Finite-Key Security Analysis of MP-QKD under Trojan-horse Attacks

Abstract Trojan-horse attack (THA) pose a serious threat to the practical security of quantum key distribution by leaking source-encoding information through back-reflected probe light. Although mode-pairing QKD (MP-QKD) can overcome the conventional rate–distance limit, its finite-key security under THA has not been systematically investigated. In this work, we develop a unified framework for analyzing the finite-key security of decoy-state MP-QKD with source-side leakage. Based on the reference-state method and the Cauchy–Schwarz inequality, we introduce a leakage parameter to quantify the deviation between practical emitted states and ideal reference states, and derive statistical constraints that enable valid decoy-state parameter estimation. We further obtain an explicit and computable finite-key secret-key-rate formula by incorporating statistical fluctuation analysis. Numerical simulations show that both side-channel leakage and finite-size effects reduce the achievable secret-key rate and maximum secure transmission distance, while the degradation becomes less significant for weak leakage or a sufficiently large number of transmitted pulses. These results provide a rigorous basis for evaluating and improving the practical security of MP-QKD systems against Trojan-horse attacks.