1 Introduction

The rich dynamics of chaos theory have provided many advantages, especially in the field of computer science [1]. One of the outputs of this positive effect is the design of chaos-based encryption algorithms [2, 3]. The use of chaotic systems in the design of substitution box (s-box) structures, which have a critical role in block encryption algorithms, has been a hot research topic [4]. The fact that these designs are an alternative to prevent side channel attacks [5] has led more researchers to focus on the subject. The researchers focused on how to improve the nonlinearity value of s-box structures generated using only chaotic system outputs [6,7,8,9,10,11, 55] to compete with designs based on mathematical transformations [12,13,14,15,16,17,18,19,20, 56, 57] and optimization algorithms [21,22,23,24,25,26,27,28,29,30,31,32,33,34, 44]. It has been tested in the designs proposed in many studies that the highest nonlinearity value that can be reached for chaotic s-box structures can be 106.75 [35,36,37,38]. This upper value for the 20.000 s-box structure has been analyzed and experimentally verified in Ref. [35].

One of the most comprehensive studies on chaos-based s-box structures has been carried out in this study. 10,000 different s-box structures have been generated for each of the seven different chaotic systems. 70,000 s-boxes have been generated in total. Six of these chaotic systems are previously known in the literature. The seventh chaotic system proposed for the first time in this study is an original contribution of the study. Another contribution of the study to the literature is that it has been shown that nonlinearity value can increase the highest known value among s-box structures generated using the newly proposed chaotic system. Two different s-box structures are presented for 107 which is the new highest nonlinearity value. One of them was obtained with the suggested map and the other was obtained with the May map. In this study, these nonlinearity values were obtained by producing a total of 70.000 different s-box structures, 10.000 for each map. However, if this value is increased, for example, if 200.000 s-boxes are produced in this way, it is thought that higher nonlinearity values can be achieved. This can be a source of motivation for researchers. However, when this number increases too much, the complexity of the algorithm approaches optimization-based methods. Side channel analyses have been carried out for these two s-box structures. Analysis results have shown that these designs can be more resistant to application-oriented attacks than mathematically based designs. Among the 70,000 s-box structures, those with a nonlinearity value of 106 and above are shared as a public dataset [60]. The dataset contains 289 different s-box structures. It is thought that this dataset will provide various advantages to cryptologists in various applications such as block ciphers, hash functions, and random number generators. In addition, a random number generator was created with the proposed map. With this map, 1 million bit-long data sets were created and subjected to various tests. The data sets, which passed all tests successfully, showed that the study can also provide various gains for random number generators.

This study was carried out with many motivations. These sources of motivation are listed below and analyzed in detail within the text.

  • To show that more complex chaotic maps can be obtained by combining existing chaotic maps.

  • Using these chaotic maps in s-box generation and creating more effective s-box structures.

  • Increasing the highest nonlinearity value in s-box structures obtained with chaotic maps.

  • To carry out side-channel analysis of the produced s-box structures. Side channel analyses are not available in most other studies in the literature.

  • Creating a large pool of s-boxes obtained through chaotic maps.

  • To show that random numbers can be generated with the outputs of the proposed chaotic map.

The rest of the study is organized as follows. In section 2, mathematical models of chaotic systems used in the study are given. In this section, the starting point and general features of the newly proposed chaotic system are also detailed. In section 3, the details of s-box generator architecture are given. In the fourth section, the results of the random number generator, performance analysis, and comparisons of the generated s-box structures are given. In the last section, the results have been interpreted and suggestions for future studies have been discussed.

2 Proposed new chaotic system

Different chaotic systems such as fractional [39, 40], hyper chaotic [41, 42], and time-delay [43, 44] have been used in the design of chaos-based encryption algorithms. However, studies have shown that the transformation function may be more important than the type of chaotic system [45]. Therefore, the study focused on discrete-time chaotic systems. Because the discrete-time chaotic system has an advantage over other types of chaotic systems thanks to its simple mathematical models. Six different chaotic systems that have been widely studied in the literature have been used in the study. Mathematical models of these systems are given in Eqs. (1)-(6), respectively [46].

Logistic map:

$$\begin{array}{ccc}{x}_{n+1}=a{x}_{n}\left(1-{x}_{n}\right),& {x}_{n}\in \left[\mathrm{0,1}\right],& a\in \left[3.5, 4\right]\end{array}$$
(1)

Tent map:

$$\begin{array}{cc}\begin{array}{cc}{x}_{n+1}=& \left\{\begin{array}{c}\begin{array}{cc}a{x}_{n}& {x}_{i}<0.5\end{array}\\ \begin{array}{cc}a\left(1-{x}_{n}\right)& {x}_{i}\ge 0.5\end{array}\end{array}\right.\end{array},& \begin{array}{cc}{x}_{n}\in \left[\mathrm{0,1}\right],& a\in \left[\mathrm{1,2}\right]\end{array}\end{array}$$
(2)

Sine map:

$$\begin{array}{ccc}{x}_{n+1}=a{\text{sin}}\left(\pi {x}_{n}\right) ,& {x}_{n}\in \left[\mathrm{0,1}\right],& a\in \left[\mathrm{0.85,1}\right]\end{array}$$
(3)

Circle map:

$$\begin{array}{ccc}{x}_{n+1}={x}_{n}+a-\frac{b}{2\pi }{\text{sin}}\left(2\pi {x}_{n}\right)mod1,& {x}_{n}\in \left[\mathrm{0,1}\right],& a\in \left[\mathrm{0,1}\right] , b\in \left[\mathrm{0,4}\pi \right]\end{array}$$
(4)

May map:

$$\begin{array}{cc}{x}_{n+1}={x}_{n}{\text{exp}}\left[a\left(1-{x}_{n}\right)\right],& a\in \left[\mathrm{0,5}\right]\end{array}$$
(5)

Gaussian map

$$\begin{array}{cc}{x}_{n+1}={\text{exp}}\left(-a{{x}_{n}}^{2}\right)+c,& \in \left[\mathrm{4.7,17}\right] , c\in \left[-\mathrm{1,1}\right]\end{array}$$
(6)

New chaotic systems have always been an interesting topic in studies related to chaos theory. Researchers have encountered different chaotic systems while searching for mathematical models that can model systems and processes most effectively in science and engineering studies. Recently, it has been shown that new chaotic systems can be obtained by combining chaotic systems. It has been shown by Wang et al. [47] that a new chaotic system can be obtained by combining the logistic map and the sin map, which have successful applications in many fields. It has been shown over an image encryption algorithm that this chaotic system (I1DS) can produce successful results in information security applications [47]. The success of the proposed encryption architecture has been associated with the complex behavior of the chaotic system. Because the proposed new chaotic system has a more complex structure than the logistic map and the sin map. Lyapunov exponents, a quantitative chaos analysis tool, have shown that the newly proposed chaotic systems contain a more complex entropy source. For example, in Fig. 1, the comparison of Lyapunov exponents of other chaotic systems with the I1DS system is given.

Fig. 1
figure 1

Lyapunov exponent for different chaotic maps [33]

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The fact that a system has a positive Lyapunov exponent is indicative of chaotic behavior. The magnitude of the positive Lyapunov exponent is interpreted as the high complexity of the system. The starting point of the study is based on the logic in Ref. [47]. Can a more complex model be obtained by combining the models in Eqs. (1)-(6)? In this context, using different combinations of expressions in the mathematical models of six chaotic systems, a model that can give the most complex behavior is tried to be obtained. At the end of the experiments, it has been experimentally observed that the richest behaviors can be obtained by using the model in Eq. (7). Then, in this study, it has been investigated whether an advantage can be obtained for cryptographic purposes using this model.

$$\begin{array}{cc}{x}_{n+1}=\frac{\mathit{exp}\left({{x}_{n}}^{2}\right)}{{\text{sin}}(2\pi {x}_{n}a)}+c,& a\in \left[\mathrm{1,5}\right] , c\in \left[-\mathrm{10,10}\right]\end{array}$$
(7)

The simple mathematical model mentioned here is not intended to refer to the chaotic system but to the simplicity of the approach to producing a chaotic system. The motivation for this study is that it can produce more complex chaotic systems. Information regarding this is stated in detail in section 4.2. The most important motivation of the study is to produce structures that can be alternatives to mathematical transformations. Mathematical transformations have better s-box performance metrics. But it is less resistant to side channel attacks, as shown in Table 6. To eliminate this structure, we aim to produce more complex chaotic systems. The method we propose is to bring these chaotic systems together with simple logic. Considering the computational complexity of the method we propose, the chaotic system generation method, it can be stated that it has a structure as simple as O(n). Because a new chaotic system is produced by bringing together pieces from existing alternatives. Therefore, the complexity of the algorithm can be expressed as O(n).

3 Chaos based S-box designs

Important milestones within the scope of cryptanalysis studies are linear and differential cryptanalysis techniques. These studies have been one of the determining factors in the transition from the DES (Data Encryption Standard) block cipher to the AES (Advanced Encryption Standard) block cipher [48]. Since s-box structures are the focus of both attack techniques, researchers have sought techniques that can be used to design s-box structures that are resistant to these attacks. A method has been proposed by Nyberg to meet all design criteria most appropriately. Indeed the AES s-box structure is a marvel of mathematical design. However, the developing technology has revealed new attack techniques and the strong mathematical model (deterministic structure) of the AES s-box structure has shown that it can turn into a disadvantage by using various side channel information [5, 49, 50].

At this point, alternative methods to mathematical transformations have begun to be investigated. It has been proposed to transform nonlinear system outputs into s-box structures. However, the problem arose that these proposals could not meet the design criteria most appropriately. For example, while the nonlinearity value of the AES s-box structure is 112, the highest achievable value for chaos-based designs is calculated as 106.75. It has been suggested to use optimization algorithms to solve this problem. But computational complexity of these design approaches is also considered a problem that needs to be addressed.

In this study, it has been re-investigated how much nonlinearity criteria can be improved for s-box structures to be generated using only chaotic system outputs. Although many studies have shown that the highest nonlinearity value is 106.75, the reason for conducting such a study again is to analyze whether the more complex behavior of the proposed new chaotic system will affect the design metrics positively. For a fair comparison with previous studies, the s-box generator method suggested in Ref. [35, 51] has been used. The flowchart of the method is given in Fig. 2.

Fig. 2
figure 2

Flowchart diagram for s-box generator based on chaotic systems

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4 Analysis results

A total of 70,000 s-box structures are obtained for chaotic systems whose mathematical expressions, initial conditions, and control parameters are given in Eqs. (1)-(7). Among these s-box structures, those with a nonlinearity value of 106 and above were recorded as the dataset. Table 1 shows s-box numbers with 106 and above nonlinearity values for each chaotic system.

Table 1 Number of s-box structures with 106 and above nonlinearity value
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To compare the obtained results more easily, the analysis results are visualized in Fig. 3.

Fig. 3
figure 3

Histogram diagram of the number of s-box structures with 106 and above nonlinearity value

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In Table 2, the average of nonlinearity values calculated for all produced s-box structures is given.

Table 2 The average of nonlinearity values for generated all s-boxes
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The results in Table 2 are very useful for seeing the big picture. In Ref. [45], it is claimed that the success of s-box design criteria is independent of the chaotic system type. Indeed, if enough s-boxes are produced, it is seen that average nonlinearity value will converge to a certain value. In other words, it can be said that the average nonlinearity value for random selection-based s-box structures is 103.5. These results confirm the hypothesis claimed in the Ref. [45]. When the results in Table 1 are examined, they point out that when chaotic systems are specifically focused, exceptional results can be achieved as in Tables 3 and 4.

Table 3 Generated s-box1 with nonlinearity value 107
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Table 4 Generated s-box2 with nonlinearity value 107
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4.1 Performance comparisons

Many metrics are available to evaluate the success of S-box designs. Ref. [52, 53] can be explored for details of these metrics and explanations of how calculations are made. The criteria considered in this study are: The first of these metrics is called the strict avalanche criteria (SAC) [61]. The basic approach to this criterion is that when a bit in the input data changes, half or nearly half of the bits in the output data change. That is, the SAC value should be close to 0.5. The second evaluation criterion is called nonlinearity. It can be said that this criterion is more important than the others. Because s-box structures are nonlinear structures. The higher the nonlinearity values, the stronger they will be. The third evaluation criterion is called the bit independence criterion (BIC) [61]. This criterion states that when any input bit is inverted, the output bits should change independently for all avalanche vector pairs. It then states that the output bits must be bidirectionally independent. That is, a bijective substitution box in an f plane must not only be nonlinear but also provide the SAC value. Finally, the criterion called the input/output XOR distribution is used. The basic approach in this criterion is that the XOR values at the input have the same probability as those at the output [62]. That is, for an s-box structure to be resistant to differential cryptanalysis, it must not allow XOR distribution. Performance comparisons for these five metrics are given in Table 5. This table includes performance analysis of different types of s-box designs such as mathematical, optimization, and chaos-based. This diversity offers the opportunity to evaluate the structures from different angles.

Table 5 Performance comparison
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It is quite easy to obtain an s-box with just chaotic maps, as in this work. However, since the nonlinearity value is low in such s-box structures, various security vulnerabilities may arise. Therefore, various approaches have emerged to improve the nonlinearity and other criteria of these structures. The most well-known of these approaches are mathematical transformations and optimization techniques. By using mathematical transformations, nonlinearity, and other criteria can reach very high values. However, these structures may show some vulnerabilities, especially against differential cryptanalysis. Another commonly used approach is optimization. This approach has been used in nearly 30 studies so far. With optimization, especially the nonlinearity value increased up to 112, which is the highest value. However, since the computational complexity of this approach is high, it may be difficult to apply in various situations.

According to the analysis results, the s-box structures in Tables 3 and 4 have higher analysis results than most of the studies according to the nonlinearity criteria. In the first published studies in the past, 106.75 was the upper limit. However, in our study here, we have shown that this value can be exceeded. It has been observed that strict avalanche criteria produce results close to ideal. The fact that the XOR distribution is 12 indicates a feature of the produced s-box structures that should be improved. Because this value is calculated as the highest 4 in the AES algorithm. In other words, the max XOR value is expected to be 4. This value is high in most studies in the literature. More new studies are needed to improve this value. Looking at the average results of the SAC value, it is seen that this value is very close to 0.5 in the proposed s-box structures. The studies given in Table 5 also meet this criterion. Therefore, when comparing s-box structures, the nonlinearity value is generally used rather than the SAC value. However, the SAC value of s-box structures must be close to 0.5. In this way, attackers will not be able to make any inferences. In the BIC criterion, the SAC value and nonlinearity value are calculated. Here, the SAC value is expected to be close to 0.5 and the nonlinearity value is expected to be as high as possible. For the proposed s-box structures, the SAC value in this criterion is very close to 0.5. The nonlinearity value can be improved here. Again, in many studies, the BIC-Non value is low.

4.2 Side channel analysis

The most important claim of random selection-based s-box structures is that they will be more resistant to side channel attacks than mathematically based s-box structures. The accuracy of this hypothesis has been demonstrated in previous analysis studies. In the Ref. [5], attacks were carried out on the standard AES algorithm using 10/20/30 different plaintext/ciphertext pairs. Then, these attacks were analyzed again by replacing the standard s-box with chaotic s-box structures. It has been shown that when chaotic s-box structures are used, fewer parts of the secret key can be obtained, in other words, they are more resistant to side channel attacks. These analysis scenarios have been repeated for the s-box structures in Tables 3 and 4. Obtained side channel attack results are given in Table 6.

Table 6 Analysis results for side channel attack
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One of the important contributions of the chaos-based cryptography literature is to develop cryptological structures that are more resistant to side-channel analysis. As part of this motivation, how more complex chaotic systems affect the success of attackers has become a new research question. The new chaotic system proposed in this study is planned to be used as a measure against side channel analyses by increasing the complexity of the system. For this reason, in this section, the resistance of the proposed structures against side channel analyses is explained. Side channel analyses are a subheading of applied cryptanalysis. Cryptanalysis approaches are divided into three brute force attacks, mathematical attacks, and social engineering. Side channel analyses are a subheading of applied cryptanalysis and are based on the principle of obtaining the secret key of the encryption algorithm based on the side channel information (sound, light, power consumption, etc.) that occurs during the operation of the algorithm on the target hardware after the implementation of the encryption algorithm on hardware. The cryptanalysis technique performed in this chapter is based on the principle of obtaining the encryption key by analyzing the power traces of the encryption algorithm. All algorithms, all tests, and all analyses have been performed on the AES algorithm. One of the most critical components of the AES algorithm is the 16*16 s-box structures. The chaotic s-box structures proposed in this study can be used instead of the AES s-box structure. In this study, the effect of different s-box structures on side channel analyses of different analyses by using the proposed chaotic s-box structures instead of the AES s-box structure is examined.

When the results in Table 6 are analyzed, the following emerges. The secret key of the AES algorithm is divided into 160 parts. It is shown how many of these 160 parts are correctly guessed. Each cell of the table assumes a value. The attack technique performed here is a plain text attack. In other words, when we have both plain text data and its encrypted message equivalents, we try to obtain the key. If we know 10 plain text data, in case of side channel attack, when we divide the key into 160 parts in AES secretion, we can guess only 6 parts correctly, while we can guess only 5 parts with the proposed approach and 2 parts with the other proposed algorithm. This shows that the proposed structure is more resistant to side channel attacks than the AES s-box structure. When the number of plain texts is increased, the attack success increases. In the maximum attack scenario, when 30 plain texts are used, it is observed that the majority of the key of the encryption algorithm is obtained. However, even in this case, it is observed in Table 4 that the proposed s-box structure achieves more successful results than the AES s-box structure.

The results of the analysis showed that the success achieved in preventing side blood attacks by using previous chaotic s-box structures could not be improved. It is important that although the nonlinearity criterion was increased, better results could not be obtained against side channel attacks. The achievement of this result is associated with the worse XOR criterion, another design metric of the proposed s-box structures. Since differential attacks perform the attack scenario based on the XOR distribution table, the focus should be on improving the nonlinearity value only. A low XOR value is an important parameter.

4.3 Random number generator

Random numbers are frequently used in computer science, especially in two fields. The first of these is modeling and simulation techniques. Another application area is cryptographic structures [63]. Complex and efficient random number generators are needed especially for cryptographic structures. Although true random number generators (TRNG) should be used here most of the time, pseudo-random number generators (PRNG) are needed especially for speed. Whether for stream ciphers or block ciphers, different keys are constantly needed. Because the only unknown thing in these algorithms is the key information. An attacker who gets the key will be able to break the algorithm easily. Therefore, the key structure must be random and strong. For this, the need for new PRNG structures is increasing day by day. With this motivation, a random number generator was created using the proposed chaotic map. With the proposed map, 3 different datasets with a length of 1 million bits were created. For testing these numbers, the NIST SP 800–22 test is most used in the literature [64]. The data sets obtained in this study were also passed through the NIST test structure. The NIST test consists of 15 active tests in total. The proposed generator successfully passed all test results for all 3 data sets. These test results are given in Table 7.

Table 7 NIST SP 800–22 test results
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5 Conclusion and discussion

In this study, a new chaotic system has been proposed. Whether the rich dynamics of the proposed new chaotic system will provide advantages for cryptology studies has been examined. The focus is on s-box structures, which have an important role in block cipher algorithms as a practical application area. Using both the proposed new chaotic system and other commonly used chaotic systems, 70,000 different chaos-based s-box structures have been generated. End of the comprehensive analysis, new results have been obtained for the literature.

  • The highest value that can be reached as the nonlinearity value of s-box structures designed using only chaotic system outputs has been calculated as 106.75 until today. It has been shown that this value can be increased further. The new highest value is calculated as 107.

  • Two s-box structures with a nonlinearity value of 107 have been found.

  • The average nonlinearity value of the produced 70,000 s-box structure has been calculated. The average value is calculated as 103.49. It has been observed that the average value converges to 103.4 in the case of too many s-boxes. This result indicates the mean nonlinearity value for random selection-based s-box structures. It also strengthens the hypothesis that the transformation function, rather than the chaotic system type, may be more effective in the performance of design metrics.

  • For s-box structures with a nonlinearity value of 107, the side channel analysis results are more resistant than the AES s-box structure, but it is worse than a previously analyzed s-box structure in Ref. [35], indicating that the XOR value is as important as the nonlinearity value.

  • The proposed map is also used as a random number generator. It successfully passed all NIST test results for 3 different data sets produced.

These results; will provide a new motivation for researchers to generate s-box structures with higher nonlinearity values in future studies. However, not only increasing the nonlinearity measurement but also decreasing the largest value in the XOR distribution table should be investigated. The proposed chaotic system is thought to have potential outputs as new design proposals for other information security applications such as block ciphers, image encryption, random number generator, and hash functions.