# A Chaotic System with Infinite Equilibria and Its S-Box Constructing Application
## 4 S-Box Generation Algorithm and Its Performance Analysis
```
1:Start
2:i = 1; sbox = [];
3:Entering system parameters \(\rightarrow(a=1.5,b=0.5,c=5)\)
4:Entering initial condition \(\rightarrow(x_{0}=0.5,y_{0}=1,z_{0}=0.5)\)
5:Determination of the appropriate value of \(\Delta\)h(0.05)
6:Solving the chaotic system using RK-4 algorithm and obtaining time series
7:while (i \(<\) 257) do
8:\(x=mod(fix(rem(ys(1),1)*10^{3}),256)\);
9:\(z=mod(fix(rem(ys(3),1)*10^{3}),256)\);
10:\(decnum=bitxor(x,z)\)
11:if (Is there decnum in S-Box = yes) then
12: Go step 7. 13:else [Is there decnum in S-Box = no]
14: sbox[i] \(\leftarrow\) decnum
15: i++;
16:endif
17:endwhile
18:sbox \(\leftarrow\) reshape(sbox,16,16)
19:Implementation of S-Box Performance Tests
20:Ready to use 16*16 chaos based S-Box
21:End
```  
**Algorithm 1** S-Box generator algorithm pseudo code  
Tables 1 and 2 show S-Boxes generated by the S-Box algorithm. The S-Box 1 presented in Table 1 is generated using the \(y\) and \(z\) phases of the chaotic system. The production of S-Box 2 presented in Table 2, \(x\) and \(z\) phases are preferred. Analysis of Nonlinearity, Strict avalanche criterion (SAC), Output bits independence criterion (BIC), Differential Approximation Probability (DP) and Linear Approximation Probability (LP) analyses are performed on the proposed S-Boxes. The performance test results are summarized in Table 3. In addition to the proposed S-Box performance results, the results of some S-Box studies in the literature are also given in Table 3.  
Nonlinearity [58] is one of the most important criteria within the S-Box evaluation criteria. It is found that the nonlinearity values of the proposed S-Box 1 are 106, 104, 104, 104, 106, 108, 110 and 108 while the nonlinearity values of the proposed S-Box 2 are 108, 106, 106, 106, 106, 106, 108, and 108. As can be seen from Table 3, the proposed S-Box 1 nonlinearity max, min, avg values are 110, 104 and 106, respectively. The nonlinearity max, min and avg values are 108, 106 and 106, respectively. The AES S-Box nonlinearity value has the best value in the literature. Other studies in the literature are attempting to produce S-Boxes with equal or better nonlinearity values. When the performance test results in Table 3 are compared, it is seen that proposed S-Boxes have the best max, avg and min values after the AES S-Box. One of the tests used in the performance evaluation of S-Boxes is the independence of output bits method developed by [PERSON] and [PERSON] [59]. In this method, S-Box is evaluated on two criteria. The BIC-SAC value refers to the relationship between the independence of the output bits and the Strict avalanche criterion. BIC-Nonlinearity examines the relation between the independence of output bits and nonlinearity. BIC-SAC and BIC-Nonlinearity values of the proposed S-Boxes in Table 3 are given. When these values and the values of the S-Boxes in the literature are examined, it is seen that the proposed S-Boxes have a good BIC-SAC value and have a BIC-Nonlinearity value lower than the AES BIC-Nonlinearity value but better than the other studies in the literature. S-Boxes has a good value in the literature studies after the [PERSON] and [PERSON] [62] algorithm S-Boxes. As a result, when all the performance test results in Table 3 are evaluated, it has been found that S-Boxes generated by the developed S-Box algorithm produce better results in many criteria compared to the literature. According to the test results, it has been shown that the proposed S-Boxes can be used for cryptographic application.