Tree Management

 

Modular exponentiation is an expensive operation in TGDH. The number of exponentiations for membership events varies, depending on the tree structure. For example, if a single member or a subtree merges to the root node of the current tree, then exactly two modular exponentiations are required. If a key tree is balanced, and a member joins to a leaf node, then the number of exponentiations is $log$₂ n where $n$ is the current number of users. Hence, it is easy to see that joining to the root always requires the minimal number of exponentiations for additive membership operations. If $n$ members join to the root, however, the resulting tree becomes unbalanced (similar to a linked list). If a member in the deepest node leaves the group, $n-1$ exponentiations are required to update the group key. However, if a key tree is fully balanced, the number of exponentiations is $log$₂ n. These examples indicate that a well-balanced key tree reduces the expected cost of leaves. Our heuristic to keep the tree balanced is to choose the insertion node of a join or merge operation as the rightmost shallowest node, which does not increase the height (see also sections 5.2 and 5.5).