Some improved bounds on the information rate of perfect secret sharing schemes

Abstract

In this paper we study secret sharing schemes for access structures based on graphs. A secret sharing scheme enables a secret key to be shared among a set of participants by distributing partial information called shares. Suppose we desire that some specified pairs of participants be able to compute the key. This gives rise in a natural way to a graph G which contains these specified pairs as its edges. The secret sharing scheme is called perfect if a pair of participants corresponding to a nonedge of G can obtain no information regarding the key. Such a perfect secret sharing scheme can be constructed for any graph. In this paper we study the information rate of these schemes, which measures how much information is being distributed as shares compared with the size of the secret key. We give several constructions for secret sharing schemes that have a higher information rate than previously known schemes. We prove the general result that, for any graph G having maximum degree d, there is a perfect secret sharing scheme realizing G in which the information rate is at least 2/(d+3). This improves the best previous general bound by a factor of almost two. © 1992 International Association for Cryptologic Research.