We study the hierarchical secure aggregation problem with groupwise keys. The problem consists of an aggregation server, $U$ relays, and $UV$ users, where each relay serves $V$ disjoint users, and each subset of $G$ users shares an independent groupwise key. Two security requirements are imposed: relay security and server security. Specifically, each relay must not learn any information about the users' inputs, and the server must not learn any additional information beyond the recovered sum of all inputs. We first show that the problem is infeasible when $G = 1$. For the feasible regime $1 < G \le UV$, we fully characterize the optimal rate region. In particular, we prove that both each user and each relay must transmit at least one symbol per input symbol. Furthermore, we characterize the minimum required groupwise key rate as $\max\left\{\frac{V}{\binom{UV}{G} - \binom{(U-1)V}{G}},\; \frac{U - 1}{\binom{UV}{G} - U \binom{V}{G}}\right\},$ where the two terms correspond to the constraints imposed by relay security and server security, respectively. For achievability, we propose an explicit linear coding scheme based on structured precoding matrices, and show that it satisfies both correctness and security requirements. The construction avoids permutation-based symmetrization by leveraging sufficiently generic matrix designs over large fields. Finally, we establish a matching converse, thereby characterizing the optimal rate region.

翻译：我们研究了基于分组密钥的分层安全聚合问题。该问题包含一个聚合服务器、$U$个中继节点和$UV$个用户，每个中继为$V$个不相交用户提供服务，且每$G$个用户共享一个独立的分组密钥。研究施加了两项安全要求：中继安全性和服务器安全性。具体而言，各中继不得获知用户输入的任何信息，服务器不得获知除恢复全部输入总和之外的任何额外信息。首先证明当$G=1$时问题不可解。在可行区域$1<G\le UV$内，我们完整刻画了最优速率区域。特别地，证明每个用户和每个中继每输入符号必须至少传输一个符号。此外，将所需最小分组密钥速率表征为$\max\left\{\frac{V}{\binom{UV}{G} - \binom{(U-1)V}{G}},\; \frac{U - 1}{\binom{UV}{G} - U \binom{V}{G}}\right\}$，其中两项分别对应中继安全性和服务器安全性的约束。在可实现性方面，我们提出基于结构化预编码矩阵的显式线性编码方案，并证明其满足正确性和安全性要求。该构造通过在大域上采用充分通用的矩阵设计，避免了基于置换的对称化方法。最后，我们建立了匹配的逆定理，从而完整刻画了最优速率区域。