On the Communication-Key Rate Region of Hierarchical Vector Linear Secure Aggregation

Motivated by heterogeneous data distributions and task-dependent aggregation requirements in federated learning, we study information-theoretic secure aggregation of linear functions over a two-hop hierarchical network. The system comprises an aggregation server, an intermediate layer of U relays, and UV users, where each relay serves a disjoint cluster of V users. Each relay observes all uplink transmissions within its cluster and forwards a coded message to the server. The server is authorized to compute a prescribed linear function F of the users' inputs with zero error, while being prevented from learning any additional information about an unauthorized linear function G. Moreover, each relay must obtain no information about any non-trivial linear function Bu of the inputs in its own cluster. We define the communication rates on both hops as the number of transmitted symbols per input symbol. By deriving matching information-theoretic converse and achievability bounds, we fully characterize the optimal communication rates and propose an explicit linear coding scheme that achieves the resulting optimal region. Our results demonstrate that hierarchical architectures can attain optimal communication rates while substantially reducing the server-side masking burden, thereby enabling scalable secure aggregation of authorized linear functions.