# Prototyping the Symmetry-Based Chaotic Communication System Using Microcontroller Unit
## 2 Materials and Methods  
### Symmetry Coefficient Modulation and Parameter Modulation  
Signal modulation is a crucial feature of communication systems, and the progress of CCSs is inextricably connected with developing novel efficient types of chaotic signal modulation. One of the recently reported techniques for CCS realization is symmetry coefficient modulation (SCM) [11], while the most known modulation type is still parameter-based modulation (PM). The main difference between SCM and parameter modulation is that SCM does not require changing the bifurcation parameter, and, therefore, appears to provide a more concealed data transfer. Information is transmitted by changing the coefficient \(s\) (so-called symmetry coefficient) of the discrete chaotic map, obtained by applying the second-order self-adjoint semi-implicit numerical method [11; 13] to the continuous prototype system. A block diagram of a chaotic communication system based on the PM and SCM approaches is shown in Figure 1. The working principles of PM and SCM are, nevertheless, similar and can be described as follows. The digital signal \(m(t)\) modulates one or more parameters (or the symmetry coefficient for the SCM case) of the transmitting generator \(X(t)\). The presence or absence of synchronization on the receiver side determines the transmitted information symbol (bit). After obtaining the difference between the received signal and the synchronized oscillator \(\Delta=X(t)-X^{*}(t)\), the recovered message \(m^{*}(t)\) can be detected by a simple envelope detector [13]. There are many synchronization techniques known from the literature. However, for simplicity, we implemented the chaos-based communication system using the classical Pecora-Carrol method [24] in this study. As a continuous prototype for discrete chaos generators on the transmitter and receiver side, the well-known Lorenz chaotic system [25]  
Figure 1: Block diagram of the communication system with parameter modulation (PM)/symmetry coefficient modulation (SCM), where \(a_{1}\) or \(s_{1}\) and \(a_{2}\) or \(s_{2}\) are the parameter or symmetry coefficient values corresponding to a certain binary symbol during binary message transmission [13]. was chosen. The Lorenz system can be described by the following system of ordinary differential equations:  
\[ \begin{split}\dot{x}&=\sigma(y-x);\\ \dot{y}&=x(r-z)-y;\\ \dot{z}&=xy-bz;\end{split}  \tag{1}\]  
where \(\sigma=10\), \(r=28\), and \(b=2.3\). Let us apply the Pecora-Carrol synchronization to the (1) to obtain the finite-difference scheme of the slave oscillator:  
\[ \begin{split}\dot{x}&=\sigma(y-x);\\ \dot{y}&=x(r-z)-y+k(y_{M}-y);\\ \dot{z}&=xy-bz;\end{split}  \tag{2}\]  
where \(k\) is the coupling strength coefficient and \(y_{M}\) is the second variable of master system. From the previous works [26; 27], it is known that the optimal sync variable for the Lorenz system is \(y\) and the value of coupling strength \(k=40\) provides the fastest synchronization. The finite-difference model for the master Lorenz system was obtained using the variable symmetry composite diagonal (VSCD) [11; 28] integration method. The finite difference schemes obtained with the VSCD method imply the composition of two first-order conjugate methods applied with \(h\cdot s\) and \(h\cdot(1-s)\) integration steps, respectively:  
\[ \begin{split} h_{1}&=hs\\ h_{2}&=h(1-s)\end{split}  \tag{3}\]  
\[ \begin{split} x_{n+s}&=x_{n}+h_{1}(\sigma(y_{n}-x_{n} )).\\ y_{n+s}&=y_{n}+h_{1}(x_{n+s}(r-z_{n})-y_{n});\\ z_{n+s}&=z_{n}+h_{1}(x_{n+s}y_{n+s}-bz_{n});\\ z_{n+1}&=z_{n+s}+h_{2}(x_{n+s}y_{n+s}-bz_{n+1});\\ y_{n+1}&=y_{n+s}+h_{2}(x_{n+s}(r-z_{n+1})-y_{n+1});\\ x_{n+1}&=x_{n+s}+h_{2}(\sigma(y_{n+1}-x_{n+1})). \end{split}  \tag{4}\]  
The implicitness in Equation (4) can be easily resolved analytically:  
\[