A single trusted entity known as a Key Server is in charge of key generation, distribution, and management in centralized key management schemes. To prevent eavesdropping and protect the exchange content, a group key is used to encrypt the group communication. This management mechanism is typically based on a binary tree structure. When the membership of a group changes dynamically, the group key must be changed, triggering a certain updated cost. This paper addresses an important problem in centralized dynamic group key management. It consists in finding a set of leaf nodes in a binary key tree to insert new members with minimal insertion cost and keeping the tree as balanced as possible. The two mentioned important objectives are combined into a unified (deterministic) optimization model whose objective function contains discontinuous step functions with binary variables, which is known to be NP-hard. We then reformulate the problem as a combinatorial optimization program with continuous objective by introducing new binary variables. Applying penalty techniques, it results in a standard DC (Difference of Convex functions) program that can be solved efficiently by DCA (DC algorithm). Besides, the insertion nodes must be the leaf nodes, we introduce a two-step algorithm to reduce the model complexity: the first is to find the set of leaf nodes, while the second is to solve the simplified optimization problem. Numerical experiments have been studied intensively to justify the merit of our proposed model as well as the corresponding DCA.

称为密钥服务器的单个可信实体负责集中式密钥管理方案中的密钥生成、分发和管理。为了防止窃听和保护交换内容，使用组密钥对组通信进行加密。这种管理机制通常基于二叉树结构。当组的成员身份动态变化时，组键必须改变，触发一定的更新成本。本文解决了集中式动态组密钥管理中的一个重要问题。它包括在二叉键树中找到一组叶节点，以最小的插入成本插入新成员，并尽可能保持树的平衡。将上述两个重要目标组合成一个统一的（确定性）优化模型，其目标函数包含具有二元变量的不连续阶跃函数，这被称为 NP-hard。然后，我们通过引入新的二元变量将问题重新表述为具有连续目标的组合优化程序。应用惩罚技术，它产生了一个标准的 DC（凸函数差分）程序，可以通过 DCA（DC 算法）有效地求解。此外，插入节点必须是叶子节点，我们引入了两步算法来降低模型复杂度：第一步是找到叶子节点的集合，第二步是解决简化的优化问题。