## 4 Simulation and Results
The comparisons are conducted based on three important evaluation parameters: the number of information leakages \(L_{rec}\) (%), the efficiency metric of information reconciliation \(\eta_{IR}\), and the number of communication rounds required during the information reconciliation step. For every observed QBER value, the proposed scheme generates the total amount of syndrome information in both the primary and additional rounds, leading to minimal practical information leakages \(L_{rec}\) during the information reconciliation step. As depicted in Figure 6a, \(L_{rec}\) of the proposed scheme closely approaches the theoretical limit. These information leakages influence the efficiency metric of information reconciliation (\(\eta_{IR}\)), as detailed in Equation (23). The proposed scheme also achieves an efficiency metric \(\eta_{IR}\) closer to the perfect information reconciliation (\(\eta_{IR}=1\)) than the other methods, as illustrated in Figure 6b. Due to the superior accuracy and minimal variability in QBER estimation with the syndrome-based method, the proposed scheme effectively determines and adapts the optimal code rate \(R_{C}^{(opt)}\). Consequently, it requires fewer communication rounds during both the syndrome-based estimation and the rate-adaptive reconciliation  
Figure 6: Performance comparison of (**a**) the number of information leakages (\(L_{rec}\)), represented as a percentage; (**b**) the efficiency metric of information reconciliation ( \(\eta_{IR}\)); and (**c**) the number of communication rounds during the information reconciliation step. In these simulations, the proposed rate-adaptive reconciliation with syndrome-based QBER estimation and subblock confirmation is evaluated and compared to other methods, including Cascade, blind, and symmetric blind. The simulation results are obtained from an average of 2000 iterations for every observed QBER point. processes, compared to both blind and symmetric blind reconciliation methods, as depicted in Figure 6c. However, the interactive reconciliation in both blind and symmetric blind methods, which can operate without a priori QBER estimation, typically require more communication rounds to ensure successful decoding. In the case of Cascade, an average of more than 40 communication rounds are required for a sifted key frame of \(10^{4}\) bits [12] to reconcile the error bits using the dichotomic search algorithm. Figure 7 presents the success rate and frame error rate (FER) of the proposed scheme, based on the simulation results of 2000 iterations for each observed QBER point. In the primary round (which adopts LDPC codes with block lengths of 64,800 bits), the rate-adaptive reconciliation achieves an average success rate of approximately 99.93% and exhibits an average FER of 7.25 \(\times\)\(10^{-4}\) for the entire observed QBER range. However, relying solely on the primary round of the rate-adaptive reconciliation does not guarantee the integrity of the identical key between the two legitimate parties. Consequently, the proposed scheme integrated rate-adaptive reconciliation with the subblock confirmation, where the reconciled key from the primary round was divided into subblocks of size \(N_{sb}\) = 16,200 bits. Subsequently, polynomial-based hashing was employed to detect unverified subblocks, which were then processed in the additional rounds using the syndrome-based estimation and rate-adaptive reconciliation with LDPC codes of block length \(N_{sb}\). After the completion of the primary and additional rounds, the simulation results of the proposed scheme demonstrate a 100% success rate with an FER of zero for the 2000 iterations, as illustrated in Figure 7. These results guarantee a 100% convergence probability when the proposed rate-adaptive reconciliation is integrated with syndrome-based error estimation and subblock confirmation within this unified procedure.