Along with the evolution of wireless technologies, users are expected to have high-throughput and low-latency indoor mobile applications, such as augmented reality and virtual reality. Therefore, the use of millimeter-wave for low-latency communications becomes attractive. Under low-latency criteria, a fast multi-link establishment is desired. In this paper, we present a novel concurrent multi-beam training jointly with a proposed multi-beam codebook. First, we propose a progressive multi-beam estimation (PMBE). This algorithm makes multiple coarse beams emerge within a few training steps. Second, we propose a Fast Fourier Transform (FFT)-based codebook which can be considered as a Discrete Fourier Transform based codebook built in a bit-reversal scrambler, thus forming a scattering multi-beam codebook. This FFT-based codebook design is optimally implemented on a sub-connected architecture, a hardware-efficient architecture. For fast multi-link establishment, we design our concurrent multi-beam training by combining the proposed PMBE jointly with the proposed FFT-based codebook, so as to further improve multiplexing gain. Since multiplexing gain can occur immediately, even under strictly-limited training steps, the high data rate can be achieved. Simulation results show that when reaching more than 95% capacity gain of exhaustive channel estimation, our work only spends 1% training steps of the exhaustive case.