We provide new insights into an open problem recently posed by Yuan-Sun [ISIT 2025], concerning the minimum individual key rate required in the vector linear secure aggregation problem. Consider a distributed system with $K$ users, where each user $k\in [K]$ holds a data stream $W_k$ and an individual key $Z_k$. A server aims to compute a linear function $\mathbf{F}[W_1;\ldots;W_K]$ without learning any information about another linear function $\mathbf{G}[W_1;\ldots;W_K]$, where $[W_1;\ldots;W_K]$ denotes the row stack of $W_1,\ldots,W_K$. The open problem is to determine the minimum required length of $Z_k$, denoted as $R_k$, $k\in [K]$. In this paper, we characterize a new achievable region for the rate tuple $(R_1,\ldots,R_K)$. The region is polyhedral, with vertices characterized by a binary rate assignment $(R_1,\ldots,R_K) = (\mathbf{1}(1 \in \mathcal{I}),\ldots,\mathbf{1}(K\in \mathcal{I}))$, where $\mathcal{I}\subseteq [K]$ satisfies the \textit{rank-increment condition}: $\mathrm{rank}\left(\bigl[\mathbf{F}_{\mathcal{I}};\mathbf{G}_{\mathcal{I}}\bigr]\right) =\mathrm{rank}\bigl(\mathbf{F}_{\mathcal{I}}\bigr)+N$. Here, $\mathbf{F}_\mathcal{I}$ and $\mathbf{G}_\mathcal{I}$ are the submatrices formed by the columns indexed by $\mathcal{I}$. Our results uncover the novel fact that it is not necessary for every user to hold a key, thereby strictly enlarging the best-known achievable region in the literature. Furthermore, we provide a converse analysis to demonstrate its optimality when minimizing the number of users that hold keys.

翻译：本文针对袁-孙[ISIT 2025]提出的一个开放问题提供了新的见解，该问题涉及向量线性安全聚合问题中所需的最小个体密钥速率。考虑一个包含$K$个用户的分布式系统，其中每个用户$k\in [K]$持有一个数据流$W_k$和一个个体密钥$Z_k$。服务器旨在计算线性函数$\mathbf{F}[W_1;\ldots;W_K]$，同时不获取关于另一个线性函数$\mathbf{G}[W_1;\ldots;W_K]$的任何信息，这里$[W_1;\ldots;W_K]$表示$W_1,\ldots,W_K$的行堆叠。该开放问题是确定$Z_k$所需的最小长度，记为$R_k$，$k\in [K]$。在本文中，我们刻画了速率元组$(R_1,\ldots,R_K)$的一个新的可达域。该域是多面体，其顶点由二进制速率分配$(R_1,\ldots,R_K) = (\mathbf{1}(1 \in \mathcal{I}),\ldots,\mathbf{1}(K\in \mathcal{I}))$刻画，其中$\mathcal{I}\subseteq [K]$满足\textit{秩增量条件}：$\mathrm{rank}\left(\bigl[\mathbf{F}_{\mathcal{I}};\mathbf{G}_{\mathcal{I}}\bigr]\right) =\mathrm{rank}\bigl(\mathbf{F}_{\mathcal{I}}\bigr)+N$。这里，$\mathbf{F}_\mathcal{I}$和$\mathbf{G}_\mathcal{I}$是由索引集$\mathcal{I}$对应的列构成的子矩阵。我们的结果揭示了一个新的事实：并非每个用户都必须持有密钥，从而严格扩大了文献中已知的最佳可达域。此外，我们提供了逆分析以证明其在最小化持有密钥的用户数量时的最优性。