Finite key analysis for discrete phase randomized BB84 protocol

Quantum key distribution (QKD) is a secure communication method that relies on the inherent randomness of quantum mechanics to ensure information-theoretic security. The first and most widely used QKD protocol is BB84, and the proof of BB84’s security is vital. The discrete phase randomized BB84 protocol is a variant of the decoy BB84 protocol. It has been proven to be promising in the development of high-speed QKD systems. However, it still lacks an analysis with a finite number of pulses. This paper presents a comprehensive security analysis of the discrete phase BB84 protocol, using two different methods under different conditions. The analysis involves simulations and optimizations to determine the optimal parameter settings. It is confirmed that for a small number of finite pulses, i.e., \(10^7\) ${10}^7$ , if the number of discrete phases exceeds 30, one can calculate the key rate by assuming that a continuous phase randomization process was in operation. On the other hand, for a relatively smaller number of discrete values, i.e., 16 discrete phases, we have developed a numerical method to calculate the key rate. We have confirmed that its performance is reduced but still acceptable with a finite number of pulses.