## 2 Theoretical Framework and Simulation Experiment Setup
In Equation (13), the phases \(\varphi_{1k}^{j}\) and \(\varphi_{2k}^{j}\) are respectively written as follows:  
\[\varphi_{1k}^{j}=\arcsin\Bigl{[}\text{Re}(IQ_{k}^{j})\Bigr{]},\quad\varphi_{2k }^{j}=\arcsin\Bigl{[}\text{Im}(IQ_{k}^{j})\Bigr{]}, \tag{14}\]  
where the terms \(IQ_{k}^{1}\)-\(IQ_{k}^{4}\) are respectively described as  
\[IQ_{k}^{1}=\cos\left(\frac{2\pi\cdot n_{k}^{1}}{2^{n}}\right)+i \cdot\sin\left(\frac{2\pi\cdot n_{k}^{1}}{2^{m}}\right), \tag{15}\] \[IQ_{x,k}^{j_{n}}=\frac{\sum\limits_{l=1}^{m/2}2^{m/2-l}\Bigl{(}2 \cdot l_{(k-1)\cdot m/2+l}^{j_{n}}-1\Bigr{)}+i\cdot\frac{m/2}{l}2^{m/2-l} \Bigl{(}2\cdot Q_{(k-1)\cdot m/2+l}^{j_{n}}-1\Bigr{)}}{2^{m/2-1}}, \tag{16}\]where the superscript \(j_{n}=2\), \(3\), \(4\). The variables \(n_{k}^{1}\), \(I\) and \(Q\) can be expressed as follows: \(n_{k}^{1}=\sum_{m}^{l=1}b_{(k-l)\cdot m+l}^{1}\cdot 2^{l-1}\), \(I_{\ell}^{i_{n}}=b_{2\ell-1}^{i_{n}}\) and \(Q_{\ell}^{i_{n}}=b_{2\ell}^{i_{n}}\), where \(\ell=1\), \(2\), \(\cdots\), \(N\). In Equations (9) and (10), the slowly varying amplitudes \(E_{inj}^{1}\)-\(E_{inj}^{4}\) of the complex electric field can be described as [15]  
\[ \begin{split} E_{inj}^{1}(t)&=\sqrt{S_{Cx}(t)\cdot I _{d1}},\quad E_{inj}^{2}(t)=\sqrt{S_{Gy}(t)\cdot I_{d2}},\\ E_{inj}^{3}(t)&=\sqrt{S_{Ex}(t)\cdot I_{d3}},\quad E _{inj}^{4}(t)=\sqrt{S_{Ey}(t)\cdot I_{d4}},\end{split}  \tag{17}\]  
where the light intensities \(I_{d1}=|E_{inj,0}^{1}|^{2}\), \(I_{d2}=|E_{inj,0}^{2}|^{2}\), \(I_{d3}=|E_{inj,0}^{3}|^{2}\) and \(I_{d4}=|E_{inj,0}^{4}|^{2}\). The terms \(E_{inj,0}^{1}\), \(E_{inj,0}^{2}\), \(E_{inj,0}^{3}\) and \(E_{inj,0}^{4}\) are the amplitudes of the continuous-wave lasers \(\text{CW}_{1}\text{--CW}_{4}\), respectively. The masked input signals \(S_{Gx}(t)\), \(S_{Gy}(t)\), \(S_{Ex}(t)\) and \(S_{Ey}(t)\) can be expressed as  
\[ \begin{split}& S_{Gx}(t)=\text{Mask}_{x}(t)\times C_{Gx}(u-L_{Gx}) \times\gamma,S_{Gy}(t)=\text{Mask}_{y}(t)\times C_{Gy}(n-L_{Gy})\times\gamma, \\ & S_{Ex}(t)=\text{Mask}_{x}(t)\times C_{Ex}(n-L_{Ex})\times\gamma,S_{Ey}(t)=\text{Mask}_{y}(t)\times C_{Ey}(n-L_{Ey})\times\gamma,\end{split}  \tag{18}\]  
where the masked signals \(\text{Mask}_{x}(t)\) and \(\text{Mask}_{y}(t)\) are chaotic signals, as presented in [24]. \(\gamma\) is a scaling factor. The discrete channel delay lengths \(L_{Gx}\), \(L_{Ex}=\tau_{x}/h\), and \(L_{Gy}\), \(L_{Ey}=\tau_{y}/h\), where \(h\) is the step size, \(\tau_{x}\) is the channel delay of the GS-PC\({}_{x}\) or ES-PC\({}_{x}\), and \(\tau_{y}\) is the channel delay of the GS-PC\({}_{y}\) or ES-PC\({}_{y}\). In such a system presented in Figure 2, chaos synchronization between each pair of PCs (i.e., GS-PC\({}_{x}\) and GS-PC\({}_{x}^{\prime}\), GS-PC\({}_{y}\) and GS-PC\({}_{y}^{\prime}\), ES-PC\({}_{x}^{\prime}\) and ES-PC\({}_{x}^{\prime}\), and ES-PC\({}_{y}^{\prime}\)) plays a key role in in security and encrypted message recovery. In the following, we use four parallel RCs to address chaos synchronization between each pair of PCs. According to lag chaotic synchronization theory, the lag synchronization solution is obtained as follows. \[ \begin{split} y_{Gx}^{\prime}(n)&=C_{Gx}(n-L_{Gx}),\quad y_{Gy}^{\prime}(n)=C_{Gy}(n-L_{Gy}),\\ y_{Ex}^{\prime}(n)&=C_{Ex}(n-L_{Ex}),\quad y_{Ey} ^{\prime}(n)=C_{Ey}(n-L_{Ey}),\end{split}  \tag{19}\]  
where the time-dependent outputs \(y_{Gx}^{\prime}\), \(y_{Gy}^{\prime}\), \(y_{Ex}^{\prime}\) and \(y_{Ey}^{\prime}\) are respectively regarded as linear functions of the GS-PC\({}_{x}\), GS-PC\({}_{y}\), ES-PC\({}_{x}\) and ES-PC\({}_{y}\) such that  
\[ \begin{split} y_{Gx}^{\prime}(n)&=\quad W_{Gx,1}b_{ out}+W_{Gx,2}C_{Gx}(n-L_{Gx})+\sum_{i=1}^{N}W_{Gx,i+2}I_{Gx,i}(n),\\ y_{Gy}^{\prime}(n)&=\quad W_{Gy,1}b_{out}+W_{Gy,2}C_{Gy }(n-L_{Gy})+\sum_{i=1}^{N}W_{Gy,i+2}I_{Gy,i}(n),\\ y_{Ex}^{\prime}(n)&=\quad W_{Ex,1}b_{out}+W_{Ex,2}C_{ Ex}(n-L_{Ex})+\sum_{i=1}^{N}W_{Ex,i+2}I_{Ex,i}(n),\\ y_{Ey}^{\prime}(n)&=\quad W_{Ey,1}b_{out}+W_{Ey,2}C_{ Ey}(n-L_{Ey})+\sum_{i=1}^{N}W_{Ey,i+2}I_{Ey,i}(n),\end{split}