## Abelian Group Theory by D. Arnold, R. Hunter, E. Walker

By D. Arnold, R. Hunter, E. Walker

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3 Security and efficiency requirements The main goal of the group key management is to assure that only authorized users are in the possession of the group key. As the group is dynamic, the group key must be changed accordingly, and this must be done securely. The security requirements for the group management are: • Forward secrecy (or forward access control): Impossibility of a former group member that has been excluded to gain access to future group keys and implicitly to future multicast group traffic.

Thus, the CAG is obtained from the diagraph of relation ≺ by eliminating all one length cycles, implied by the reflexive property, and all the arcs implied by the transitive property. The result is a directed, intransitive, acyclic, and non-reflexive graph, as shown in Figure 4-1. Since ≺ has a greatest element in rc, it can be shown that CAG has exactly one node, the rc, with in-degree zero. Moreover, it follows from the properties of ≺ and the construction of CAG that for any node in CAG, different than rc, there is at least one directed path from rc to that node.

The controller initiates all the sessions, generates and distributes the keys, maintains information about the categories, and accordingly adds and removes users and categories to and from the multicast group. Although the central controller can be a potential performance bottleneck, this architecture is more desirable in systems where one party must retain the control such as in multicast software delivery [HS00, Han01]. Moreover, methods of replication can be used to unload the controller. 36 CATEGORY-BASED GROUP KEY MANAGEMENT We assume that categories are excluded based on the accessibility relation and we formulate the requirement that is used in the design of our solution: • ∀ci , c j ∈ C : c j ∈ E ∧ ci ≺ c j → ci ∈ E ; • No coalition of users from excluded categories can gain access to future group keys.