**4. Performance Analysis of Proposed Method**

The study is based on a general s-box generator algorithm to examine the effect of the proposed postprocessing technique on the s-box performance. A flowchart of the s-box generator algorithm is given in Figure 3. The details of this algorithm and the program prepared for the Windows operating system can be accessed from [7,14]. Researchers can generate s-box structures using the original method, and verify their performance improvements for new s-box structures modified using the postprocessing technique through the program in [14].

The effect of the proposed method on the performance criteria was analyzed in this section. As explained, there are five basic criteria for s-box performance analysis. The bijective criterion is guaranteed by the proposed method. Therefore, this criterion is not included in the analysis tables. Two main categories can be used to classify chaotic systems. These categories are discrete and continuous-time chaotic systems. Discrete-time systems are first-order difference equations. Continuous-time chaotic systems are at least third-order differential equations [8]. An analysis of six different chaotic systems was carried out using three different chaotic systems for both chaotic system classes. Twenty-five different s-box structures were generated for each chaotic system class. Logistic

map, sine map, and circle map are used as discrete-time chaotic systems. Performance comparisons for original and improvement s-box structures are given in Tables 2–4 respectively. Similarly, performance comparisons for the original and improved s-box structures generated for each of the continuous-time Lorenz, Labyrinth Rene Thomas system, and Chua systems are given in Tables 5–7, respectively. To show the success of the proposed method, care was taken to ensure that the average nonlinearity property of all the original s-box structures used in the analysis was less than 103. Performance improvement was observed in all the s-box structures given in the analysis tables.


**Table 2.** Performance comparisons for original and improved s-boxes based on a logistic map.


**Table 3.** Performance comparisons for original and improved s-boxes based on a sine map.

**Table 4.** Performance comparisons for original and improved s-boxes based on a circle map.



**Table 5.** Performance comparisons for original and improved s-boxes based on a Lorenz system.

**Table 6.** Performance comparisons for original and improved s-boxes based on the Labyrinth Rene Thomas system.



**Table 7.** Performance comparisons for original and improved s-boxes based on a Chua circuit.

The statistical properties of the chaotic data used in the s-box generation process are not included in this section. In [35], it is shown that the performance criteria of the s-box structures to be generated using the data which do not show chaotic behavior may be better than the s-box structures generated from chaotic data. In addition, in the code given in Table 1, the initial condition of the logistic map used as the chaotic system was chosen randomly. In other words, the proposed method provides performance improvement, regardless of the statistical properties of the entropy source. This is another strength of the proposed method.

#### **5. Conclusions**

Chaotic systems will provide various opportunities for cryptology sciences. Among these, a successful design approach is chaos-based s-box designs. However, the fact that chaos-based s-boxes are worse in terms of performance criteria than designs based on mathematical transformations is a serious problem. This problem is addressed in the study. The question of whether performance improvements of chaos-based designs can be achieved using various postprocessing methods was explored. In the study, the zigzag transformation method, which has a very simple structure, was used. It was observed that the proposed method provides performance improvements in chaos-based s-box structures that have performance characteristics that can be evaluated below average. Since the performance criteria of the chaos-based s-box structures are very close to each other, comparisons were made using the nonlinearity measurement, which is a criterion that can reflect the difference in the best way. In a literature review for the s-box, it was observed that the average value for the nonlinearity value is 103. Therefore, care was taken to ensure that the average nonlinearity value of all the s-box values used in the analysis was below 103. In line with these conditions, 150 different s-box structures were generated. The generated s-box structures were obtained from six different chaotic systems selected from two different chaotic system classes. The reason for using different chaotic systems was to show that the proposed method can be successful for all chaotic systems. All these s-box structures are explicitly presented for the examination of other researchers on a web page [39].

If a general evaluation is made, the advantages of the proposed method are listed below.


Despite these advantages, the proposed postprocessing idea should be based on a more robust foundation in future studies. Some possible avenues for future studies are listed below.


**Author Contributions:** F.A. and F.Ö. Wrote and edited the manuscript. All authors have read and agreed to the published version of the manuscript.

**Acknowledgments:** The authors gratefully thank to the Referee for the constructive comments and recommendations which definitely help to improve the readability and quality of the study.

**Conflicts of Interest:** The authors declare no conflict of interest.
