## 3 Results and Discussions
This is attributed to the fact that the \(NMSE_{Gx}\) and \(NMSE_{Gy}\) for the GS-PC\({}_{x}\) and GS-PC\({}_{y}\) are lower compared to the \(NMSE_{Gx}\) and \(NMSE_{Gy}\) for the ES-PC\({}_{x}\) and ES-PC\({}_{y}\), respectively. Here, by optimizing some key parameter values of the reservoir QD spin-VCSEL, \(\rho_{Gx}\), \(\rho_{Gy}\), \(\rho_{Ex}\) and \(\rho_{Ey}\) are taken as \(0.9856\), \(0.9851\), \(0.9495\) and \(0.9581\), respectively. The optimized parameters are as follows: \(T=10\) ps; \(\theta=40\) fs; \(\eta=4\); \(k_{f}=30\) ns\({}^{-1}\); and \(k_{inj}=30\) ns\({}^{-1}\). By performing high-quality chaos synchronization between each pair of PCs (i.e., GS-PC\({}_{x}\) and GS-PC\({}^{\prime}_{x}\), GS-PC\({}_{y}\) and GS-PC\({}^{\prime}_{y}\), ES-PC\({}_{x}\) and ES-PC\({}^{\prime}_{x}\), and ES-PC\({}^{\prime}_{y}\)) using the reservoir QD spin-VCSEL, one of the messages QPSK, 4 QAM, 8 QAM and 16 QAM can be decoded by synchronously dividing a reservoir-generated chaos and a delayed chaos masked message. The temporal traces of the delayed encoding message (\(S_{1}(n-L_{Gx})\), or QPSK), the delayed chaos masked message (\(U_{Gx}(n-L_{Gx})\)), and the decoding message (\(S^{{}^{\prime}}_{1}(n)\)) are displayed in Figure 7(a\({}_{1}\)-a\({}_{3}\)). As observed from Figure 7(a\({}_{1}\)-a\({}_{3}\)), the temporal trajectory of \(S_{1}(n-L_{Gx})\) is very similar to that of (\(S^{{}^{\prime}}_{1}(n)\)). Furthermore, \(U_{Gx}(n-L_{Gx})\) exhibits a chaotic state. Figure 7(a\({}_{4}\)-a\({}_{6}\)) present the temporal trajectories of \(S_{2}(n-L_{Gy})\) (4 QAM), \(U_{Gy}(n-L_{Gy})\) and (\(S^{{}^{\prime}}_{2}(n)\)). As seen from these figures, the temporal trajectory of \(S_{2}(n-L_{Gy})\) is basically identical to that of (\(S^{{}^{\prime}}_{2}(n)\)), while \(U_{Gy}(n-L_{Gy})\) shows a chaotic state. Moreover, as displayed in Figure 7(a\({}_{7}\)-a\({}_{12}\)), the temporal trajectories of \(S_{3}(n-L_{Ex})\) (8 QAM) and \(S_{4}(n-L_{Ey})\) (16 QAM) are almost the same as those of (\(S^{{}^{\prime}}_{3}(n)\)) and (\(S^{{}^{\prime}}_{4}(n)\)), respectively. \(U_{Ex}(n-L_{Ex})\)and \(U_{Ey}(n\!-\!L_{Ey})\) both exhibit a chaotic state. Moreover, as in Figure 8, we present the eye-diagrams for these four decoded messages (\(S_{1}^{{}^{\prime}}(n)\), \(S_{2}^{{}^{\prime}}(n)\), \(S_{3}^{{}^{\prime}}(n)\) and \(S_{4}^{{}^{\prime}}(n)\)). One sees from this figure that the \"eyes\" sizes of the eye-diagrams of these decoded messages are enough large, indicating that the decoded messages of the system have a relatively large tolerance error for noise and jitter and have good quality. However, the superposition of multiple decoded messages causes the signal line of each eye-diagram to become thicker and appear fuzzy. The reason is that very small synchronization errors may be converted into noise and superimposed on the signal line of the eye-diagram. These results indicate that the encoding messages QPSK, 4 QAM, 8 QAM and 16 QAM can be effectively masked in a chaotic carrier and successfully recovered using reservoir computing. The bit error rate (\(BER\)) is a commonly utilized metric to gauge the quality of data transmission in optical chaos-based secure communication systems [15]. The \(BER\) is defined as the ratio of the number of errored bits to the overall number of bits transmitted. Figure 9 showcases the dependences of the \(BER\)s for the decoded messages (\(S_{1}^{{}^{\prime}}(t)\), \(S_{2}^{{}^{\prime}}(t)\), \(S_{3}^{{}^{\prime}}(t)\) and \(S_{4}^{{}^{\prime}}(t)\)) and their associated baseband signals (\(b_{1}^{{}^{\prime}}\), \(b_{2}^{{}^{\prime}}\), \(b_{3}^{{}^{\prime}}\) and \(b_{4}^{{}^{\prime}}\)) on two key parameters (\(k_{inj}\) and \(k_{f}\)). As evidenced by Figure 9(a\({}_{1}\),a\({}_{2}\)), the \(BER\)s for \(S_{1}^{{}^{\prime}}(t)\), \(S_{2}^{{}^{\prime}}(t)\), \(S_{3}^{{}^{\prime}}(t)\) and \(S_{4}^{{}^{\prime}}(t)\) exhibit oscillatory behavior as \(k_{inj}\) is adjusted within the range of 0.1 ns\({}^{-1}\) to 50 ns\({}^{-1}\).