Conjunctive hierarchical secret sharing by finite geometry

Secret sharing is a general method for distributing sensitive data among the participants of a system such that only a collection of predefined qualified coalitions can recover the secret data. One of the most widely used special cases is threshold secret sharing, where every subset of participants of size above a given number is qualified. In this short note, we propose a general construction for a generalized threshold scheme, called conjunctive hierarchical secret sharing, where the participants are divided into disjoint levels of hierarchy, and there are different thresholds for all levels, all of which must be satisfied by qualified sets. The construction is the first method for arbitrary parameters based on finite geometry arguments and yields an improvement in the size of the underlying finite field in contrast with the existing results using polynomials.