Joint Compartmented Threshold Access Structures

A secret sharing scheme is a method of distributing a secret value among members of a group such that only certain coalitions of these participants can find the secret. A subset of users that can recover the secret is called a qualified coalition, and the set of all qualified coalitions is called the access structure. An access structure is called monotone if every coalition containing a qualified coalition as a subset is also a qualified coalition. An important class of access structures is the compartmented threshold access structure, where the user set is partitioned into compartments, and a qualified subset has to satisfy a certain threshold at each compartment as well as the overall threshold. Such access structures may be desirable to guarantee fair representation across different sections of a community. Compartmented access structures were introduced in [6], and several secret sharing schemes realizing such access structures were proposed in [1, 3, 7]. Ideality and perfectness are two important criteria for a secret sharing scheme in terms of efficiency and security, respectively. A secret sharing scheme is said to be ideal if the size of the share assigned to each participant is no larger than the size of the secret; and it is said to be perfect if an unqualified coalition can gain no information about the secret. It is shown that all monotone access structures can be realized by a perfect secret sharing scheme [4]. Thus, an important question for an access structure is whether it is possible to find a secret sharing scheme that is both ideal and perfect. Traditionally, a compartmented access structure is assumed to consist of disjoint compartments [6, 1, 3, 7]. We generalize this concept such that the compartments are not necessarily disjoint, and refer to such an access structures as a joint compartmented threshold access structure (JCTAS). In this paper, we give necessary conditions for the existence of an ideal and perfect scheme for JCTASes. Then we propose an ideal and almost surely perfect construction for these access structures.