Information-theoretic topological secure aggregation (TSA)\cite{zhang2026information_regular} enables distributed users to compute neighborhood sums over arbitrary networks without revealing individual inputs, while remaining communication-efficient. It has broad applications, including secure model aggregation in decentralized federated learning (FL). Existing TSA formulations rely on arbitrarily correlated keys generated by a trusted key server, which introduces a single point of failure. In this paper, we instead study TSA with \tit{pairwise} secret keys, where each user pair $(i,j)$ shares an independent key $S_{i,j}$. Such keys can be established through inter-user communication, eliminating the need for a key server and improving robustness. Focusing on a ring topology with $K$ users, we characterize the minimum per-user communication rate: \tit{to securely compute one bit of the desired input sum, each user must send at least $1$ bit to its neighbors when $K=3,4$, and at least $2$ bits for all $K\ge 5$}. The higher rate in larger networks arises because each user must simultaneously satisfy two independent key-alignment constraints from its two neighborhoods, which cannot be resolved within a single broadcast symbol under pairwise key independence. We propose a linear pairwise-masking scheme that achieves these rates and prove its optimality via tight entropic converse bounds that exploit the dependency structure of the keys. Notably, for all $K\ge 4$, only a subset of the $\binom{K}{2}$ pairwise keys -- specifically, those between users at ring distance $2$ -- is sufficient to achieve optimality, revealing a nontrivial role of topological sparsity in secure aggregation.

翻译：信息论拓扑安全聚合（TSA）\cite{zhang2026information_regular}使分布式用户能够在任意网络上计算邻域和，同时保护个人输入隐私并保持通信高效。该技术具有广泛的应用，包括去中心化联邦学习中的安全模型聚合。现有TSA方案依赖可信密钥服务器生成任意相关密钥，这引入了单点故障。本文研究基于\emph{配对}密钥的TSA，其中每对用户$(i,j)$共享独立密钥$S_{i,j}$。此类密钥可通过用户间通信建立，从而消除密钥服务器需求并提升鲁棒性。针对包含$K$个用户的环形拓扑，我们刻画了最小每用户通信速率：\emph{为安全计算一比特目标输入和，当$K=3,4$时每个用户需向邻居发送至少1比特，当$K\ge5$时需至少2比特}。大型网络中更高的速率源于每个用户需同时满足两个邻域独立的密钥对齐约束，这在配对密钥独立性条件下无法通过单个广播符号解决。我们提出线性配对掩码方案实现该速率，并通过利用密钥依赖结构的紧熵逆界证明其最优性。值得注意的是，当$K\ge4$时，仅需$\binom{K}{2}$对配对密钥的一个子集（具体为环形距离为2的用户间密钥）即可实现最优，揭示了拓扑稀疏性在安全聚合中的非平凡作用。