# A Novel Intermittent Jumping Coupled Map Lattice Based on Multiple Chaotic Maps
## 3 Analysis of Dynamic Behaviors
### Inter-Lattice Independence Analysis  
Some CML-based image cryptosystems may use multiple lattices simultaneously for driving the permutation and diffusion phases. In such circumstance, the higher the independence between two lattices, the stronger the ability to combat information leakage. In this paper, a Mutual-Information-based Entropy Distance (MIED) is defined to measure the independence between two quantized sequences. The mathematical formula is:  
\[MIED(\textbf{c}^{i},\textbf{ c}^{j})=MAX-I(\textbf{c}^{i};\textbf{ c}^{j}), \tag{16}\]  
where the superscript \(i\) (or \(j\)) stands for the lattice's index, while the symbol \(MAX\) is a constant, representing the maximum possible value of \(H(\textbf{c}^{i})\) (or \(H(\textbf{c}^{j})\)). In this simulation study, we quantize a state variable into ten levels as before, so that \(MAX\)\(=log_{2}10\approx 3.3219\). In Equation (16), \(I(\textbf{c}^{i};\textbf{ c}^{j})\) is the mutual information between \(\textbf{c}^{i}\) and \(\textbf{c}^{j}\), and its definition is written as:  
\[I(\textbf{c}^{i};\textbf{ c}^{j})=\sum_{c^{i}=0}^{9}\sum_{c^{j}=0}^{9}p(c^{i},\,c^{j})log_{2}\frac{p(c^{i},\,c^{j})}{p(c^{i})\cdot p(c^{j})}. \tag{17}\]  
The nonnegativity of mutual information ensures that \(0\leq MIED(\textbf{c}^{i},\textbf{ c}^{j})\leq MAX\) holds for any pair of lattices \(i\) and \(j\). The upper and lower bounds, namely \(MAX\) and \(0\), correspond to the maximal and minimal degrees of independence, respectively. Figure 7 shows the resulting values of MIED for the inter-lattice independence analysis, in which four combinations of settings for \(\mu\) and \(e\), namely \(\{3.49,\,0.1\}\), \(\{3.49,\,0.9\}\), \(\{3.99,\,0.1\}\) and \(\{3.99,\,0.9\}\), are considered. When \(\mu=3.49\), the values of MIED, in general, form a middle-level plane containing fluctuations of various patterns. Interestingly, for the NCML system, the punctate leakages are expanded to box-like pits as \(e\) increases. This is because a greater coupling coefficient usually enhances the correlations between a pair of lattices \(i\) and \(j\), thereby yielding a higher value of \(I(\textbf{c}^{i};\textbf{ c}^{j})\). Moreover, as shown in the second column of Figure 7b, the box-like pits are arranged with a regular period. This arises from the fact that the Arnold cat map, which is used in the NCML system, forces some pairs of lattices to interact with each other repeatedly during the iterative procedure. Unfortunately, this defect is inherited by the NDCML system, as shown in Figure 7d. When \(\mu=3.99\), the values of MIED constitute a plateau lying close to _MAX_. The groove along the principal diagonal corresponds to the self-independence of a specific lattice, namely MIED(\(\mathbf{c}^{i}\), \(\mathbf{c}^{i}\)), where \(i=1,\ 2,\cdots,\ L\). Clearly, a flat plateau with a narrow groove indicates good performance of the inter-lattice independence. For the CML system, increasing \(e\) reinforces the interactions between two lattices, and thus results in a wider groove and four sunken regions around the corners. A similar phenomenon also happens in the DCML system, implying that a dynamical coupling coefficient is insufficient to eliminate the coupling-caused correlations. For the NCML and NDCML systems, there exist multiple diagonal grooves embedded into the flat plateau due to the use of Arnold cat map. From the first two columns of Figure 7e, we find that the IJCML system can reach higher value of MIED when \(\mu=3.49\) and \(e=0.9\). More significantly, the IJCML system forms the flattest plateau with a unit-width groove when \(\mu=3.99\). These comparisons demonstrate that the IJCML system has a larger parameter space for generating the independent sequences randomly. Figure 7: Resulting values of MIED for the inter-lattice independence analysis. The four settings of \(\{\mu,e\}\), from left to right, are \(\{3.49,\ 0.1\}\), \(\{3.49,\ 0.9\}\), \(\{3.99,\ 0.1\}\), and \(\{3.99,\ 0.9\}\), respectively.