Research Article  Open Access
A Novel Technique for the Construction of Safe Substitution Boxes Based on Cyclic and Symmetric Groups
Abstract
In the literature, different algebraic techniques have been applied on Galois field to construct substitution boxes. In this paper, instead of Galois field , we use a cyclic group in the formation of proposed substitution box. The construction proposed Sbox involves three simple steps. In the first step, we introduce a special type of transformation of order 255 to generate . Next, we adjoin to and write the elements of in matrix to destroy the initial sequence . In the step, the randomness in the data is increased by applying certain permutations of the symmetric group on rows and columns of the matrix. In the last step we consider the symmetric group , and positions of the elements of the matrix obtained in step 2 are changed by its certain permutations to construct the suggested Sbox. The strength of our Sbox to work against cryptanalysis is checked through various tests. The results are then compared with the famous Sboxes. The comparison shows that the ability of our Sbox to create confusion is better than most of the famous Sboxes.
1. Introduction
The foundation of modern cryptography was laid by Shannon [1]. Cryptography is the science of converting the secret information into dummy data so that it could reach the destination safely without leakage of the information. The modern cryptography is divided into several branches. However, symmetric key cryptography and public key cryptography are the two main areas of study. In symmetric key cryptography, the same key is used at both ends to encrypt and decrypt data/information, but in public key cryptography two different keys, public and private keys, are used. It is wellknown that, in symmetric key cryptography the substitution box is a standout and basic ingredient, which performs substitution. In block ciphers, it is widely used to make the relationship between the ciphertext and the key unclear and vague. Due to these important applications of substitution box many algorithms have been developed to construct safer and more reliable Sboxes. Substitution boxes are used for the strong design of block encryption algorithms. Sbox is the only nonlinear component for most of the block encryption algorithms such as international data encryption algorithm (IDEA), advanced encryption standard (AES), and data encryption standard (DES) [2]. Substitution boxes yield a DESlike cryptosystem with the perplexity property depicted by Shannon. In [3], it is shown that for weaker Sboxes, DES can be easily broken. It means that the security of DESlike cryptosystems is merely determined by the quality of the Sboxes used. Thus, in order to develop secure cryptosystems, the formation of safe Sboxes is a main focus of the researcher. To examine the strength of Sboxes, nonlinearity test, bit independent criterion, strict avalanche criterion, linear approximation probability analysis, differential uniformity test, and majority logic criterion are used. In the literature, there are many Sbox construction methods such as inversion mapping, power polynomial, heuristic methods, and pseudorandom methods [4]. Incursions on the Sbox component of data encryption standard (DES) damage the design process of advanced encryption standard (AES) [3, 5]. Therefore, the substitution box component of AES is designed to ensure the security of the data/information in the presence of differential and linear cryptanalysis attacks [6].
Recently, since proposed algebraic attacks have been succeeded in some loops of AES, researchers have focused on alternative construction methods for substitution box [21]. Therefore, substitution box construction techniques based on group theory have been applied for alternative substitution box designs.
2. Algebraic Structure of Proposed Substitution Box
Let us denote a set of positive integers less than by ; that is, . Consider a transformation defined byIt can be easily verified that has order 255; that is, for any , Thus for all , generates a cyclic group =. In this paper, we have taken
Step I. First we simply present the elements ofin matrix (see Table 2). Cayley graph of is shown in Figure 1. In this way, the initial sequence is destroyed. If this matrix is conceded as Sbox, its nonlinearity is 103.75, which is acceptable. Now we move to step II to create more randomness.
Step II. Since we have presented our data in matrix, that is, a matrix with rows and columns, the randomness can be increased by interchanging the positions of the rows and columns. Algebraically, it is achieved by applying permutations of the symmetric group on the matrix. Since order of is , therefore corresponding to one matrix (Sbox) formed after applying one permutation on rows, number of new Sboxes can be created by applying all the permutations on columns. Thus by this technique, we can construct different Sboxes. We choose two particular types of permutations of the symmetric group such that one of them is applied on the rows and the other on columns. This action increases the diffusion capability of the cipher. The permutations are as follows.The resulting Sbox (see Table 3) has nonlinearity of 106.25. In step III, we further enhance its working capability.
Step III. Recently, we have noticed that certain permutations of the symmetric group are amazingly constructive. In this step, we apply a permutations of (see Table 1) on the data/matrix obtained after step II to construct a very strong Sbox (see Table 4).




3. Security Analysis
In this section, a point by point exploration of the suggested Sbox is presented. Furthermore, we have made a comparison with the famous Sboxes, such as AES Sbox, Xyi Sbox, Skipjack Sbox, S8 AES Sbox, Residue Prime Sbox, APA Sbox, and Gray Sbox. The illustration of various analysis applied on these substitution boxes is given. It is seen that our Sbox meets all the standards near the ideal status.
3.1. Nonlinearity
The key objective of the substitution box is to provide assistance in giving nonlinear change from unique data to the encoded information. The measure of nonlinearity presented by the cipher considered as the most important part in the entire process of encryption. It is defined asHere is the Walsh Spectrum. The average values of the nonlinearity of newly constructed Sbox is 112. A comparison between the nonlinearity of the suggested Sbox and multiple renowned substitution boxes is given in Table 5.

3.2. Bit Independence Criterion
Webster and Tavares firstly demonstrated bit independence criterion [22]. A function fulfils the BIC requirements if , the output bits j and k, where , change independently by inverting the input bit . In cryptographic systems, the BIC is a very important characteristic because by increasing independence between bits, it is very hard to decipher and predict the scheme of the system. The outcomes of nonlinearity of BIC are presented in Table 6. In order to find the independence properties a comparison of the bits, created by the eight basic functions, with each other is established. The relationship between the outcomes of change in input bit and the change in jth and kth output bits is identified. In the first phase the ith bit is varied from to n by keeping and bits fixed. Next, the values of j and k are altered from to n. Furthermore, the minimum and average values of BIC along with square deviation of the proposed Sboxes are presented in Table 7. The average and minimum values of BIC of the proposed Sbox are . The square deviation of the newly created substitution box is 0. All these results are better than most of the wellknown Sboxes and similar to AES, S_{8} AES, and Gray Sboxes.


3.3. Strict Avalanche Criterion Analytically
Tavares and Webster introduced strict avalanche criterion [22]. In this criterion, the output bits are examined after changing a single input bit. In ideal condition, by changing a single input bit, half of the output bits change their shape. In [23] an effective technique is presented to check whether a complete substitution box satisfies the SAC or not. The results of SAC of the suggested Sbox (see Table 8) are nearly equal to , which shows its strength.

3.4. Linear Approximation Probability
In this analysis, the imbalance of an event is examined. It is useful in finding the maximum value of an imbalance of the output in an event. Let us denote the input and output masks by and , respectively. Then mathematically, linear approximation probability is defined as follows.In above expression denotes the set of all possible values in domain and is the number of elements of the Sbox.
The maximum LP value is 0.0625, which is matching with the best known Sboxes such as Gray, APA, and AES. In Table 9, a comparison of the results of this analysis, between our Sbox and some famous Sboxes, is given.

3.5. Differential Uniformity
Differential uniformity is another important method of block cipher cryptanalysis. It was introduced by Biham and Shamir to break block ciphers [3]. It exploits certain events of I/O differences and represents the maximum likelihood of generating an output differential = when the input differential is = . In this analysis, the XOR distribution between the inputs and outputs of substitution box is computed. Mathematically, it is defined aswhere denotes cardinality and is set of all inputs [3, 24, 25]. By using the approach introduced in [3], an input/output XOR distribution matrix of size is calculated for suggested Sbox and is provided in Table 10. As a general Sbox design guideline, the maximum differential uniformity has to be kept as low as possible to withstand differential attacks. The highest value of differential uniformity for suggested Sbox is 4, which is compared with some wellknown Sboxes in Table 11 to show the strength of suggested Sbox.


4. Majority Logic Criterion
In majority logic criterion, statistical analyses are performed to examine the statistical strength of the Sbox in image encryption application [26]. The encryption process creates a distortion in the image, these kinds of distortions determine the strength of the algorithm. Therefore, it is necessary to investigate the statistical properties through various analyses. These analyses are correlation, entropy, contrast, homogeneity, and energy. The suggested Sboxes can further be used for encryption and multimedia security. We have used two JPEG images, Pepper and Baboon, for MLC analysis. The results of these analyses in comparison with the other wellknown Sboxes are depicted in Table 12. Figure 2 shows the result of image encryption with proposed Sbox. The histograms of the original image and the encrypted images of Baboon and Pepper are shown in Figure 3. These results indicate that the proposed Sbox is suitable for encryption applications and is adequate enough to become part of the algorithms designed for the secure transmission of information/data.

(a)
(b)
(a)
(b)
5. Conclusion
In this study, we introduce a group theoretic technique to form strong Sboxes. The cyclic group instead of a Galois field is used to destroy the initial sequence . The construction of Sbox involves three simple steps:(i)First present the elements of in matrix.(ii)Next, apply two permutations of on rows and column of the matrix. It will significantly improve the performance of the Sbox.(iii)In the last step, a permutation of is applied on the matrix (obtained in step (ii)) to form proposed Sbox.
The results acquired from different analyses show that the performance of our Sbox against various algebraic attacks is much better than most of wellknown Sboxes and similar to AES, S_{8} AES, and Gray Sboxes. Therefore, our Sbox meets all the requirements and is considered as a strong Sbox for the secure communication.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research project was supported by a grant from the Research Center of the Center for Female Scientific and Medical Colleges, Deanship of Scientific Research, King Saud University.
References
 C. E. Shannon, “Communication theory of secrecy systems,” Bell Labs Technical Journal, vol. 28, no. 4, pp. 656–715, 1949. View at: Publisher Site  Google Scholar  MathSciNet
 L. R. Knudsen and M. J. B. Robshaw, The Block Cipher Companion, Springer, Berlin, 2011.
 E. Biham and A. Shamir, “Differential cryptanalysis of DESlike cryptosystems,” Journal of Cryptology, vol. 4, no. 1, pp. 3–72, 1991. View at: Publisher Site  Google Scholar
 T. W. Cusick and P. Stanica, Cryptographic Boolean functions and applications, Academic Press, San Diego, CA, USA, 2009. View at: MathSciNet
 T. Helleseth, “Linear cryptanalysis method for des cipher,” in Advances in Cryptology—EUROCRYPT, vol. 765 of Lecture Notes in Computer Science, pp. 386–397, Springer, Berlin, Germany, 1993. View at: Publisher Site  Google Scholar
 J. Daemen and V. Rijmen, The design of RijndaelAES: the advanced encryption standard, Springer, Berlin, 2002.
 A. Razaq, A. Yousaf, U. Shuaib, N. Siddiqui, A. Ullah, and A. Waheed, “A Novel Construction of Substitution Box Involving Coset Diagram and a Bijective Map,” Security and Communication Networks, vol. 2017, 2017. View at: Google Scholar
 M. T. Tran, D. K. Bui, and A. D. Doung, “Gray Sbox for advanced encryption standard,” in Proceedings of the International Conference on Computer Intel Security, vol. 1, pp. 253–258, 2008. View at: Google Scholar
 A. Gautam, G. S. Gaba, R. Miglani, and R. Pasricha, “Application of Chaotic Functions for Construction of Strong Substitution Boxes,” Indian Journal of Science and Technology, vol. 8, no. 28, pp. 1–5, 2015. View at: Publisher Site  Google Scholar
 I. Hussain, T. Shah, H. Mahmood, M. A. Gondal, and U. Y. Bhatti, “Some analysis of Sbox based on residue of prime number,” Proceedings of the Pakistan Academy of Sciences, vol. 48, no. 2, pp. 111–115, 2011. View at: Google Scholar
 I. Hussain, T. Shah, and H. Mahmood, “A new algorithm to construct secure keys for AES,” International Journal of Contemporary Mathematical Sciences, vol. 5, no. 2528, pp. 1263–1270, 2010. View at: Google Scholar  MathSciNet
 X. Y. Shi, Hu. Xiao, X. C. You, and K. Y. Lam, “A method for obtaining cryptographically strong 8*8 Sboxes,” in Proceedings of the International Conference on Advanced Information Networking and Applications, vol. 2, pp. 14–20, 2002. View at: Google Scholar
 Skipjack and Kea: Algorithm Specifications Version, 1998, http://csrc.nist.gov/CryptoToolkit/.
 A. H. Alkhaldi, I. Hussain, and M. A. Gondal, “A novel design for the construction of safe Sboxes based on TDERC sequence,” Alexandria Engineering Journal, vol. 54, pp. 65–69, 2015. View at: Publisher Site  Google Scholar
 G. Chen, Y. Chen, and X. Liao, “An extended method for obtaining Sboxes based on threedimensional chaotic baker maps,” Chaos, Solitons & Fractals, vol. 31, no. 3, pp. 571–579, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 G. Tang, X. Liao, and Y. Chen, “A novel method for designing Sboxes based on chaotic maps,” Chaos, Solitons & Fractals, vol. 23, no. 2, pp. 413–419, 2005. View at: Publisher Site  Google Scholar
 M. Khan, T. Shah, and M. A. Gondal, “An efficient technique for the construction of substitution box with chaotic partial differential equation,” Nonlinear Dynamics, vol. 73, no. 3, pp. 1795–1801, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 A. Belazi, M. Khan, A. A. A. ElLatif, and S. Belghith, “Efficient cryptosystem approaches: Sboxes and permutation–substitutionbased encryption,” Nonlinear Dynamics, vol. 87, no. 1, pp. 337–361, 2017. View at: Publisher Site  Google Scholar
 A. Ullah, S. S. Jamal, and T. Shah, “A novel construction of substitution box using a combination of chaotic maps with improved chaotic range,” Nonlinear Dynamics, vol. 88, no. 4, pp. 2757–2769, 2017. View at: Publisher Site  Google Scholar
 M. Khan, T. Shah, and S. I. Batool, “Construction of Sbox based on chaotic Boolean functions and its application in image encryption,” Neural Computing and Applications, vol. 27, no. 3, pp. 677–685, 2016. View at: Publisher Site  Google Scholar
 G. V. Bard, Algebraic Cryptanalysis, Springer, Berlin, 2009. View at: Publisher Site
 A. Webster and S. Tavares, “On the design of Sboxes,” in Advancesin Cryptology: Proc. of Crypto’85 Lecture Notes in Computer Science, pp. 523–534, 1986. View at: Google Scholar
 J. Pieprzyk and G. Finkelstein, “Towards effective nonlinear cryptosystem design,” IEE Proceedings Part E Computers and Digital Techniques, vol. 135, no. 6, pp. 325–335, 1988. View at: Publisher Site  Google Scholar
 H. A. Ahmed, M. F. Zolkipli, and M. Ahmad, “A novel efficient substitutionbox design based on firefly algorithm and discrete chaotic map,” Neural Computing and Applications, pp. 1–10. View at: Google Scholar
 E. Al Solami, M. Ahmad, C. Volos, M. Doja, and M. Beg, “A New Hyperchaotic SystemBased Design for Efficient Bijective SubstitutionBoxes,” Entropy, vol. 20, no. 7, p. 525, 2018. View at: Publisher Site  Google Scholar
 I. Hussain, T. Shah, M. A. Gondal, and H. Mahmood, “Generalized Majority Logic Criterion to Analyze the Statistical Strength of SBoxes,” Zeitschrift für Naturforschung A, vol. 67, no. 5, pp. 282–288, 2012. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2018 Abdul Razaq et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.