On the Capacity Region of Individual Key Rates in Vector Linear Secure Aggregation

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∈ [K] holds a data stream Wk and an individual key Zk. A server aims to compute a linear function F[W1;ldots;WK] without learning any information about another linear function G[W1;ldots;WK], where [W1;ldots;WK] denotes the row stack of W1,ldots,WK. The open problem is to determine the minimum required length of Zk, denoted as Rk, k∈ [K]. In this paper, we characterize a new achievable region for the rate tuple (R1,ldots,RK). The region is polyhedral, with vertices characterized by a binary rate assignment (R1,ldots,RK) = (1(1 ∈ I),ldots,1(K∈ I)), where Isubseteq [K] satisfies the textitrank-increment condition: rankleft(bigl[FI;GIbigr]right) =rankbigl(FIbigr)+N. Here, FI and GI are the submatrices formed by the columns indexed by 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.