Abstract
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 W_k and an individual key Z_k. A server aims to compute a linear function F[W₁;…;W_K] without learning any information about another linear function G[W₁;…;W_K], where [W₁;…;W_K] denotes the row stack of W₁,…,W_K. The open problem is to determine the minimum required length of Z_k, denoted as R_k, k∈ [K]. In this paper, we characterize a new achievable region for the rate tuple (R₁,…,R_K). The region is polyhedral, with vertices characterized by a binary rate assignment (R₁,…,R_K) = (1(1 ∈ I),…,1(K∈ I)), where I⊆ [K] satisfies the \textit{rank-increment condition}: rank([F_I;G_I]) =rank(F_I)+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.