On the information rate of secret sharing schemes

We derive new limitations on the information rate and the average information rate of secret sharing schemes for access structure represented by graphs. We give the first proof of the existence of access structures with optimal information rate and optimal average information rate less than 1/2 + Ε, where Ε is an arbitrary positive constant. We also consider the problem of testing if one of these access structures is a substructure of an arbitrary access structure and we show that this problem is NP-complete. We provide several general lower bounds on information rate and average information rate of graphs. In particular, we show that any graph with n vertices admits a secret sharing scheme with information rate Ω((log n)/n).