Privacy-Enhanced Zero-Order Federated Learning via xMK-CKKS over Wireless Channels

Homomorphic encryption (HE) enables privacy-preserving aggregation in federated learning (FL) by allowing the server to operate on encrypted data without decryption. Existing HE-over-the-air (OTA) methods mainly rely on single-key HE schemes and require channel estimation or pre-equalization to compensate for wireless fading. However, single-key HE remains vulnerable to honest-but-curious (HBC) clients holding the shared secret key, while multi-key HE provides stronger client-level security by assigning each device its own secret key. We propose a four-phase protocol that enables the aggregation of xMK-CKKS over a shared wireless channel without channel estimation. The protocol retransmits partial public keys and ciphertexts through the same channel realization, so that the dominant large-modulus encryption terms cancel algebraically during decryption. We integrate this protocol with zero-order FL over slowly varying LoS-dominant channels, where each device transmits a single encrypted scalar per round and the communication/encryption overhead is independent of the model dimension. We show that the residual noise induced by encryption and wireless aggregation preserves the standard convergence rate O(1/sqrt K) up to a negligible noise floor, where K is the number of communication rounds. The protocol assumes a non-trusted server and is secure against HBC clients, preventing any client from recovering the local updates of other participants. Numerical results on MNIST and CIFAR-10 validate the theoretical analysis.