Small imperfect randomness restricts security of quantum key distribution

Quantum key distribution protocol requires true random numbers to prepare and measure quantum states, but the randomness may be weakened by applying the imperfect input random numbers or imperfect quantum devices. In this work, we propose the randomness attack model in the practical quantum key distribution system, and we analyze the final secret key rate under the randomness attack. The security analysis result demonstrates that the imperfect randomness can be amplified by considering the signal loss, thus the quantum key distribution system may be attacked even if the randomness has very small part of imperfection. Combining the practical experimental parameters with the small nonrandom probability parameters $0.1%$ and $1%$, we prove that the maximal transmission distance will be sharply reduced from 140 km to 60 and 12 km, respectively.