## 2 Preliminaries
### Partitioned Elementary Cellular Automata  
The Cellular Automata (CA) [35] is a dynamic model with discreteness in the time and space dimensions. The mathematical expression of a CA is shown as:  
\[CA=(A^{N},\sum f,E), \tag{4}\]  
in which \(A^{N}\) denotes a cell space with \(N\) dimension. \(\sum\) is a discrete finite set to represent the states of the cell. \(f\) is a mapping function generally referred to as the local transition rule. \(E\) represents the set of neighbors of the cell. ECA is a simple CA in one dimension [36]. In the ECA, each cell has two state values: 0 and 1, meaning that \(\sum\) is denoted as \([0,1]\). When the radius of CA is equal to 1, the \(i-1\)th cell and the \(i+1\)th cell can be considered as the \(i\)th cell's neighbors. Furthermore, the periodicity of the ECA's border condition is indicated as follows:  
\[\left\{ \begin{array}{l}i+1=1,i=M\\ i-1=M,i=1\end{array} \right., \tag{5}\]  
in which we designate \(M\) as the total quantity of cells. The result after an iteration of a cell in ECA is jointly influenced by the three cells in the current iteration, which are this cell itself and its two neighbors. It can be formulated as follows:  
\[S_{t+1}(i)=f_{r}(S_{t}(i-1),S_{t}(i),S_{t}(i+1)), \tag{6}\]  
in which \(S_{t}(i)\) stands for the state value of the \(i\)th cell, and the \(t\) in it refers to the time dimension. \(f\) represents a Boolean state function that specifies a local transition rule \(r\), implementing a mapping from the set of states \(\{111,110,101,100,011,010,001,000\}\) to the set of states \(\{0,1\}\), so it can be inferred that there are 256 rules in total. For instance, when \(r=102,f\) can be represented in Table 1.  
In Table 1, the value of \(S_{t+1}(i)\) is '011000110', which can be expressed as a decimal as '102'. Previous researchers have categorized all ECAs into five types [34; 37; 38; 39]: null rules, fixed-point rules, periodic rules, locally chaotic rules and global chaotic rules. We construct the 2D-DPRCML system using any two ECAs with global chaotic rules. Table 2 lists all the global chaotic rules. The Partitioned Elementary Cellular Automata (PECA) is a high-dimensional dynamic system that consists of at least two ECAs with different local transition rules. The 2D-PECA is composed of two different ECAs, and it can be described by Equation (7):  
\[\left\{ \begin{array}{l}x_{t+1}(i)=f_{rx}[x_{t}(i-1),x_{t}(i),x_{t}(i+1)]\\ y_{t+1}(i)=f_{ry}[y_{t}(i-1),y_{t}(i),y_{t}(i+1)]\end{array} \right., \tag{7}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(S_{t}(i-1)\),\(S_{t}(i)\),\(S_{t}(i+1)\) & **111** & **110** & **101** & **100** & **011** & **010** & **001** & **000** \\ \hline \(S_{t+1}(i)\) & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: The function \(f\) with a local transition rule \(r=102\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Class** & **Rule Number** \\ \hline Global chaotic & 18(183), 22(151), 30(86, 135, 149), 45(75, 89, 101), 60(102, 153, 195), \\  & 90(165), 105, 106(120, 169, 225), 129(126), 137(110, 124, 193), \\  & 146(182), 150, 161(122) \\ \hline \end{tabular}
\end{table} 
Table 2: The global chaotic rules. The rules in parentheses are similar to the representation rules preceding the parentheses. where \(x\) and \(y\) represent two ECAs with different local transition rules \(rx\) and \(ry\), respectively. \(x\) and \(y\) contain \(M\) cells each. \(i\) and \(t\) represent the space and time dimensions, respectively, as in Equation (6). For example, we assign \(rx=102\), \(ry=105\), \(M=100\) and the initial values of two ECAs in 2D-PECA are random. The iterative result of the 2D-PECA is shown in Figure 1. The blue cells refer to cells with a status of 1, and the white cells refer to cells with a status of 0. In Figure 1, both ECAs \(x\) and \(y\) are in chaotic states when the local transition rules are 102 and 105, respectively. It is obvious that they possess pseudo-randomness.