Complexity Analysis

 

This section analyzes the communication and computation costs for join, leave, merge and partition protocols of TGDH. Our analysis considers the following costs: number of rounds, number of messages, number of exponentiations, and height of the key tree. Each cost can be further classified into serial and total cost. The former assumes parallelization within each protocol round and represents the greatest cost incurred by any participant in a given round. The total cost is simply the sum of all participants' costs in a given round.

We compare our protocol to the authenticated contributory group key agreement scheme GDH.2 described in [3]. To the best of our knowledge, GDH.2 of the Cliques protocol suite is the only other protocol that provides contributory authenticated group key agreement, supports dynamic membership events, and handles both partitions and merges. However, GDH.2 has very different communication semantics. (In particular, merge is relatively expensive in GDH.2, whereas partition is relatively expensive in TGDH.)

The overhead of TGDHdepends on the structure of the key tree. The number of exponentiations depends on the height and density of the key tree, the depth of the joining point (or leaving node), and the number of leaving members in case of partition. For this reason, we analyze the protocol overhead by fixing the maximum number of members.

Figure 11(a) compares the number of exponentiations for a join event in TGDH(with a fully-balanced tree) and GDH.2. As expected, TGDHcosts $O(logn)$ exponentiations. The graph also shows that the number of exponentiations in TGDHdepends on the position of the joining points. Figure 11(b) shows the number of serial exponentiations when joining to a random tree of $N$ members. The average nears a constant value, since the joining point in a random tree is near the root. Based on these graphs, the computation overhead of TGDH is lower than that GDH.2. The computation cost of leave is also smaller than GDH.2. A comparison of the communication cost is summarized in table 1. The communication cost for join in TGDH is slightly more expensive than that of GDH.2.

The cost of partition is particularly interesting, since TGDHis a multi-round protocol whereas a partition in GDH.2 is handled in a single round. We measure this cost after generating a uniform random partition. For this experiment, we consider partitions that split the group into two sub groups, since this case reflects the highest cost. To split the group, we first pick a random number $x$ uniformly in the interval . We then pick $x$ members randomly to form each sub group. Next, we compute the cost of partition for both groups which gives us two sample costs. The average and maximum numbers are computed from these samples.

As figure 11(c) illustrates, the number of serial exponentiations is small, compared to $N-d$ exponentiations of GDH.2, where $d$ is the number of leaving members. The communication cost (total number of messages in figure 11(d) and the number of rounds in figure 11(e)) for partition is noticeably larger than that of GDH.2. Note that TGDHis less efficient for subtractive events, but more efficient for additive events. Figure 11(f) shows the average and maximum height of the key tree after merge operations. The results indicates the logarithmic height of the key tree.

        
Figure 11: Cost Comparison

 

Operations Join Leave

Protocol GDH.2 TGDH GDH.2 TGDH

Rounds 2 2 1 1

Broadcasts 1 2 1 1

Total messages 2 2 1 1

Max bandwidth $N$ $2N$ $N-1$ $2N-2$