Federated learning (FL) enables collaborative model training across distributed nodes without exposing raw data, but its decentralized nature makes it vulnerable in trust-deficient environments. Inference attacks may recover sensitive information from gradient updates, while poisoning attacks can degrade model performance or induce malicious behaviors. Existing defenses often suffer from high communication and computation costs, or limited detection precision. To address these issues, we propose LSHFed, a robust and communication-efficient FL framework that simultaneously enhances aggregation robustness and privacy preservation. At its core, LSHFed incorporates LSHGM, a novel gradient verification mechanism that projects high-dimensional gradients into compact binary representations via multi-hyperplane locally-sensitive hashing. This enables accurate detection and filtering of malicious gradients using only their irreversible hash forms, thus mitigating privacy leakage risks and substantially reducing transmission overhead. Extensive experiments demonstrate that LSHFed maintains high model performance even when up to 50% of participants are collusive adversaries while achieving up to a 1000x reduction in gradient verification communication compared to full-gradient methods.