The optimal information ratio (OIR) of an access structure is the infimum of the information ratios of all secret sharing schemes realizing it. When the infimum is taken over all linear secret sharing schemes, it is called the optimal linear information ratio (OLIR). We review the problem of finding the OIRs and OLIRs of graph access structures on six participants. Study of such access structures was initiated by van Dijk et al. (1998) [43]. In a sequence of works, the OIRs of nine access structures, out of the 18 initially unsolved non-isomorphic ones, were determined. Very recently, the known lower bounds on the OLIRs of the other nine cases have been improved (Farràs et al., 2018 [20]), determining the OLIRs for another two cases. In this paper, for each of the remaining seven cases, we provide a new upper bound on the OIR which matches that of the recently found lower bound on the OLIR. Improved upper bounds are achieved by constructing a linear secret sharing scheme, using a suitable decomposition technique for each access structure. A new decomposition technique, called (λ,ω)-weighted decomposition, is also introduced which is a generalization of all known decomposition methods.