# A Novel Method for Constructing the S-Box Based on Spatiotemporal Chaotic Dynamics
## 2 Preliminaries  
The CML system [29] is presented as:  
\[y_{t}(s+1)= (1-\alpha)\mu y_{t}(s)(1-y_{t}(s))+\frac{\alpha}{2}\{\mu y_{a}(s)( 1-y_{a}(s))+\mu y_{c}(s)(1-y_{c}(s))\}, \tag{1}\]  
where \(s\) is the time parameter (\(s=1\), \(2\), \(3\), \(\ldots\)), \(\alpha\) is the coupling coefficient (\(0\leq\alpha\leq 1\)), \(\mu\in(0,4]\), \(l\), \(a\) and \(c\) are the lattices (\(1\leq l\), \(a,c\leq N\)), \(a=l+1,c=l-1\), and \(N\) is a count of all the lattices. The CML system is a spatiotemporal chaotic dynamics, which is coupled in adjacent lattices.  
Different from the CML chaotic dynamics, the coupling method of the NCML chaotic system is non-adjacent. The NCML system [28] is presented as:  
\[y_{l}(s+1)= (1-\alpha)\mu y_{l}(s)(1-y_{l}(s))+\frac{\alpha}{2}\{\mu y_{m}(s)(1-y_ {m}(s))+\mu y_{n}(s)(1-y_{n}(s))\}, \tag{2}\]  
where \(m,n\) are also the lattices (\(1\leq m,n\leq N\)) and the relations of \(l\), \(m\), \(n\) satisfy the Arnold cat map expressed by:  
\[\begin{bmatrix}m\\ n\end{bmatrix}=\begin{bmatrix}1&b\\ d&bd+1\end{bmatrix}\begin{bmatrix}l\\ l\end{bmatrix}(\text{mod }N). \tag{3}\]  
The NCML system has good chaotic features when the parameters \(\mu\), \(b\), and \(d\) are assigned with proper values.  
1. Lyapunov exponents evaluate the divergence of nearby orbits and provide a qualitative view of the dynamical system. If a system has at least one positive Lyapunov exponent, the system is certainly in chaotic behaviors. The Lyapunov exponent of each lattice in the NCML system is positive, and between 0.0203 and 0.4351. This means that the NCML system's dynamics is more complicated, which can resist the degradation under finite precision computation in modern computers.
2. In the bifurcation diagram of the NCML system, there is no periodic window, as shown in Figure 1a. By contrast, there are periodic windows in the bifurcation diagram of the CML system, as shown in Figure 1b. Therefore, the NCML system is more suitable for cryptography than the CML system due to no periodic windows.
3. The chaotic trajectory of NCML system is as random as that of the CML system, as shown in Figure 2. The random chaotic sequences are suitable for the construction of a random S-box.
4. The NCML chaotic system has very small mutual information values between chaotic trajectories. Mutual information can be used to evaluate the independence of two chaotic trajectories (named \(s1\) and \(s2\)), which is defined as: \[I(s1,s2)=H(s1)-H(s1/s2).\] (4)  
The lower value of mutual information of \(s1\) and \(s2\) means the higher independence of chaotic trajectories. As shown in Figure 3b, most of the mutual information values are equal to zero, which indicates that most of the chaotic trajectories generated by the N  
Figure 1: Bifurcation diagram. (**a**) The Non-adjacent Coupled Map Lattices (NCML) chaotic system; (**b**) the CML chaotic system.  
independent and cannot be recovered by other trajectories. For the CML chaotic system, its mutual information values between chaotic trajectories are indicated in Figure 3a. It can be noted that the NCML chaotic system holds smaller mutual information values between chaotic trajectories than the CML chaotic system. The independent chaotic trajectories are suitable for cryptography because they cannot be restored by other trajectories.