## 4 Analysis on the Features of 2D-DPRCML
### The NIST Test  
The National Institute of Standards and Technology (NIST) SP800-22 suite, as an important test tool, is commonly used to assess the randomness of sequences, and it consists of 15 subtests [45]. In these tests, \(p\)-value refers to the degree of randomness of the tested sequence. \(p=1\) denotes that the sequence is perfectly random, and \(p=0\) denotes that the sequence is totally non-random. The significance level \(\alpha\) is typically a decimal number between 0.001 and 0.01. In this paper, we set \(\alpha=0.01\). If the \(p\)-value is higher than 0.01, then the tested sequence is random, and the reliability of the test is 99%. Otherwise, the tested sequence is non-random.  
In our test, \(\mu\) is set to 4, and \(\varepsilon\) is set to 0.375. We let 2D-DPRCML iterate 50,000 times to obtain the test sequence, and then we eliminate the first 10,000 points in the sequence to minimize the influence of starting values. We selected the sequence \(X(x(1),\ x(2),\ \ldots,\ x(31,250))\) produced by the lattice \(x(6,5)\) for testing, and the decimals in the sequence are quantized to 32 bit unsigned numbers as follows:  
\[y(t)=x(t)\times(2^{32}-1), \tag{20}\]  
in which the length of \(Y(y(1),\ y(2),\ldots,\ y(31,250))\) is \(10^{6}\) bits, which is used for testing. Table 3 displays the testing results.  
\begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **Sub-Tests** & \(p\)**-Value** & **Result** \\ \hline
1 & Frequency & 0.547174 & Pass \\
2 & Block frequency & 0.067007 & Pass \\
3 & Runs & 0.617330 & Pass \\
4 & Longest runs of ones & 0.405138 & Pass \\
5 & Rank & 0.305160 & Pass \\
6 & FFT & 0.349265 & Pass \\
7 & Non-overlapping template & 0.463683 & Pass \\  & matching (average) & 0.455667 & Pass \\
8 & Overlapping template matching & & \\  & (all ones) & 0.455667 & Pass \\
9 & Universal statistical & 0.430132 & Pass \\
10 & Linear complexity & 0.605678 & Pass \\  & Serial 1 & 0.953191 & Pass \\
11 & Serial 2 & 0.799506 & Pass \\
12 & Approximate entropy & 0.629571 & Pass \\  & Cumulative sums (forward) & 0.695145 & Pass \\
13 & Cumulative sums (reverse) & 0.883695 & Pass \\
14 & Random excursions & & \\  & \(x=-4\) & 0.486479 & Pass \\  & \(x=-3\) & 0.735904 & Pass \\  & \(x=-2\) & 0.622806 & Pass \\  & \(x=-1\) & 0.82144 & Pass \\  & \(x=1\) & 0.527469 & Pass \\  & \(x=2\) & 0.413015 & Pass \\  & \(x=3\) & 0.604139 & Pass \\  & \(x=4\) & 0.631688 & Pass \\
15 & Random excursions variant & & \\  & \(x=-9\) & 0.520664 & Pass \\  & \(x=-8\) & 0.552538 & Pass \\  & \(x=-7\) & 0.715424 & Pass \\  & \(x=-6\) & 0.802102 & Pass \\ \hline \end{tabular}
\end{table}
Table 3: Test results of NIST 800-22.  
It can be seen from Table 3 that the 2D-DPRCML's output sequence passes all the subtests. In addition, we also test the sequences generated by the lattices \(x(1,3)\), \(x(5,4)\) and \(x(10,7)\), and all of them pass the randomness test. For comparison purposes, we performed a pass frequency-based test [45] on the sequences produced by the four systems, and the results are presented in Table 4.  
In Table 4, it is clear that the first three systems' output sequences rarely pass the frequency test, which is the reason why the sequences generated by these chaotic systems cannot be directly used in cryptosystems [12,46]. In general, when chaotic systems are applied to cryptosystems, bit-extraction algorithms and nonlinear functions are commonly used to obtain better randomness and uniformity. However, the proposed 2D-DPRCML system exhibits excellent chaotic properties even without using the above methods, which is a good indication of its promising applications in cryptography.