On the Composition of Matroids and Ideal Secret Sharing Schemes

In an ideal secret sharing scheme, the access structure is uniquely determined by its minimal sets Δs. The purpose of this paper is to characterise Δs. We introduce the concept of strong connectivity and show that under this equivalence relation, an ideal secret sharing scheme decomposes into threshold schemes. We also give a description of the minimal sets that span the strong connectivity classes. As a result we obtain a necessary condition on the types of subsets that are allowed in an ideal access structure as well as an upper bound on the number of such access structures.