We proposed and numerically studied a method for chaotic synchronization of three-section distributed Bragg reflection (DBR) semiconductor lasers driven by a common noise. The analysis of cross-correlation and correlation dimension shows that a pair of parameter-matched DBR lasers can achieve chaos synchronization when the injection strength of noise light is within 0.34-0.83. More importantly, chaos synchronization is very sensitive to the mismatch of parameters in grating region and phase region of DBR laser, and DBR laser has a large space for hardware parameters. By modulating the current in the grating region, the chaos synchronization on-off keying can be realized, and the synchronization recovery time is only about 4 ns. Using chaos synchronization of DBR laser, we are expected to achieve high-speed and secure chaotic laser key distribution.