A Combinatorial Approach to Threshold Schemes

We investigate the combinatorial properties of threshold schemes. Informally, a (t, w)-threshold scheme is a way of distributing partial information (shadows) to w participants, so that any t of them can easily calculate a key, but no subset of fewer than t participants can determine the key. Our interest is in perfect threshold schemes: no subset of fewer than t participants can determine any partial information regarding the key. We give a combinatorial characterization of a certain type of perfect threshold scheme. We also investigate the maximum number of keys which a perfect (t, w)-threshold scheme can incorporate, as a function of t, w, and the total number of possible shadows, v. This maximum can be attained when there is a Steiner system S(t, w, v) which can be partitioned into Steiner systems S(t − 1. w, v). Using known constructions for such Steiner systems, we present two new classes of perfect threshold schemes, and discuss their implementation.