## 3 Introduction of 2D-DPRCML  
The 2D-PECA is employed for dynamic pseudo-random coupling and perturbing the proposed 2D-DPRCML system. We assign two ECAs in 2D-PECA as \(x\) and \(y\), respectively. A single ECA contains \(M\) cells. The 2D-DPRCML system contains \(R~{}\times~{}L\) lattices. The proposed system is described as follows:  
\[ \begin{array}{ll}x_{n+1}(i,j)&=\big{\{}(1-\varepsilon)f(x_{n}(i,j))+\frac{ \varepsilon}{4}[f(x_{n}(a_{1},b_{1}))+f(x_{n}(a_{1},b_{2}))+f(x_{n}(a_{2},b_{1} ))\\ &\qquad\qquad\qquad\qquad\qquad\qquad+f(x_{n}(a_{2},b_{2}))]+p_{n}\times\delta _{n}(i,j)\big{\}}\text{mod}1,\end{array}  \tag{8}\]  
\[n=\text{floor}(\frac{t}{R\times L}+1), \tag{9}\]  
\[\left\{ \begin{array}{l}a_{1}=(\text{bin2 dec}(S^{t}_{x}(c_{1}c_{2}c_{3}\cdots c _{32}))\text{mod}R)+1\\ a_{2}=(\text{bin2 dec}(S^{t}_{x}(c_{33}c_{34}c_{35}\cdots c_{64}))\text{mod}R)+ 1\\ b_{1}=(\text{bin2 dec}(S^{t}_{y}(c_{1}c_{2}c_{3}\cdots c_{32}))\text{mod}L)+1\\ b_{2}=(\text{bin2 dec}(S^{t}_{y}(c_{33}c_{34}c_{35}\cdots c_{64}))\text{mod}L)+1 \end{array} \right., \tag{10}\]  
Figure 1: The Partitioned Elementary Cellular Automata. \[\left\{ \begin{array}{l}a_{2}=((a_{1}+1)\text{mod}R)+1,\text{if}\ a_{1}=a_{2}\\ b_{2}=((b_{1}+1)\text{mod}L)+1,\text{if}\ b_{1}=b_{2}\end{array} \right. \tag{11}\]  
In Equation (8), \(i=1,\,2,\,3,\,\ldots,\,R\) and \(j=1,\,2,\,3,\,\ldots,\,L\) denote the system's row and column indexes, respectively. Both \(n\) and \(t\) stand for time series. In addition, \(n=n+1\) when all the lattices are iterated once. \(t=t+1\) when 2D-PECA is iterated once. From Equation (8), we can find that the updated value of each lattice is influenced by itself and the four coupling lattices during the iteration of the system. Such a process occurs in all lattices in each iteration. In other words, the coupling process enables the influence of each lattice to gradually spread to the other lattices as the system iterates, which can be regarded as the mutual perturbation between the individual lattices. This perturbation enables the system to have a more complex dynamical behavior. The relationship between \(n\) and \(t\) is shown in Equation (9), where the floor (\(x\)) corresponds to the largest integer less than or equal to \(x\). \(\varepsilon\) (\(\varepsilon\in(0,\,1)\)) stands for the coupling coefficient. A modulo operation is denoted by mod and returns the remainder of a division. \(x\) mod 1 is associated with obtaining the decimal component of \(x\), and keeps the outcome of Equation (8) in the interval (0, 1). Moreover, \(a_{1}\), \(a_{2}\), \(b_{1}\) and \(b_{2}\) are pseudo-randomness coordinates of coupled lattices calculated in Equation (10), whereby \(a\) and \(b\) represent the row and column indexes, respectively. \(S_{x}^{t}\) and \(S_{y}^{t}\) are iterative results of \(x\) and \(y\) in the 2D-PECA separately at \(t\) times. \(c_{i}\) is the \(i\)th cell in an ECA, and its value is in {0,1}, thus the iterative result \(c_{1}c_{2}c_{3}\,\ldots\,c_{M}\) can be expressed as a binary number as in Equations (10) and (12), or as a vector in GF(2) as in Equation (13). bin2 dec(\(\bullet\)) is the conversion from binary to decimal. In addition, the purpose of Equation (11) is to handle the conflict. \[p_{n}=\left\{ \begin{array}{l}\frac{\text{bin2 dec}(S_{x}^{t}(c_{M-31}c_{M-30} c_{M-32}\cdots c_{M}))}{2^{2^{2}}-1},\text{if}n\in\text{odd}\\ \frac{\text{bin2 dec}(S_{y}^{t}(c_{M-31}c_{M-30}c_{M-32}\cdots c_{M}))}{2^{2^{2 }}-1},\text{if}n\in\text{even}\end{array} \right., \tag{12}\]  
\[\delta_{n}=[S_{x}^{t}(c_{1},c_{2},c_{3},\cdots,c_{R})^{T}\times S_{y}^{t}(c_{1 },c_{2},c_{3},\cdots,c_{L})-0.5]\times 2 \tag{13}\]  
Equation (12) is the formula for calculating the perturbation \(p_{n}\), in which the operation is determined by the time \(n\). The result of \(p_{n}\) is also in the interval (0, 1). There are two reasons for designing different expressions for the perturbation coefficient for even and odd clock cycles: the first is to improve data utilization.