  Random number plays a key role in information science, especially in
cryptography. Based on the probabilistic nature of quantum mechanics, quantum
random number generators can produce genuine randomness. In particular, random
numbers can be produced from laser phase fluctuations with a very high speed,
typically in the Gbps regime. In this work, by developing a physical model, we
investigate the origin of the randomness in quantum random number generators
based on laser phase fluctuations. We show how the randomness essentially stems
from spontaneous emissions. The laser phase fluctuation can be quantitatively
evaluated from basic principles and also qualitatively explained by the
Brownian motion model. After taking account of practical device precision, we
show that the randomness generation speed is limited by the finite resolution
of detection devices. Our result also provides the optimal experiment design in
order to achieve the maximum generation speed.
