  In secure group communications, users of a group share a common group key to
prevent eavesdropping and protect the exchange content. A key server
distributes the group key as well as performs group rekeying whenever the
membership changes dynamically. Instead of rekeying after each join or leave
request, we use batch rekeying to alleviate the out-of-sync problem and improve
the efficiency. In this paper, we propose an optimization approach to the
problem of updating group key in the Logical Key Hierarchy (LKH) structure with
batch rekeying. A subtree of new nodes can be appended below a leaf node or is
replaced the position of leaving node on the binary key tree. The latter has a
lower updating key cost than the former since when a member leaves, all the
keys on the path from the root to the deletion node must be updated anyway. We
aim to minimize the total rekeying cost, which is the cost of deletion and
insertion members while keeping the tree as balanced as possible. The mentioned
problem is represented by a unified (deterministic) optimization model whose
objective function contains discontinuous step functions with binary variables.
Thanks to an exact penalty technique, the problem is equivalently reformulated
as a standard DC (Difference of Convex functions) program that can be solved
efficiently by DCA (DC algorithm). Numerical experiments have been studied
intensively to justify the merit of our proposed approach as well as the
corresponding DCA.
