Secure bound of quantum key distribution with nonuniform random seed in privacy amplification

Abstract

Privacy amplification is an essential step in quantum key distribution for distilling a secure key, and it conventionally assumes that the random numbers used to select the hash function are strictly uniformly distributed. The impact of nonuniform random numbers in privacy amplification on secure bounds has rarely been discussed. The quantum leftover hash lemma is expanded to the case of nonuniform random numbers and its security proved on the basis of the min-entropy, giving a corrected secure bound that carries an explicit penalty of twice the gap between the seed length and the seed min-entropy. Simulation indicates that the secure bound is seriously affected when the average min-entropy of the random numbers falls below 0.95, and measurements of four commonly used random number generators show that they do not achieve perfect min-entropy.