We demonstrate a fiber-optic source of random numbers in which random-number generation (RNG) is implemented via an inherently quantum process—detection of correlated photon pairs produced by spontaneous four-wave mixing (FWM) inside an optical fiber. The detection times of individual photon pairs, generated in our RNG scheme via phase-matched FWM in a highly nonlinear photonic-crystal fiber with a suitable dispersion profile, are converted into megabit-size binary sequences, serving as strings of random numbers. The randomness of these bit strings is verified by the National Institute of Standards and Technology Statistical Test Suite. We identify FWM regimes in which mode-locked short-pulse laser sources can provide a high peak power needed to drive photon-pair generation in optical fibers without an excessive loss of randomness due to unwanted patterns in the RNG photon-pair output.