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.

Secret sharing is a method to distribute a secret amongst participants, such that only some predefined coalitions, called qualified sets can recover the secret. The set of the qualified subsets is called the access structure, and if all minimally qualified sets have two elements then the access structure is representable by a graph. The information ratio of an access structure is the amount of information the most overloaded participant must remember per secret bit. In this paper, we compute the information ratio of the unicyclic graph, i.e., connected graphs that are arising from a tree by adding an edge between two non-connected vertices.

Abstract Bipartite secret sharing schemes realize access structures in which the participants are divided into two parts, and all the participants in the same part play an equivalent role. Such a bipartite structure can be described by the collection of its minimal points. The complexity of a scheme is the ratio between the maximum share size given to the participants and the secret size, and the Shannon complexity of a structure is the best lower bound provided by the entropy method. Within this work, we compute the Shannon complexity of regular bipartite structures and provide optimal constructions for some bipartite structures defined by 2 and 3 points.

We consider the information ratio of graph based secret sharing schemes in a special case of graphs in which vertices of degree at least 3 are not connected by an edge. We prove that – after some trivial reduction of the original graph – the information ratio depends on the maximal value of the difference of the degree and the number of triangles containing a given vertex of the reduced graph. This result can be considered as a common generalization of previous results on the information ratio of special trees and graphs of girth at least 6.

Perfect secret sharing scheme is a method to distribute a secret information s among participants such that only predefined coalitions, called qualified subsets of participants can recover the secret, while any other coalitions, the unqualified subsets, cannot determine anything about the secret. The most important property is the efficiency of the system, which is measured by the information ratio. It can be shown that for graphs the information ratio is at most (δ + 1)/2 where δ is the maximum degree of the graph. Blundo et al. constructed a family of δ-regular graphs with information ratio (δ + 1)/2 on at least c · 6δvertices. We improve this result by constructing a significantly smaller graph family on c · 2δvertices achieving the same upper bound both in the worst and the average case.