Asymptotically Optimal Tree-based Group Key Management Schemes
  In key management schemes that realize secure multicast communications encrypted by group keys on a public network, tree structures are often used to update the group keys efficiently. Selcuk and Sidhu have proposed an efficient scheme which updates dynamically the tree structures based on the withdrawal probabilities of members. In this paper, it is shown that Selcuk-Sidhu scheme is asymptotically optimal for the cost of withdrawal. Furthermore, a new key management scheme, which takes account of key update costs of joining in addition to withdrawal, is proposed. It is proved that the proposed scheme is also asymptotically optimal, and it is shown by simulation that it can attain good performance for nonasymptotic cases. 
Hideyuki Sakai and Hirosuke Yamamoto