Combinatorial Models for Perfect Secret Sharing Schemes

In this paper, we review combinatorial models for secret sharing schemes. A detailed comparison of several existing combinatorial models for secret sharing schemes is conducted. We pay particular attention to the ideal instances of these combinatorial models. We show that the models under examination have a natural hierarchy, but that the ideal instances of these models have a different hierarchy. We demonstrate that, in the ideal case, the combinatorial structures underlying the combinatorial models are essentially independent of the model being used. Furthermore, we show that the matroid
associated with an ideal scheme is uniquely determined by the access structure of the scheme and is independent of the model being used. Using this result, we present a combinatorial classification of
ideal threshold schemes.