## 3 CMPDFS Communication Scheme for WSNs
### The Communication Scheme for WSNs  
Here, we propose a secure communication scheme based on CMPDFS for WSNs. The diagram of the proposed communication scheme is presented in Figure 2. The master system \(L1\) produced by one sensor node represents the sender end. The slave system \(L2\), which is constructed by the sink node, represents the receiver end. The coupled signal \(\mathbf{z}\), produced by the sender, is transmitted to the receiver. \(H\) is the message signal, and \(D\) is the proportional matrix. Figure 2: The diagram of communication scheme based on CMPDFS. Considering the noise in the communication channel, we obtain the actual transmitted signal  
\[\mathbf{s}^{\prime}(t) = \mathbf{s}(t)+A\varepsilon(t) \tag{7}\] \[= \mathbf{H}(t)+\mathbf{D}\mathbf{g}(\mathbf{y}(t),\mathbf{z}(t))+A \varepsilon(t)\]  
where \(A\varepsilon(t)\) represents the noise in a given channel generated by all noise sources and \(A\) denotes the amplitude of the noise. The communication process comprises two parts. First, the initial value of the coupled signal \(\mathbf{z}(0)\) is broadcasted to the sink node. For communication, \(\mathbf{z}(0)\) is applied to provide the sink node with a command to begin sending the information. For synchronization, \(\mathbf{z}(0)\) is used to construct the decryptor at the sink node. In the second part, after receiving the correct return signal \(\mathbf{z}(0)\) from the sink node, the sensor node begins to encrypt the message signal \(\mathbf{H}(t)\) (which is the plaintext in fact) and produces the encrypted message \(\mathbf{s}(t)\). It is sent to the sink node with noise (we denote \(\mathbf{s}^{\prime}(t)\)=\(\mathbf{s}(t)+A\varepsilon(t)\)). The encryption method of the sender end consists of both chaotic encryption and chaotic masking. At the receiving end, the controller \(\mathbf{v}\) is designed using Equation (5) which includes \(\mathbf{s}=\mathbf{s}^{\prime}-A\varepsilon(t)\) (we filter \(\mathbf{s}^{\prime}\) and obtain the effective transmitted signal \(\mathbf{s}\)). As CMPDFS is initiated, \(\mathbf{x}(t)\) synchronizes \(\mathbf{H}(t)+\mathbf{D}\mathbf{y}(t)\). Therefore, the recovered signal is \(\mathbf{H}_{g}(t)=\mathbf{H}(t)=\mathbf{x}(t)-\mathbf{D}\mathbf{y}(t)\). The CMPDFS communication scheme is different from traditional chaotic masking in a few key ways. Firstly, the method for recovering the signal is different. The recovered signal in our communication scheme based on CMPDFS is \(\mathbf{H}_{g}=\mathbf{x}-\mathbf{D}\mathbf{y}\), while the recovered signal in traditional chaotic masking based on CS is \(\mathbf{H}_{g}=\mathbf{s}^{\prime}-\mathbf{y}\). As \(\mathbf{s}^{\prime}(t)\) includes noise, the recovered signal in traditional chaotic masking is inherently inaccurate. Traditional chaotic masking also requires that the amplitude of the chaotic signals always largely outweigh the amplitude of plaintext and noise. In the proposed communication scheme, if the amplitude of chaotic signals is insufficient, we can select a large proportional matrix \(\mathbf{D}\), which is large enough to outweigh the amplitude of plaintext and noise. Therefore, the accuracy of the recovered signal in the proposed method is higher than traditional chaotic masking and the BER is theoretically zero. Secondly, the actual transmitted signal is \(\mathbf{s}^{\prime}(t)=\hat{\mathbf{H}}+\mathbf{D}\mathbf{g}(\mathbf{y}, \mathbf{z})+A\varepsilon(t)\), which is the sum of all noise sources, the derivative of the plaintext (message signal), and the function of the state variables. Chaotic encryption, chaotic masking, and noise masking are developed to increase communication security. Any state variable from the sender end can be chosen. Additionally, \(\mathbf{s}^{\prime}(t)\) is a complex variable signal and includes the complicated calculations of complex numbers, which provide more than twice the security of real variables. Moreover, it is possible to transmit two message signals by utilizing the real part and imaginary part concurrently.