## 2 Complex Modified Projective Difference Function Synchronization (CMPDFS)  
The general form of a controlled coupled complex chaotic system can be expressed by \(n\) dimensional differential equations as follows:  
\[L1:\left\{ \begin{array}{lcl}\dot{\mathbf{y}}&=&\mathbf{g}(\mathbf{y},\mathbf{z} ),\\ \dot{\mathbf{z}}&=&\mathbf{p}(\mathbf{y},\mathbf{z}),\\ L2:\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\mathbf{z})+\mathbf{v},\end{array} \right. \tag{1}\]  
where \(\mathbf{y}=(y_{1},y_{2},\cdots,y_{q})^{\mathrm{T}}\) and \(\mathbf{z}=(z_{1},\,z_{2},\,\cdots,\,z_{n-q})^{\mathrm{T}}\) are the observable complex state vectors of the drive system \(L1\). The state vector \(\mathbf{z}\) is treated as the coupling vector. \(\mathbf{x}=(x_{1},\,x_{2},\,\cdots,\,x_{q})^{\mathrm{T}}\) is the complex state vector of the response system \(L2\), which is controllable and reconstructed using the controller \(\mathbf{v}\). The designed controller is represented by \(\mathbf{v}=\mathbf{v}^{\prime}+j\mathbf{v}^{\prime}\), where \(\mathbf{v}^{\prime}=(v_{1}^{\prime},\,v_{2}^{\prime},\,\cdots,v_{n}^{\prime}) ^{\mathrm{T}}\) is the real part of a complex variable, and \(\mathbf{v}^{i}=(v_{1}^{i},\,v_{2}^{i},\,\cdots,\,v_{n}^{i})^{\mathrm{T}}\) is the real part of a complex variable. **Definition 1**. : _With regard to two state vectors \(\mathbf{x}(t)\), \(\mathbf{y}(t)\) and a desired difference function vector \(\mathbf{H}(t)\), if the square of error satisfies_  
\[\lim_{t\rightarrow+\infty}\|\mathbf{e}(\mathbf{t})\|^{2} = \lim_{t\rightarrow+\infty}\|\mathbf{x}(\mathbf{t})-\mathbf{H}( \mathbf{t})-\mathbf{Dy}(\mathbf{t})\|^{2} \tag{2}\] \[= \lim_{t\rightarrow+\infty}\|\mathbf{x}^{\prime}(t)-\mathbf{H}^{ \mathrm{t}}(\mathbf{t})-\mathbf{D}^{\prime}\mathbf{y}^{\prime}(t)+\mathbf{D}^ {i}\mathbf{y}^{i}(t)\|^{2}\] \[+ \|\mathbf{x}^{\prime}(t)-\mathbf{H}^{\mathrm{t}}(\mathbf{t})- \mathbf{D}^{\prime}\mathbf{y}^{i}(t)-\mathbf{D}^{\prime}\mathbf{y}^{\prime}(t )\|^{2}\] \[= 0,\]  
_where \(\|\cdot\|\) represents the Euclidean norm, then \(\mathbf{x}(\mathbf{t})\) and \(\mathbf{y}(\mathbf{t})\) will reach CMPDFS with \(\mathbf{H}(t)\) and modified projective matrix \(\mathbf{D}\). \(\mathbf{H}(t)=\{h_{1}(t),h_{2}(t),\cdots,h_{q}(t)\}^{\mathrm{T}}\) is a bounded vector, and \(h_{1}(t)\), \(h_{2}(t)\), \(\cdots\), \(h_{q}(t)\) are difference function factors. \(h_{1}(t):C\to C\) (\(l=\ 1,\,2,\cdots,\,q\)) are bounded complex functions. \(\mathbf{D}=diag\{d_{1},d_{2},\cdots,d_{q}\}\), \(d_{l}\in C\) are bounded complex numbers._  
In [14], [PERSON] and [PERSON] discussed the phase synchronization (PHS) of two CVCSs, and designed a special controller to hold the error at a constant value. CS signifies that there is no difference between the two state variables while PHS indicates that the differences are constant. Therefore, the following remarks can be made. **Remark 1**. : \(CS\) _is a special case of CMPDFS with \(\mathbf{H}(t)=0\) and \(\mathbf{D}(t)=1\)._  
**Remark 2**. : _PHS (under controller in [14]) is a special case of CMPDFS with \(\mathbf{H}(t)=Constant\) and \(\mathbf{D}(t)=1\)._  
**Remark 3**. : _MDFS is a special case of CMPDFS with \(\mathbf{D}(t)=1\)._ \begin{table}
\begin{tabular}{c c} \hline
**The Symbol** & **Meanings** \\ \hline
**e** & The error vector \\ \(\mathbf{H}(t)\) & The difference function vector \\
**D** & The modified projective matrix which is the private key \\
**s** & The transmitted signal without noise \\ \(\mathbf{s}^{\prime}\) & The transmitted signal with noise \\ \(\mathbf{A}(\varepsilon)\) & The noise where A is the amplitude of the noise \\ \(\mathbf{H_{g}}(t)\) & The recovered signal \\
**k** & The control strength matrix \\ \(\mathbf{w}(\varepsilon)\) & The probability density function of Gaussian distribution \\ \(E_{H^{\prime}}\) & The square of error in the real part of the recovered signal \\ \(E_{H^{\prime}}\) & The square of error in the imaginary part of the recovered signal \\ \hline \end{tabular}
\end{table}