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 an non-trusted server and is secure against HBC clients, preventing any client from recovering the local updates of other participants. Numerical results on MNIST validate the theoretical analysis.

翻译：同态加密（HE）通过在无需解密的情况下允许服务器对加密数据进行操作，实现了联邦学习（FL）中的隐私保护聚合。现有的空中计算（OTA）同态加密方法主要依赖单密钥HE方案，并需要信道估计或预均衡来补偿无线信道衰落。然而，单密钥HE仍易受持有共享密钥的诚实但好奇（HBC）客户端的攻击，而多密钥HE通过为每个设备分配独立密钥提供了更强的客户端级安全性。我们提出了一种四阶段协议，可在无需信道估计的情况下通过共享无线信道实现xMK-CKKS的聚合。该协议通过相同的信道实现重传部分公钥和密文，使得主导的大模数加密项在解密过程中通过代数方式相互抵消。我们将此协议与慢变视距主导信道下的零阶FL相结合，其中每轮每个设备仅传输单个加密标量，且通信/加密开销与模型维度无关。研究表明，加密和无线聚合引入的残余噪声将标准收敛速率\(O(1/\sqrt{K})\)保持至可忽略的本底噪声水平，其中\(K\)为通信轮数。该协议假设服务器不可信，且能抵御HBC客户端攻击，防止任何客户端恢复其他参与者的本地更新。MNIST上的数值结果验证了理论分析。