Hierarchical Secure Aggregation with Heterogeneous Security Constraints and Arbitrary User Collusion

In hierarchical secure aggregation (HSA), a server communicates with clustered users through an intermediate layer of relays to compute the sum of users' inputs under two security requirements -- server security and relay security. Server security requires that the server learns nothing beyond the desired sum even when colluding with a subset of users, while relay security requires that each relay remains oblivious to the users' inputs under collusion. Existing work on HSA enforces homogeneous security where titall inputs must be protected against titany subset of potential colluding users with sizes up to a predefined threshold. Such a homo formulation cannot capture scenarios with tithet secty reqs where diff users may demand various levels of protection. In this paper, we study hierarchical secure aggregation (HSA) with heterogeneous security requirements and arbitrary user collusion. Specifically, we consider scenarios where the inputs of certain groups of users must remain information-theoretically secure against inference by the server or any relay, even if the server or any relay colludes with an arbitrary subset of other users. Under server security, the server learns nothing about these protected inputs beyond the prescribed aggregate sum, despite any such collusion. Under relay security, each relay similarly obtains no information about the protected inputs under the same collusion model. We characterize the optimal communication rates achievable across all layers for all parameter regimes. Furthermore, we study the minimum source keys required at the users to ensure security. For this source key requirement, we provide tight characterizations in two broad regimes determined by the security and collusion constraints, and establish a general information-theoretic lower bound together with a bounded-gap achievable scheme for the remaining regime.