## 4 Performance Tests of the Proposed PRNG
### Key Sensitivity  
In order to evaluate the sensitivity of the proposed PRNG to small changes in the keys, we use the following metrics: the number of changing pixel rate (NPCR), the unified average changed intensity (UACI) [84] and the average absolute difference (AAD) [85]. For these tests, we select a set of 1000 keys, so that \(K_{i}\) and \(K_{i+1}\) (\(i=1,2,3,\,...,\,1000\)) differ by a single bit between them. According to the structure of the proposed PRNG, we assume in the key for the proposed PRNG that only \(z_{0}\) changes in the least significant bit (LSB), and this small change is quantified by \(\eta_{0}\). Therefore, we select each initial condition considering that \(z_{0}^{t+1}=z_{0}^{t}+\eta_{0}\), where \(t=1,2,3,...,1000\). Now, to calculate the NPCR, UACI, and ADD, we use Equations (13), (15) and (16), respectively, considering two cases. That is, the pseudorandom sequences are analyzed by reading 16-bits or 8-bits numbers.  
In this way, NPCR is represented by Equation (13).  
\[NPCR(p,q)=\frac{1}{n}\sum_{i=0}^{n}D_{i}(p,q), \tag{13}\]  
where,  
\[D(p,q)=\left\{\begin{array}{ll}0\;\;if\;\;S_{i}^{p}=S_{i}^{q}\\ 1\;\;if\;\;S_{i}^{p}\
eq S_{i}^{q}\end{array}\right., \tag{14}\]  
UACI is represented by Equation (15).  
\[UACI(p,q)=\frac{1}{n}\left[\sum_{i=0}^{n}\left(\frac{\left|S_{i}^{p}-S_{i}^{q} \right|}{2 bits-1}\right)\right], \tag{15}\]  
where \(bits=8\) when the pseudorandom sequences are read in 8-bit numbers, and \(bits=16\) when they are read in 16-bit numbers.  
And, AAD is represented by Equation (16).  
\[AAD(p,q)=\frac{1}{n}\sum_{i=1}^{n}|S_{i}^{p}-S_{i}^{q}|, \tag{16}\]  
Importantly, pseudorandom sequences of 1,000,000 numbers were generated for NPCR and UACI, and sequences of 2,000,000 numbers were generated for AAD. In all cases, the sequences are made up of 16-bit numbers. Figure 12 shows the statistical distribution for NPCR(\(p\),\(q\)), UACI(\(p\),\(q\)), and AAD(\(p\),\(q\)). Figure 12a,c,e show NPCR\({}_{16}\)(\(p\),\(q\)), UACI\({}_{16}\)(\(p\),\(q\)),  
Figure 11: Statistical distribution for the different correlation coefficients.  
ADD\({}_{16}\)(_p,q_) when the pseudorandom sequences are read in 16-bits numbers. On the other hand, Figure 12b,d,f show NPCR\({}_{8}\)(_p,q_), UACI\({}_{8}\)(_p,q_), ADD\({}_{8}\)(_p,q_) when the sequences are read in 8-bits numbers. Note that, when the pseudorandom sequences are observed as sequences with 16-bits numbers, the all calculated values for the NPCR(_p,q_) are 0.999977, for the UACI(_p,q_) the mean value is 0.33349, and for the AAD(_p,q_) the mean value is \(2.18455\times 10^{4}\). In a similar way, when the pseudorandom sequences are observed as sequences with 8-bits numbers, the calculated mean of the NPCR(_p,q_) is very close to 0.99608, for UACI(_p,q_) is close to 0.33460, and for AAD(_p,q_) is 85.3289, which is very close to ideal value reported by Wang et al. in 2016 [52].  
Figure 12: Statistical distribution for the sensitivity metrics estimated from 1000 pseudorandom sequences considering that the PRNG keys are very close to each other: (**a**) NPCR\({}_{16}\)(_p,q_), (**b**) NPCR\({}_{8}\)(_p,q_), (**c**) UACI\({}_{16}\)(_p,q_), (**d**) UACI\({}_{8}\)(_p,q_), (**e**) AAD\({}_{16}\)(_p,q_), and (**f**) AAD\({}_{8}\)(_p,q_).