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Selective chaotic maps Tiki-Taka algorithm for the S-box generation and optimization

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Abstract

Cryptography often involves substituting (and converting) the secret information into dummy data so that it could reach the desired destination without leakage. Within symmetric key cryptography, substitution-box (S-box) is often adopted to perform the actual block cipher substitution. To address the nonlinear requirement of cryptography (i.e., ensuring the generated S-box is sufficiently robust against linear and differential cryptanalysis attacks), many chaos-based metaheuristic algorithms have been developed in the literature. This paper introduces a new variant of a metaheuristic algorithm based on Tiki-Taka algorithm, called selective chaotic maps Tiki-Taka algorithm (SCMTTA). Unlike competing works (which typically integrates a single chaotic map into a particular metaheuristic algorithm), SCMTTA assembles five chaotic maps (i.e., tent map, logistic map, Chebyshev map, singer map and sine map) as part of the algorithm itself in order to further enhance ergodicity and unpredictability of the generated solution. Based on a simple penalized and reward mechanism, one best performing chaotic map will be selected in the current cycle, while the poor performing one will miss its current turn. Experimental results on the case study related to the generation of 8 × 8 substitution-box demonstrate that the proposed SCMTTA gives competitive performance against other existing works due to its ability to adaptively modify its chaotic behavior based on the performance feedback of the current search process.

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Acknowledgements

The work reported in this paper is funded by the Universiti Malaysia Pahang Grant: An Automatic Researcher Profiling System for UMP employing UMPIR data (RDU 192211).

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Authors and Affiliations

  1. Faculty of Computing, College of Computing and Applied Sciences, Universiti Malaysia Pahang, 26600, Pekan, Pahang, Malaysia

    Kamal Z. Zamli & Abdul Kader

  2. Department of Computer Science and IT, Faculty of Information Technology, University of Malakand, Chakdara, Khyber Pakhtunkhwa, Pakistan

    Fakhrud Din

  3. Computers Technologies Engineering Department, Dijlah University College, Baghdad, Iraq

    Hussam S. Alhadawi

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  1. Kamal Z. Zamli
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Correspondence to Kamal Z. Zamli.

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Appendix

Appendix

S-box evaluation criteria [48] include bijectivity, nonlinearity, strict avalanche criteria (SAC), bit independence criteria (BIC), differential approximation probability (DP), linear approximation probability (LP) and transparency order (TO).

1.1 Bijective

A n × n S-box can fulfill the bijective property [49] if the Boolean functions \(f_{i} \left( {1 \le i \le n} \right)\) of an S-box satisfy the following equation:

$$wt\left( {\mathop \sum \limits_{{i = 1}}^{n} a_{i} f_{i} } \right) = 2^{{n - 1}}$$

where \(wt()\) is the Hamming weight, \(a_{i} \in \left\{ {0,~1} \right\}\), and \(\left( {a_{1} ,a_{2} ,~.~.~.~,a_{n} } \right)~ \ne \left( {0,~0,~.~.~.~,~0} \right)\). Every \(f_{i}\) is basically required to be 0/1 balanced and the S-box is bijective.

1.2 Nonlinearity

The nonlinearity \(N_{{\text{f}}}\) of \(f\left( x \right)\) is defined as in Eq. (2) where \(f\left( x \right):F_{2}^{n} \to F_{2}\) and \(f\left( x \right)\) is a Boolean function as follows:

$$N_{f} = \mathop {\min }\limits_{{l \in L_{n} }} d_{H} \left( {f,l} \right)$$

where \(L_{n}\) denotes an affine function set, \({\text{d}}_{H} \left( {f,l} \right)\) represents the Hamming distance between \(f\) and \(l\).

In practical application, Walsh spectrum is adopted to calculate the nonlinearity score, which is defined as follows:

$$S_{{\left( f \right)}} \left( \omega \right) = \mathop \sum \limits_{{x \in GF\left( {2^{n} } \right)}} \left( { - 1} \right)^{{f\left( x \right) \oplus x \cdot \omega }}$$

where \(\omega\) is the element of \(GF\left( {2^{n} } \right)\), \(x \cdot \omega\) is the dot product between \(x\) and \(\omega\).

Therefore, the nonlinearity \(N_{{\text{f}}}\) is evaluated as follows:

$$N_{f} = 2^{{n - 1}} - \frac{1}{2}\left( {\mathop {\max }\limits_{{w \in GF\left( 2 \right)^{n} }} \left| {S_{f} \left( w \right)} \right|} \right) = 128.0{\text{ }} - \frac{1}{2}\left( {\mathop {\max }\limits_{{w \in GF\left( 2 \right)^{n} }} \left| {S_{f} \left( w \right)} \right|} \right)~\,{\text{for}}~\,n = 8$$

In this case, the objective function is to maximize \(N_{f}\). For 8 × 8 S-box, the best nonlinearity score is 112.

1.3 Strict avalanche criteria (SAC)

A function can satisfy SAC if the probability of altering the output bits is half whenever a single input bit is complemented [50]. Generally, the SAC of an S-box is checked by employing a dependency matrix. The S-box is deemed to almost satisfy the SAC if each element and the mean value of the matrix are both close to the ideal value of 0.5.

1.4 Bit independence criterion (BIC)

BIC implies that for a given set of avalanche vectors, all avalanche variables should be pair-wise independent. The avalanche vectors are generated by complementing a single plaintext bit. The correlation coefficient between a pair of avalanche variables is required to measure the degree of independence. For two avalanche variables A and B:

$$\rho \left\{ {A,B} \right\} = \frac{{\text{cov} \left\{ {A,B} \right\}}}{{\sigma \left\{ A \right\}\sigma \left\{ B \right\}}}$$

where the correlation coefficient and covariance of \(A\) and \(B\) are denoted by \(\rho \left\{ {A,B} \right\}\) and \(\text{cov} \left\{ {A,B} \right\}\), respectively, where \(\text{cov} \left\{ {A,B} \right\} = E\left\{ {AB} \right\}~ - ~E\left\{ A \right\}~ \times ~E\left\{ B \right\}\) and \(\sigma ^{2} \left\{ A \right\} = ~E\left\{ {A^{2} } \right\}~ - ~\left( {E\left\{ A \right\}} \right)^{2}\).

Assume \(f_{1} ,f_{2} , \ldots\) denote the Boolean functions of S-box. According to Webster and Tavares in [50], \(f_{i} \oplus f_{k} ~\left( {j \ne k,~1 \le j,~k \le n} \right)\) should be highly nonlinear and fulfill the avalanche criterion if the S-box met the output BIC. Thus, by calculating the SAC and the nonlinearity of \(f_{i} \oplus f_{k}\), BIC can be verified.

1.5 Differential approximation probability (DP)

Ideally, the S-box should have differential uniformity. The differential uniformity is achieved via the differential input mapping to differential output mapping. To ensure the uniform mapping probability, an input differential \(\Delta {x}_{i}\) should uniquely map to an output differential \(\Delta {y}_{i}\) for each i. In this case, the XOR distribution between inputs and outputs of S-box must be determined. Mathematically, the measurement of differential uniformity of a given S-box is the differential approximation probability DP defined as follows:

$${\text{DP}}\left( {\Delta x - \Delta y} \right) = \left( {\frac{{\# \{ x \in X|S\left( x \right) \oplus S\left( {x \oplus \Delta x} \right) = \Delta y\} }}{{2^{m} }}} \right)$$

where \(X\) is the set of \(2^{m}\) possible input values.

The smaller the DP score, the stronger the ability of the S-box to resist the differential cryptanalysis attacks, and vice versa.

1.6 Linear approximation probability (LP)

The maximum value of the imbalance of an event is represented by the linear approximation probability LP. The parity of input and output bits selected by the mask Γx and Γy, respectively, is equal. Matsui in [51] defines the linear approximation probability (or probability of bias) of a given S-box as follows:

$${\text{LP}} = \mathop {\max }\limits_{{\Gamma x,\Gamma y \ne 0}} \left| {\frac{{\# \{ x \in X|x \cdot \Gamma x = S\left( x \right) \cdot \Gamma y\} ti}}{{2^{m} }} - \frac{1}{2}} \right|$$

where \(\Gamma x\) and \(\Gamma y\) are the input and output masks, respectively; \(X\) is the set of \(2^{m}\) possible inputs.

1.7 Transparency order (TO)

Transparency order metric is a resistance measuring metric for the S-box to differential power analysis (DPA) attacks [52]. Some cryptographically strong Boolean functions or S-boxes have been found to be poorly robust in DPA attacks such as AES reverse mappings [53]. The mathematical formulation of transparency order \(\tau _{S}\) for an S-box [52, 53] is presented as follows:

$$\tau _{S} = \mathop {\max }\limits_{{\beta \in F_{2}^{n} }} \left( {\left| {n - 2{\text{HW}}\left( \beta \right)} \right| - \frac{1}{{2^{{2n}} - 2^{n} }}\mathop \sum \limits_{{\alpha \in F_{2}^{{n^{*} }} }} \left| {\mathop \sum \limits_{{\begin{array}{*{20}c} {v \in F_{2}^{n} } \\ {{\text{HW}}\left( v \right) = 1} \\ \end{array} }}^{m} \left( { - 1} \right)^{{v.\beta }} W_{{\alpha ,S}} \left( {0,v} \right)} \right|} \right)$$

where \({\text{HW}}\left( \beta \right)\) denotes the Hamming weight of \(\beta\), and \(W_{{\alpha ,S}} \left( {u,v} \right)\) is calculated as follows:

$$W_{{\alpha ,S}} \left( {u,v} \right) = \sum \left( { - 1} \right)^{{v.\left\{ {S\left( x \right) \oplus S\left( {x \oplus \alpha } \right)} \right\} \oplus u.x}}$$

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Zamli, K.Z., Kader, A., Din, F. et al. Selective chaotic maps Tiki-Taka algorithm for the S-box generation and optimization. Neural Comput & Applic 33, 16641–16658 (2021). https://doi.org/10.1007/s00521-021-06260-8

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