Fundamental Limits of Hierarchical Secure Aggregation with Cyclic User Association

Secure aggregation is motivated by federated learning (FL) where a cloud server aims to compute an averaged model (i.e., weights of deep neural networks) of the locally-trained models of numerous clients, while adhering to data security requirements. Hierarchical secure aggregation (HSA) extends this concept to a three-layer hierarchical network, where clustered users communicate with the server through an intermediate layer of relays. In HSA, beyond conventional server security, relay security is also enforced to ensure that the relays remain oblivious to the users' inputs (an abstraction of the local models in FL). Existing study on HSA assumes that each user is associated with only one relay, limiting opportunities for coding across inter-cluster users to achieve efficient communication and key generation. In this paper, we consider HSA with a cyclic association pattern where each user is connected to $B$ consecutive relays in a wrap-around manner. We propose an efficient aggregation scheme which includes a message design for the inputs inspired by gradient coding-a well-known technique for efficient communication in distributed computing-along with a highly non-trivial security key design. We also derive novel converse bounds on the minimum achievable communication and key rates using information-theoretic arguments.