Number of renewed nodes. As defined in Section II, a node is said to be renewed if it is a non-leaf node and its associated keys are renewed. This metric measures the communication cost since the new blinded keys of the renewed nodes have to be broadcast to the whole group. For simplicity, we assume the following in the analysis: The existing key tree is completely balanced prior to the interval-based rekeying event. Every existing member has the same leave probability. The computation of the blinded group key of the root node is counted in the blinded key computations. With this assumption, the number of blinded key computations simply equals the number of renewed nodes, provided that the blinded key of each renewed node is broadcast only once. Let N be the number of members originally in the com- munication group, L (where 0 L N ) be the number of members that wish to leave the group, and J 0 be the number of new members that want to join the group. Let T denote the existing tree which contains N members. The level of a node v is l = blog2(v + 1) , where v is the node ID, and the maximum level of T is h. Based on the first assumption, E alg i.e., the key tree is initially balanced, we know that N = 2h. Also, let Xxxx be the number of renewed nodes and Ealg be the number of exponentiations for the particular algorithm alg. The performance measure Ealg is composed of two parts: and broadcasts the blinded key BK1. M6 renews the secret s alg and Eb , which respectively represent the number of key K2 and broadcasts the blinded key BK2. (iv) Finally, all members can compute the group key. exponentiations of calculating the secret keys (which is done by all members) and that of calculating the blinded keys = E (which is done by sponsors only). We have tree. No nodes are pruned in this case since all leaving leaf Ealg s alg + E
Number of renewed nodes a node is said to be re- In the following analysis, we only consider the number of secret key computations Es . alg
Number of renewed nodes. A node is said to be renewed if it is a nonleaf node and its associated keys are renewed. This metric measures the communication cost since the new blinded keys of the renewed nodes have to be broadcast to the whole group. The Figures 3 and 4 illustrate the average number of exponentiations and the average number of renewed nodes under different numbers leaving members. It is observed that Queue-batch outperforms the other two interval-based algorithms in all cases. There is a significant computation/communication reduction when the peer group is very dynamic (i.e., high number of members that wish to join or leave the communication group). The results show that the interval-based algorithms outperform the individual rekeying approach in terms of both computation and communication costs and that Queue- batch performs the best among the three interval-based algorithms. Queue-batch performs much better when the join and leave events occur very frequently. It also demonstrates its robustness in keeping the key tree balanced and its capability in minimizing the number of rounds required to update the group key. Queue-batch performs much better than the other two algorithms when the group is highly dynamic ie) in case of frequent joins and frequent leaves. When the number of join events is high, Queue-batch benefits from the pre- processing of join events in the Queue-subtree phase. In
Number of renewed nodes. As defined in Section II, a node is said to be renewed if it is a non-leaf node and its associated keys are renewed. This metric measures the communication cost since the new blinded keys of the renewed nodes have to be broadcast to the whole group. For simplicity, we assume the following in the analysis: • The existing key tree is completely balanced prior to the interval-based rekeying event. • Every existing member has the same leave probability. • The computation of the blinded group key of the root node is counted in the blinded key computations. With this assumption, the number of blinded key computations simply equals the number of renewed nodes, provided that the blinded key of each renewed node is broadcast only once. ≥ ≤ ≤ | ∫
