  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.
