Finite-key analysis of decoy model Semi-quantum key distribution based on four-state protocol

Semi-quantum key distribution allows a full quantum user Alice and a classical user Bob to share a pair of security keys guaranteed by physical principles. Semi-quantum key distribution is proposed while verifying its robustness. Subsequently, its unconditional security of semi quantum key distribution system is verified theoretically. In 2021, the feasibility of semi quantum key distribution system based on mirror protocol was verified experimentally. However, the feasibility experimental system still uses the laser pulse with strong attenuation. It has been proved in the literature that the semi-quantum key distribution system still encounters the risk of secret key leakage under photon number splitting attack. Therefore, the actual security of key distribution can be further reasonably evaluated by introducing the temptation state and conducting the finite-key analysis in the key distribution process. In this work, for the model of adding one-decoy state only to Alice at the sending based on a four state semi-quantum key distribution system, the length of the security key in the case of finite-key is analyzed by using Hoeffding inequality, and then the formula of the security key rate is obtained. It is found in the numerical simulation that when the sample size is \begin{document}$ {10}^{5} $\end{document}, the security key rate of \begin{document}$ {10}^{-4} $\end{document}, which is close to the security key rate of the asymptotic limits, can be obtained in the case of close range, It is very important for the practical application of semi quantum key distribution system.