Non-perfect secret sharing over general access structures

Abstract

In a secret sharing protocol, a dealer shares the secret such that only the subsets of players in the (monotone) access structure can reconstruct the secret, while subsets of players that are not in the access structure cannot reconstruct the secret. The sharing is perfect if the players of any set not in the access structure have no information about the secret. Non-perfect secret sharing slackens the requirement as: the players of any set not in the access structure can have some information about the secret but cannot reconstruct the secret. All known schemes in the literature for non-perfect secret sharing are directed toward specific classes of the access hierarchy like threshold, ramp, multiple-level hierarchy etc. In this work, we initiate the study of a more general non-perfect secret sharing. We model the access hierarchy via a weighted lattice. We first give a necessary condition and a sufficient condition for the existence of a secret sharing scheme for any given weighted lattice (that defines the access hierarchy). Subsequently, we provide a framework for designing non-perfect secret sharing schemes, using generalized monotone span programs (GenMSPs). We also show how to construct new non-perfect secret sharing schemes by composition of known GenMSPs, and design an exemplary secret sharing algorithm that is based on and illustrates the above framework.