In this paper, we investigate the beam training problem in the multi-user millimeter wave (mmWave) communication system, where multiple reconfigurable intelligent surfaces (RISs) are deployed to improve the coverage and the achievable rate. However, existing beam training techniques in mmWave systems suffer from the high complexity (i.e., exponential order) and low identification accuracy. To address these problems, we propose a novel hashing multi-arm beam (HMB) training scheme that reduces the training complexity to the logarithmic order with the high accuracy. Specifically, we first design a generation mechanism for HMB codebooks. Then, we propose a demultiplexing algorithm based on the soft decision to distinguish signals from different RIS reflective links. Finally, we utilize a multi-round voting mechanism to align the beams. Simulation results show that the proposed HMB training scheme enables simultaneous training for multiple RISs and multiple users, and reduces the beam training overhead to the logarithmic level. Moreover, it also shows that our proposed scheme can significantly improve the identification accuracy by at least 20% compared to existing beam training techniques.