Security and Communication Networks

Security and Communication Networks / 2018 / Article

Research Article | Open Access

Volume 2018 |Article ID 4987021 | 9 pages | https://doi.org/10.1155/2018/4987021

A Novel Technique for the Construction of Safe Substitution Boxes Based on Cyclic and Symmetric Groups

Academic Editor: Stelvio Cimato
Received04 Jun 2018
Accepted19 Sep 2018
Published04 Oct 2018

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 S-box 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 S-box. The strength of our S-box to work against cryptanalysis is checked through various tests. The results are then compared with the famous S-boxes. The comparison shows that the ability of our S-box to create confusion is better than most of the famous S-boxes.

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 well-known 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 S-boxes. Substitution boxes are used for the strong design of block encryption algorithms. S-box 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 DES-like cryptosystem with the perplexity property depicted by Shannon. In [3], it is shown that for weaker S-boxes, DES can be easily broken. It means that the security of DES-like cryptosystems is merely determined by the quality of the S-boxes used. Thus, in order to develop secure cryptosystems, the formation of safe S-boxes is a main focus of the researcher. To examine the strength of S-boxes, 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 S-box construction methods such as inversion mapping, power polynomial, heuristic methods, and pseudorandom methods [4]. Incursions on the S-box 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 S-box, 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 (S-box) formed after applying one permutation on rows, number of new S-boxes can be created by applying all the permutations on columns. Thus by this technique, we can construct different S-boxes. 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 S-box (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 S-box (see Table 4).


(1225221169782551361736214656119229114117174

1432471051619713920120512415103801332287413
16612722653219181209452516043232160239719
64231208189811525421315075822711123011227
18430212241248170172353224920777699525281
2221499273572316261892202111759133157223
681591454831911930102249018312611613437
144244192352532332161871961981048447155178106
34128101206501489424519934097171165125189
1956312131644291371295220379123177182176
3996215238671072102517914124231243418200
1861101991521086512237598511811346120142185
201471902836153140513599214973818855
4224010916714522361511222242181321638618048
1311948810261562461682141005866620422130
51202158172234161)(447215476138217112)(7025087)


121148218716553116239174106491816241

2491662531518321825514120492170236109136120199

175102244464485133118471832216815712820260

9522820121625051219122161231312227171178195

405924715222422014423952348420716764150101

11130186176188114123229130108160125208154187238

6612420924514098194254111381423521418089

153226662017715820325219276141126711020072

16324816926164172424273232782121823222575

451792518421515240932318899418557147190

145243181655549980237191311041687743222

1151192463631591171248233196251197010349

132971491272813414369817122714615610719390

5017322320518113373310510198217627986230

2351262349613538206741562916219755129100

189367421013912417221321113155825813710


481624112114821875316521161743991106

104774322214524318165545809919123716831

112110200721532266620158177252203761926714

2517010349115119246315963121172334819196

146107193901329714912713428691437181156227

681282026017510224446854411813318347157221

2361361201992491662531512188314125592204109170

207641501014059247152220224239144345216784

2217117819595228201216512501222192316127131

12515418723811130186176114188229123108130208160

1422141808966124209140245194981125435138

21232225751632481692617164422422327318278

945714719045179251841521593240882311859

2177986230501732232051131833371010562198

1625512910023512623496381357206564119729

821371018936742101241392131721321158155


14212522089219632511581494612614628208144218

2459189171202401591667916512873241261377
1188378992282113818312461171702172076075
145231171225539242154134199562132141114753
2551484162712441972031331003018818514093253
17269119151121801395723365351114323813266
2077201173841559117974321931762916480113
59235136526417531921918615688616961110
5124314182271011215819114345114225152254153
2448222701055020625721276751122159096
135181195161941749236102102361302164086248
239229541023321244129161184205226341872020
18217823242106190204871221034910715249124234
16314137237211209221382501981158516268108224
416729524710919625213981048116223160177
230231681314712327823197761572001508194

3. Security Analysis

In this section, a point by point exploration of the suggested S-box is presented. Furthermore, we have made a comparison with the famous S-boxes, such as AES S-box, Xyi S-box, Skipjack S-box, S8 AES S-box, Residue Prime S-box, APA S-box, and Gray S-box. The illustration of various analysis applied on these substitution boxes is given. It is seen that our S-box 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 S-box is 112. A comparison between the nonlinearity of the suggested S-box and multiple renowned substitution boxes is given in Table 5.


S boxes0Ave

Suggested S-box112112112112112112112112112
Coset Diagram S-box [7]108106108108108104106106106.75
Gray [8]112112112112112112112112112
Arun [9]10810610498102102987499
Prime [10]94100104104102100989499.5
S8 AES [11]112112112112112112112112112
Xyi [12]106104106106104106104106105
AES [6]112112112112112112112112112
Skipjack [13]104108108108108104104106106.75
Alkhaldi [14]10810410610610298104108104
Chen [15]100102103104106106106108104.3
Tang [16]100103104104105105106109104.5
Khan [17]10210810610210610610698104.25
Belazi [18]106106106104108102106104105.25

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 S-boxes are presented in Table 7. The average and minimum values of BIC of the proposed S-box are . The square deviation of the newly created substitution box is 0. All these results are better than most of the well-known S-boxes and similar to AES, S8 AES, and Gray S-boxes.


Rows/Columns01234567

0-112112112112112112112
1112-112112112112112112
2112112-112112112112112
3112112112-112112112112
4112112112112-112112112
5112112112112112-112112
6112112112112112112-112
7112112112112112112112-


S-boxesMinimum valueAverageSquare deviation

Suggested S-box1121120
Gray1121120
Arun921033.5225
Prime94101.713.53
S8 AES1121120
Xyi98103.782.743
AES1121120
Skipjack102104.141.767

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 S-box (see Table 8) are nearly equal to , which shows its strength.


Rows/Columns01234567

00.48440.46880.48440.54690.46880.56250.54690.5313
10.45310.53130.51560.54690.46880.46880.48440.5469
20.50000.46880.46090.46880.51560.45310.54690.5625
30.53130.52340.53130.51560.45310.46880.52340.4375
40.56250.48440.46880.51560.54690.54690.56250.4844
50.50000.48440.51560.56250.48440.54690.48440.5156
60.48440.51560.50000.48440.48440.48440.46880.5625
70.54690.56250.45310.46880.51560.48440.53130.4844

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 S-box.

The maximum LP value is 0.0625, which is matching with the best known S-boxes such as Gray, APA, and AES. In Table 9, a comparison of the results of this analysis, between our S-box and some famous S-boxes, is given.


S-boxesSuggested S-boxAESSkipjackPrimeGrayArunS8 AESXyi

Max value144144156162144164144168
Max LP0.0620.0620.1090.1320.0620.21090.0620.156

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 S-box and is provided in Table 10. As a general S-box 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 S-box is 4, which is compared with some well-known S-boxes in Table 11 to show the strength of suggested S-box.


4444444444444444

4444444444444444
4444444444444444
4444444444444444
4444444444444444
4444444444444444
4444444444444444
4444444444444444
4444444444444444
4444444444444444
4444444444444444
4444444444444444
4444444444444444
4444444444444444
4444444444444444
444444444444444- - - - -


S-boxesSuggested S-boxAESGraySkipjackChenKhanS8 AESTangXyi

Max DU44412121641012

4. Majority Logic Criterion

In majority logic criterion, statistical analyses are performed to examine the statistical strength of the S-box 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 S-boxes 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 well-known S-boxes are depicted in Table 12. Figure 2 shows the result of image encryption with proposed S-box. 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 S-box is suitable for encryption applications and is adequate enough to become part of the algorithms designed for the secure transmission of information/data.


S-boxesCorrelationEntropyContrastHomogeneityEnergy

Pepper Image

Plain Text0.93837.59090.27600.90240.1288

Suggested S-box -0.01347.98428.69690.40450.0174

Atta [19]−0.00437.98238.67270.40760.0173

Skipjack0.12057.75617.70580.47080.0239

Khan [20]0.01037.95628.31290.42190.0180

Belazi−0.01127.92338.14230.46480.0286

Baboon Image

Plain Text0.67827.12730.71790.76690.1025

Suggested S-box-0.00607.98208.64880.40620.0174

AES0.05547.25317.55090.46620.0202

Prime0.08556.93117.62360.46400.0202

Xyi0.04177.25318.31080.45330.0196

Skipjack0.10257.25317.70580.46890.0193

Khan [14]−0.05127.96128.12130.40110.0210

Belazi0.0119 07.92528.03910.4428.02219

5. Conclusion

In this study, we introduce a group theoretic technique to form strong S-boxes. The cyclic group instead of a Galois field is used to destroy the initial sequence . The construction of S-box 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 S-box.(iii)In the last step, a permutation of is applied on the matrix (obtained in step (ii)) to form proposed S-box.

The results acquired from different analyses show that the performance of our S-box against various algebraic attacks is much better than most of well-known S-boxes and similar to AES, S8 AES, and Gray S-boxes. Therefore, our S-box meets all the requirements and is considered as a strong S-box 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

  1. 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
  2. L. R. Knudsen and M. J. B. Robshaw, The Block Cipher Companion, Springer, Berlin, 2011.
  3. E. Biham and A. Shamir, “Differential cryptanalysis of DES-like cryptosystems,” Journal of Cryptology, vol. 4, no. 1, pp. 3–72, 1991. View at: Publisher Site | Google Scholar
  4. T. W. Cusick and P. Stanica, Cryptographic Boolean functions and applications, Academic Press, San Diego, CA, USA, 2009. View at: MathSciNet
  5. 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
  6. J. Daemen and V. Rijmen, The design of Rijndael-AES: the advanced encryption standard, Springer, Berlin, 2002.
  7. 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
  8. M. T. Tran, D. K. Bui, and A. D. Doung, “Gray S-box for advanced encryption standard,” in Proceedings of the International Conference on Computer Intel Security, vol. 1, pp. 253–258, 2008. View at: Google Scholar
  9. 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
  10. I. Hussain, T. Shah, H. Mahmood, M. A. Gondal, and U. Y. Bhatti, “Some analysis of S-box 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
  11. 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. 25-28, pp. 1263–1270, 2010. View at: Google Scholar | MathSciNet
  12. X. Y. Shi, Hu. Xiao, X. C. You, and K. Y. Lam, “A method for obtaining cryptographically strong 8*8 S-boxes,” in Proceedings of the International Conference on Advanced Information Networking and Applications, vol. 2, pp. 14–20, 2002. View at: Google Scholar
  13. Skipjack and Kea: Algorithm Specifications Version, 1998, http://csrc.nist.gov/CryptoToolkit/.
  14. A. H. Alkhaldi, I. Hussain, and M. A. Gondal, “A novel design for the construction of safe S-boxes based on TDERC sequence,” Alexandria Engineering Journal, vol. 54, pp. 65–69, 2015. View at: Publisher Site | Google Scholar
  15. G. Chen, Y. Chen, and X. Liao, “An extended method for obtaining S-boxes based on three-dimensional chaotic baker maps,” Chaos, Solitons & Fractals, vol. 31, no. 3, pp. 571–579, 2007. View at: Publisher Site | Google Scholar | MathSciNet
  16. G. Tang, X. Liao, and Y. Chen, “A novel method for designing S-boxes based on chaotic maps,” Chaos, Solitons & Fractals, vol. 23, no. 2, pp. 413–419, 2005. View at: Publisher Site | Google Scholar
  17. 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
  18. A. Belazi, M. Khan, A. A. A. El-Latif, and S. Belghith, “Efficient cryptosystem approaches: S-boxes and permutation–substitution-based encryption,” Nonlinear Dynamics, vol. 87, no. 1, pp. 337–361, 2017. View at: Publisher Site | Google Scholar
  19. 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
  20. M. Khan, T. Shah, and S. I. Batool, “Construction of S-box 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
  21. G. V. Bard, Algebraic Cryptanalysis, Springer, Berlin, 2009. View at: Publisher Site
  22. A. Webster and S. Tavares, “On the design of S-boxes,” in Advancesin Cryptology: Proc. of Crypto’85 Lecture Notes in Computer Science, pp. 523–534, 1986. View at: Google Scholar
  23. 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
  24. H. A. Ahmed, M. F. Zolkipli, and M. Ahmad, “A novel efficient substitution-box design based on firefly algorithm and discrete chaotic map,” Neural Computing and Applications, pp. 1–10. View at: Google Scholar
  25. E. Al Solami, M. Ahmad, C. Volos, M. Doja, and M. Beg, “A New Hyperchaotic System-Based Design for Efficient Bijective Substitution-Boxes,” Entropy, vol. 20, no. 7, p. 525, 2018. View at: Publisher Site | Google Scholar
  26. I. Hussain, T. Shah, M. A. Gondal, and H. Mahmood, “Generalized Majority Logic Criterion to Analyze the Statistical Strength of S-Boxes,” Zeitschrift für Naturforschung A, vol. 67, no. 5, pp. 282–288, 2012. View at: Publisher Site | Google Scholar

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.


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