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# An Innovative Technique for Constructing Highly Non-Linear Components of Block Cipher for Data Security against Cyber Attacks

1 Department of Mathematics, Division of Science and Technology, University of Education, Lahore, Pakistan

2 Department of Mathematics, University of Management and Technology, Sialkot Campus, Sialkot, 51310, Pakistan

3 School of Mathematical Sciences, Zhejiang Normal University, Jinhua, 321004, China

4 Department of Mathematics, University of Management and Technology, Lahore, Pakistan

5 Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, Roma, 00186, Italy

6 Department of Statistics and Operations Research, College of Science, King Saud University, P. O. Box 2455, Riyadh, 11451, Saudi Arabia

7 Faculty of Engineering and Architecturer-Kore University of Enna, Enna, 94100, Italy

* Corresponding Author: Rana Muhammad Zulqarnain. Email:

*Computer Systems Science and Engineering* **2023**, *47*(2), 2547-2562. https://doi.org/10.32604/csse.2023.040855

**Received** 01 April 2023; **Accepted** 06 June 2023; **Issue published** 28 July 2023

## Abstract

The rapid advancement of data in web-based communication has created one of the biggest issues concerning the security of data carried over the internet from unauthorized access. To improve data security, modern cryptosystems use substitution-boxes. Nowadays, data privacy has become a key concern for consumers who transfer sensitive data from one place to another. To address these problems, many companies rely on cryptographic techniques to secure data from illegal activities and assaults. Among these cryptographic approaches, AES is a well-known algorithm that transforms plain text into cipher text by employing substitution box (S-box). The S-box disguises the relationship between cipher text and the key to guard against cipher attacks. The security of a cipher using an S-box depends on the cryptographic strength of the respective S-box. Therefore, various researchers have employed different techniques to construct high order non-linear S-box. This paper provides a novel approach for evolving S-boxes using coset graphs for the action of the alternating group over the finite field and the symmetric group . The motivation for this work is to study the symmetric group and coset graphs. The authors have performed various analyses against conventional security criteria such as nonlinearity, differential uniformity, linear probability, the bit independence criterion, and the strict avalanche criterion to determine its high cryptographic strength. To evaluate its image application performance, the proposed S-box is also used to encrypt digital images. The performance and comparison analyses show that the suggested S-box can secure data against cyber-attacks.## Keywords

Modern technical advancements and their successful application in real life have resulted in a massive increase in the volume of data exchanged. Due to the confidential characteristics of information, it is important to develop ways to reduce the risk of improper utilization. A user’s data must be modified before transmission so that it is worthless to an attacker. Cryptography is employed to securely store and transmit data, ensuring that only authorized individuals have access to the original information. By utilizing cryptographic techniques, organizations can protect their sensitive data from unauthorized access. For many decades, basic cryptographic systems have been used in a variety of fields. Various companies and governments have used it in the past to conceal confidential data from adversaries. However, a substantial number of safe and encrypted conversations occur online every day. Cryptographic encryption techniques can be divided into two distinct categories: symmetric and asymmetric encryption. Symmetric encryption involves the use of a single key to both encrypt and decrypt data, while asymmetric encryption requires two separate keys, one for encryption and one for decryption. Both of these encryption techniques are essential for ensuring the security of sensitive data and communications. Modern symmetric encryption systems, which use the same keys for encryption and decryption operations, require fewer processing resources and are more practicable than old encryption algorithms. There are two types of symmetric encryption schemes: stream ciphers and block ciphers [1]. Because of their ease of implementation and ability to offer much-needed cryptographic strength, symmetric block ciphers are among the most extensively utilized algorithms for this purpose [2,3].

The most popular form of block encryption is Advanced Encryption Standard (AES), which employs substitution and permutation operations. To convert plain text into cipher text, the AES block cipher uses a symmetric key and a variable number of rounds. On the input data block, each round is composed of permutation and substitution operations. In substitution processes, input blocks are substituted with output blocks using substitution boxes (S-boxes) [4]. The S-box is a basic characteristic of modern block ciphers that creates confused cipher text from the provided plaintext [5]. As the only nonlinear component of modern block ciphers, an S-box provides a complicated link between the plaintext and the cipher text. This suggests that unsafe cryptosystems are prompted by weak substitution boxes.

As a result, the development of resilient S-boxes is a critical aspect in the evolution of efficient and safe cryptosystems. So, the researchers in this field have concentrated on the development of innovative strategies for creating cryptographically secure S-boxes. Various ideas and methods for building Substitution boxes have emerged in recent years. In [6], the author employed the I-Ching operator to generate the S-box. When tested using several algebraic criteria, the resultant S-box offers good cryptographic features. Authors in [7] gave the innovative technique to construct the strong S-box using quantic fractional transformation, which is further used in image encryption protection. The outcomes of the projected S-box are outstanding and strong against linear and differential attacks. In [8], authors present the novel technique to construct the substitution box by using dynamic polynomial mapping and constructing the large number of S-boxes. The results are good enough to withstand against linear and differential attacks. In [9], the authors describe a revolutionary modular strategy for building a huge number of S-boxes by gently modifying the parameters in a newly constructed transformation.

Razzaq et al. [10] provide a unique approach for generating the 462422016 various numbers of AES-like S-boxes based on the notion of a coset graph and the actions of a symmetric group and a permutation group. Razzaq et al. [11] built the S-box using the concepts of triangle groups (2, 3, 8), symmetric groups, and coset graphs. The resultant S-box has a nonlinearity of 113.75, which is higher than the standard AES S-box. Yousaf et al. [12] build the S-box using the action of a finite Abelian group, and the resulting S-box possesses optimum properties. Shahzad et al. [13] build the S-box using the action

The technique of constructing the S-boxes by using the concept of an alternating group and coset diagram is presented in this article. The following is the main contribution in this paper:

1) A novel group theoretic and graphical construction of S-box based on the orbits of a coset graph, alternating group

2) Symmetric group

3) S-box evaluated through standard S-boxes criteria that show outstanding results against linear and differential attacks.

The remainder of the paper is structured as follows: Section 2 comprises the basic concept and definitions related to the symmetric groups and coset graphs, while the algebraic structure of the generation of suggested S-boxes is discussed in Section 3. Section 4 evaluates and compares the strength of the newly suggested S-boxes to previous well-known S-boxes. Section 5 represents the result and discussion portion. Conclusion and future work are presented in Section 6.

This section discusses several fundamental ideas and terms related to coset graphs, alternating groups, and symmetric groups for the generation of S-boxes.

2.1 Modular Group and Coset Diagrams

The modular group M is an infinite, non-cyclic, and non-abelian group composed of two generators,

Consider a set

The coset graph of

The coset graph emerges as a result of natural action of

2.2 Triangle Group and Alternating Group

The triangle group is a group that can be represent in the form:

2.3 Mushtaq Parametrization Scheme

Let us discuss the Mushtaq technique briefly (for proof and detail, see [20]).

Firstly, set

Table 1 represent the relation between the value of

3 Algebraic Structure of S-Box

The coset graph for the symmetric group

From the above permutation representation, one can see that

The coset graph of alternating group

1) Find the cycle

2) Apply

3) Repeat step 1.2 for the next cycle containing the smallest element of

4) Write all the elements in a tabular form, and then apply a mapping

After that, omitting the initial 0, disregard all vertices bigger than 255 and

In the initial S-box, the mean non-linearity value is 101.25; however, this value may be enhanced by using a particular permutation of the symmetric group to construct a robust S-box. In this scenario, the 256 cells of Table 2 are subjected to a specific permutation of the symmetric group

3.2 Permutation of Symmetric Group of Order 256

The permutations of

(1, 179, 164, 242, 33, 34, 123, 73, 53, 77, 28, 243, 38, 192, 233, 108, 125, 91, 224, 107, 193, 5, 47, 212, 190, 55, 180, 177, 75, 69, 27, 114, 35, 196, 162, 102, 74, 170, 134, 141, 221, 195, 148, 173, 81, 40, 166, 14, 119, 106, 110, 67, 3, 236, 160, 60, 248, 101, 11, 225, 138, 208, 235, 137, 83, 117, 42, 189, 30, 109, 4, 87, 57, 120, 46, 113, 56, 23, 19, 231, 252, 147, 59, 95, 45, 16, 122, 43, 150, 214, 103, 7, 171, 37, 78, 249, 44, 93, 201, 176, 229, 21, 17, 163, 85, 68, 241, 65, 142)(2, 187, 24, 63, 172, 254, 105, 140, 18, 240, 198, 204, 184, 144, 89, 100, 70, 218, 54, 124, 12, 20, 52, 6, 175)(8, 145, 94, 222, 13, 191, 217, 128, 188, 232, 155, 251, 215, 98, 197, 255, 203, 230, 206, 158, 61, 131, 50, 10, 143, 49, 130, 127, 194, 58, 199, 167, 174, 228, 92, 223, 133, 22, 72, 80, 71, 213, 121, 48, 165, 115, 245, 186, 99, 112, 64, 149, 154, 104, 116, 181, 129, 51, 161, 39, 88, 227, 209, 66, 153, 237, 86, 146, 76, 211, 62, 152, 168, 29, 111, 247, 26, 207, 136, 159, 200, 183, 135, 79, 97, 126, 157, 185, 15, 205, 234, 219, 250, 139, 118, 151, 256, 216)(9, 84, 96, 32, 25, 246, 244, 36, 169, 31, 156)(41, 132, 178, 226, 90, 182, 82, 253)(220, 238, 239)(202, 210).

In this part, the proposed innovative approach and suggested S-box as shown in Table 3 against widely accepted traditional methods is examined. To examine the cryptographic strength of the S-box, performance standards have been developed. Several key analyses are used to determine the robustness of the S-box, including nonlinearity, differential approximation probability, bit independence criteria, strict avalanche criteria, and linear approximation probability.

To obtain the original plaintext from an S-box constructed in such a way that the plaintext and cipher text have a linear mapping, it is simple to conduct a linear cryptanalysis attack. An S-box must be constructed with a strong nonlinear mapping among its input and output to withstand this assault. A test based on this criterion was introduced in 1988 by Pieprzyk et al. [21]. Using Eq. (4.1), one can determine the nonlinearity of an n-bit Boolean function.

The suggested S-box nonlinearity outcomes for eight balanced Boolean functions are 112, 112, 112, 112, 112, 112, 112, and 110, with a minimum of 110, a maximum of 112, and an average of 111.75. An assessment of the mean nonlinearity of the resultant S-box compared to those of other recent S-boxes is shown in Fig. 3. As can be observed, the final S-box has the necessary capacity to preserve the linearity, making linear cryptanalysis difficult for the attacker.

4.2 Strict Avalanche Criterion (SAC)

The strict avalanche criteria [22,23], are fundamental characteristic for every cryptographic S-box, stating that modifications in input and output bit values affect the strict avalanche criteria (SAC). When a single bit alters the input outcomes in a transfer of 1/2 of the output bits, an S-box encounters the SAC. An S-box with a SAC score close to 0.5 has reasonable ambiguity. Table 4 shows the dependency matrix containing the SAC values of the proposed S-box and maximum, minimum values of SAC are displays in the columns of this table. The average SAC value of the S-box is equal to 0.4988. This SAC number demonstrates that the suggested S-box satisfies the SAC property satisfactorily.

4.3 Bit Independent Criterion (BIC)

The bit independence criteria require pairwise comparisons of variables to determine their independence. According to this criterion [22,23], inverting an

In current block ciphers, the cryptologist aims to provide enough bit diffusion and uncertainty to prevent cryptanalysis. These requirements can be met by strong S-boxes by providing a nonlinear mapping between input and output. A low linear probability (LP) S-box suggests a greater nonlinear mapping and confers resistance to linear cryptanalysis. This criterion determines the greatest value of an event’s imbalance. Matsui [39] developed this analysis, and a mathematical formula for calculating the LP value of the S-box is presented below.

where

4.5 Differential Probability (DP)

Differential cryptanalysis is thought to be a valuable approach for obtaining the original plaintext. Variations in the plaintext and ciphertext are discovered throughout this attempt. Biham et al. proposed this test [40]. The differential uniformity of a Boolean function is calculated by requiring that the XOR values of each output have the same probability as the XOR values of each input. The formula for calculating DP is provided below.

Table 7 shows the comparison of differential probability values of the suggested S-box and several other S-boxes. Table 8 explains the differential uniformity values of the suggested S-boxes. Fig. 4 depicts a graphical analysis of the DP score of the suggested S-box and several other S-boxes.

High nonlinearity is an important criterion for constructing good cryptosystems. In comparison to the other S-boxes shown in Fig. 1, the created S-box’s mean NL score of 111.75 is relatively high. According to this NL score, linear attacks are exceedingly difficult to succeed against the S-box. The SAC score is 0.4988 as shown in Table 4, which is nearly equivalent to the SAC’s ideal score. With regard to nonlinearity, the resultant S-box’s mean BIC score is 103.64 as shown in Table 5, and the resultant S-box has an LP score of 0.1328, which is quite good when compared to existing S-boxes. The complete comparison of DP, BIC-NL, LP, and SAC demonstrate in Table 7. When these outcomes are compared to existing S-boxes, it is clear that the resultant S-box meets the typical S-box security protocols.

In this study, a group-theoretical and graphical method for creating the high nonlinear component of the AES block cipher was presented. This approach is simple, innovative, and dynamic in nature. The preliminary

Acknowledgement: Researchers Supporting Project number (RSP2024R167), King Saud University, Riyadh, Saudi Arabia.

Funding Statement: This Project is funded by King Saud University, Riyadh, Saudi Arabia.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

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*Computer Systems Science and Engineering*,

*47*

*(2)*, 2547-2562. https://doi.org/10.32604/csse.2023.040855

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*et al*., "An Innovative Technique for Constructing Highly Non-Linear Components of Block Cipher for Data Security against Cyber Attacks,"

*Comput. Syst. Sci. Eng.*, vol. 47, no. 2, pp. 2547-2562. 2023. https://doi.org/10.32604/csse.2023.040855