Repeated Monte Carlo evaluation of Eve’s effective bit-error probability in free-space quantum key distribution (FSO-QKD) is computationally demanding, while conventional regression models may produce values outside the admissible interval (0 < P_e < 0.5). This paper proposes a structurally bounded closed-form surrogate that combines a scaled sigmoid function with polynomial representations of the legitimate-link QBER and Eve’s radial offset. The resulting models use only eight and six parameters and achieve test (R^2) values of 0.9847 and 0.9898 for the two evaluated attack modes, respectively. Over 100,000 sampled operating points, the proposed surrogate produces no output-range violations, whereas the unconstrained polynomial yields violation rates of 11.99% and 10.89%. It also exhibits greater stability under input perturbations. The proposed model therefore provides a compact, accurate, and probability-preserving approximation of the reference Monte Carlo simulator. The estimated (P_e) is an auxiliary receiver-level metric rather than a complete QKD security measure.

Quantum key distribution (QKD),surrogate modeling,free-space optical (FSO),bounded regression,eavesdropping analysis