## Cryptology ePrint Archive: Report 2005/059

Secret sharing schemes on graphs

Laszlo Csirmaz

Abstract: Given a graph $G$, a perfect secret sharing scheme based on $G$ is a method to distribute a secret data among the vertices of $G$, the participants, so that a subset of participants can recover the secret if they contain an edge of $G$, otherwise they can obtain no information regarding the secret. The average information rate is the ratio of the size of the secret and the average size of the share a participant must remember. The information rate of $G$ is the supremum of the information rates realizable by perfect secret sharing schemes.

We construct a graph on $n$ vertices with average information rate below $4/\log n$. We obtain this result by determining, up to a constant factor, the average information rate of the $d$/dimensional cube.

Category / Keywords: foundations / secret sharing, polymatroid, information theory