Strongly Multiplicative Hierarchical Threshold Secret Sharing

We consider multi-party computation (MPC) in a hierarchical setting, where participants have different capabilities depending on their position in the hierarchy. First, we give necessary conditions for multiplication of secrets in a hierarchical threshold linear secret sharing scheme (LSSS). Starting with known ideal constructions, we then propose a modified scheme with improved multiplication properties. We give sufficient conditions for the new scheme to be (strongly) multiplicative and show that our construction is almost optimal in the number of required participants. Thus, we obtain a new class of strongly multiplicative LSSS with explicit ideal constructions. Such LSSS are also useful outside the MPC setting, since they have an efficient algorithm for reconstructing secrets in the presence of errors.