关于具有组密钥的分层安全聚合容量的研究
On the Capacity of Hierarchical Secure Aggregation with Groupwise Keys
AI总结：
本文研究了具有组密钥的分层安全聚合问题，证明了当G=1时问题不可行，并在1<G≤UV的可行范围内完全刻画了最优速率区域，提出了基于结构化预编码矩阵的线性编码方案，并建立了匹配的反向界。
摘要：
我们研究了具有组密钥的分层安全聚合问题。该问题由聚合服务器、U个中继和UV个用户组成，其中每个中继服务于V个不相交的用户，且每个G个用户的子集共享一个独立的组密钥。两个安全要求被施加：中继安全和服务器安全。具体来说，每个中继不得学习任何关于用户输入的信息，而服务器不得学习除所有输入恢复总和之外的任何额外信息。我们首先证明当G=1时该问题不可行。对于可行范围1 < G ≤ UV，我们完全刻画了最优速率区域。特别是，我们证明每个用户和每个中继必须至少每个输入符号传输一个符号。此外，我们刻画了所需的最小组密钥速率为max{V/(C(UV,G) - C((U-1)V,G)), (U - 1)/(C(UV,G) - U*C(V,G))}，其中两个项分别对应中继安全和服务器安全所施加的约束。对于可实现性，我们提出了一种基于结构化预编码矩阵的显式线性编码方案，并证明该方案满足正确性和安全性要求。该构造通过利用足够通用的大域矩阵设计避免了基于排列的对称化。最后，我们建立了匹配的反向界，从而刻画了最优速率区域。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
