Nonperfect secret sharing schemes and matroids

Abstract

This paper shows that nonperfect secret sharing schemes (NSS) have matroid structures and presents a direct link between the secret sharing matroids and entropy for both perfect and nonperfect schemes. We define natural classes of NSS and derive a lower bound of |Vi| for those classes. “Ideal” nonperfect schemes are defined based on this lower bound. We prove that every such ideal secret sharing scheme has a matroid structure. The rank function of the matroid is given by the entropy divided by some constant. It satisfies a simple equation which represents the access level of each subset of participants.