  We present a real-time and fully integrated quantum random number generator
(QRNG) by measuring laser phase fluctuations. The QRNG scheme based on laser
phase fluctuations is featured for its capability of generating ultra
high-speed random numbers. However, the speed bottleneck of a practical QRNG
lies on the limited speed of randomness extraction. To close the gap between
the fast randomness generation and the slow post-processing, we propose a
pipeline extraction algorithm based on Toeplitz matrix hashing and implement it
in a high-speed field-programmable gate array. Further, all the QRNG components
are integrated into a module, including a compact and actively stabilized
interferometer, high-speed data acquisition, and real-time data post-processing
and transmission. The final generation rate of the QRNG module with real-time
extraction can reach 3.2 Gbps.
