# A Chaotic System with Infinite Equilibria and Its S-Box Constructing Application
## 4 S-Box Generation Algorithm and Its Performance Analysis  
Chaotic systems are widely used in cryptographic studies because of their high randomness, rich dynamical properties and high sensitivity to initial conditions [48; 49; 50; 51; 52; 53; 54; 55]. S-Box is one of the most main structures used in block cipher algorithms [56]. S-Box structures with strong cryptographic features make cryptography very resistant to attack. In the literature, there are studies using different methods for S-Box production [21; 23; 24; 26; 27; 57]. S-Box structures are one of the most important components used in encryption algorithms. S-Box operations provide confusion as a nonlinear component in encryption algorithms and thus a powerful S-Box structure with good performance values is very important for strong encryption. The sub-byte operation is performed by using S-Box. In this study, high random number generation capabilities of the chaotic system are used in the design of S-Box algorithm. In this way, S-Boxes are produced with higher performance characteristics. The advantage of the proposed method provided is that the S-Box structures are obtained with less processing load. Numerous S-Boxes can be produced using different initial values and parameters. As a result of performance tests, it is shown that S-Boxes with high performance characteristics will be used in cryptographic operations for secure communication. S-Box structures are used in different applications for secure communication systems, such as personal area communication, wireless sensor network and medical device connectivity. The S-Box can be used as a component or key expansion operations in encryption algorithms. In addition, S-Box structures are frequently used in text, image, video encryption and embedded system applications. In this section, a new S-Box generation algorithm is presented with a light processing load using the introduced chaotic system with infinite equilibria. Performance tests are performed on the S-Boxes generated by the S-Box algorithm. The performance evaluation of the generated S-Boxes is made by comparing with the reported S-Boxes in the literature. Algorithm 1 shows the pseudo code of the S-Box generation algorithm. Following the entry of the initial conditions and system parameters of the chaotic system in the S-Box generation algorithm, the appropriate sampling step is determined and the chaotic system is solved by the RK-4 numerical analysis method. The float values are obtained by solving the chaotic system. Until the 256 unique values required for the S-Box are generated, the system continues to generate values. By applying \(rem\) and \(fix\) operations on the float numbers obtained from the chaotic  
Figure 8: Measured phase portraits of the real circuit in (**a**) \(X-Y\) plane, (**b**) \(X-Z\) plane, and (**c**) \(Y-Z\) plane. system, the value in the first three digits of the generated float value is obtained after the comma. The obtained values are maintained in the 0-255 range by applying \(mod\) 256 operation. The values (first three digits after the comma) obtained from the \(x\) and \(z\) phases are subjected to \(bitxor\) processing to obtain a new decimal value (\(decnum\)). If the new decimal value is already generated in the S-Box, this value is discarded. If it is not in the S-Box, it is added to the S-Box. In this way, a unique 256 value generation is provided in the range of 0-255. The generated 256-element S-Box is converted into \(16\times 16\) matrix and S-Box performance tests are applied. In the pseudo code in Algorithm 1, the generation is given to be performed over the \(x\) and \(z\) phases. In practice, however, the generation of S-Box is carried out using different surpluses and tests are carried out. Two S-Boxes with good test results are proposed and performance test results are given in our work.