This paper investigates the fundamental limits of information-theoretic decentralized secure aggregation (DSA) with user dropouts. We consider a fully decentralized network where $K$ users communicate over broadcast channels without a trusted aggregation server. Each user holds a private input and aims to recover the sum of the surviving users' inputs (users may drop) while ensuring that no additional information about individual inputs is revealed to that user, even if it can collude with other users. A two-round communication protocol is considered, where we assume at least $U$ users survive and each user can collude with at most $T$ other users. For this setting, the optimal communication rate region is fully characterized: we show that DSA is infeasible if $U\le T+1$; otherwise, the optimal rate region is given by $R_1\geq 1$ and $R_2\geq \frac{1}{U-T-1}$, where $R_1$ and $R_2$ denote the first- and second-round communication rates, respectively. The proposed aggregation scheme is based on correlated secret keys constructed from $(T+1)$-private maximum distance separable (MDS) matrices, which simultaneously provide robustness against user dropouts and security against collusion. We also derive tight converse bounds that establish the optimality of the proposed scheme. Our result shows that the optimal second-round communication rate depends only on the effective redundancy level $U-T-1$ regardless the total number of users.

翻译：本文研究了支持用户退出的信息论去中心化安全聚合（DSA）的基本极限。我们考虑一个完全去中心化的网络，其中 $K$ 个用户通过广播信道进行通信，且不存在可信的聚合服务器。每个用户持有私有输入，旨在恢复幸存用户输入之和（用户可能退出），同时确保即使该用户与其他用户合谋，也不会泄露单个输入的任何额外信息。我们考虑两轮通信协议，假设至少 $U$ 个用户幸存，且每个用户最多可与 $T$ 个其他用户合谋。针对该设定，我们完整刻画了最优通信速率区域：若 $U\le T+1$，则 DSA 不可行；否则，最优速率区域由 $R_1\geq 1$ 和 $R_2\geq \frac{1}{U-T-1}$ 给出，其中 $R_1$ 和 $R_2$ 分别表示第一轮和第二轮的通信速率。所提出的聚合方案基于从 $(T+1)$-私有最大距离可分（MDS）矩阵构造的相关密钥，该方案同时提供了对用户退出的鲁棒性和对抗合谋的安全性。我们还推导了严格的逆界，证明了所提方案的最优性。结果表明，最优的第二轮通信速率仅取决于有效冗余水平 $U-T-1$，而与用户总数无关。