Optimal trees for minimizing average individual updating cost

Key tree is a popular model to maintain the security of group information sharing by using a tree structure to maintain the keys held by different users. Previously, researchers proved that to minimize the worst case updating cost in case of single user deletion, one needs to use a special 2–3 tree. In this paper, we study the average case for user update. We prove that in the optimal tree, the branching degree of every node can be bounded by 3 and furthermore the structure of the optimal tree can be pretty balanced. We also show the way to construct the optimal tree when there are loyal users in the group. Finally we discuss about the weighted case where different users have different probabilities to be the first one leaving the group. We design a polynomial time algorithm to construct the optimal tree when the number of different probabilities is a constant.