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This paper introduces a four-phase protocol enabling privacy-enhanced zero-order federated learning (FL) using the xMK-CKKS multi-key homomorphic encryption scheme over wireless channels without channel estimation. The protocol leverages retransmission of partial public keys and ciphertexts to cancel large-modulus encryption terms during decryption, achieving secure aggregation. Theoretical analysis and experiments on MNIST demonstrate that the proposed method preserves the \(O(1/\sqrt{K})\) convergence rate while providing security against a colluding server and up to \(N-1\) clients.
Secure federated learning over wireless is now possible without channel estimation, thanks to a clever retransmission scheme that algebraically cancels encryption noise.
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 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 clients sharing the same secret key. In addition, compromising a single client may compromise the security of the entire network, while multi-key HE schemes provide stronger client-level security by assigning each device its own secret key. We propose a four-phase protocol that enables xMK-CKKS, a famous multi-key HE scheme, aggregation 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 prove that the decoded encryption noise preserves the \(O(1/\sqrt{K})\) convergence rate up to a negligible noise floor. The protocol is secure against an honest-but-curious server colluding with up to \(N-1\) clients, and numerical results on MNIST validate the analysis.