Nested multipartite secret sharing

Quite recently, Tassa introduced an ideal and perfect secret sharing scheme realizing conjunctive hierarchical threshold access structures motivated by the problem of sharing a private key among three employees of a bank, at least one of whom must be a department manager, for the purpose of signing an electronic funds transfer. We ask the natural question concerning “What if there are two branches of banks that are needed to be involved in the signing process?” In such a case, one might encounter the presence of two distinct hierarchies involved in the same access structure. In this paper, being motivated by such a sample scenario, we describe a new generalization, what we name nested multipartite access structures, which may involve the well-known compartmented or hierarchical access structures as substructures. The corresponding generic scheme we describe employs multivariate interpolation and is ideal, linear and perfect with probability 1 - O(q^-1) on a finite field F_q. We describe the scheme in particular for the trivariate case as an example. Such an approach is hopefully useful not only for the initial motivating example, but also for a variety of interesting scenarios. In particular, we propose a non-nested generalization for the conventional compartmented access structures, which depicts a stronger way of controlling the additional t - (t₁ + ... + t_m) participants.