On Multiplicative Secret Sharing Schemes Realizing Graph Access Structures

In this paper we consider graph access structures and we show that such an access structure can be realized by a (strongly) multiplicative monotone span program if and only if the privacy structure contains at least three (resp. four) maximal sets. Thus, we obtain a new family of access structures with an explicit construction that is both ideal and strongly multiplicative. Until now only three families of strongly multiplicative access structures are known: the Shamir’s threshold secret sharing scheme as well as the recently proposed quasi-threshold construction by Chen and Cramer and the hierarchical threshold construction by Kasper et al. Another way to construct access structures from graphs has been shown by Liu et al. Namely, the access structure is based on the connectivity of a given graph. The authors have constructed an ideal multiplicative MSP computing the proposed access structure. We provide refined characterization of this access structure.