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There are numerous applications that require secure group communication. Much recent attention has been focused on secure multicasting over the Internet. When such systems are required to manage large groups which undergo frequent fluctuations in group membership, the need for efficient encryption key management becomes critical. This paper presents a combinatorial formulation of the multicast key management problem that is applicable not only to the specific problem of multicast key management, but also to the general problem of managing keys for any type of trusted group communication, regardless of the underlying transmission method between group participants. Specifically, we describe exclusion basis systems, show exactly when they exist, and demonstrate that such systems represent improvements over the current binary tree-based key management systems and other related systems.