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A Mathematical Approach for Generating a Highly Non-Linear Substitution Box Using Quadratic Fractional Transformation

Abid Mahboob1, Muhammad Asif2, Rana Muhammad Zulqarnain3,*, Imran Saddique4, Hijaz Ahmad5, Sameh Askar6

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, 51310, Pakistan
3 School of Mathematical Sciences, Zhejiang Normal University, Jinhua, 321004, China
4 Department of Mathematics, University of Management and Technology, Lahore, 54770, Pakistan
5 Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, Roma, 39-00186, Italy
6 Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

* Corresponding Author: Rana Muhammad Zulqarnain. Email: email

(This article belongs to this Special Issue: Multimedia Encryption and Information Security)

Computers, Materials & Continua 2023, 77(2), 2565-2578.


Nowadays, one of the most important difficulties is the protection and privacy of confidential data. To address these problems, numerous organizations rely on the use of cryptographic techniques to secure data from illegal activities and assaults. Modern cryptographic ciphers use the non-linear component of block cipher to ensure the robust encryption process and lawful decoding of plain data during the decryption phase. For the designing of a secure substitution box (S-box), non-linearity (NL) which is an algebraic property of the S-box has great importance. Consequently, the main focus of cryptographers is to achieve the S-box with a high value of non-linearity. In this suggested study, an algebraic approach for the construction of 16 × 16 S-boxes is provided which is based on the fractional transformation and finite field. This technique is only applicable for the even number exponent in the range (2-254) that are not multiples of 4. Firstly, we choose a quadratic fractional transformation, swap each missing element with repeating elements, and acquire the initial S-box. In the second stage, a special permutation of the symmetric group is utilized to construct the final S-box, which has a higher NL score of 112.75 than the Advanced Encryption Standard (AES) S-box and a lower linear probability score of 0.1328. In addition, a tabular and graphical comparison of various algebraic features of the created S-box with many other S-boxes from the literature is provided which verifies that the created S-box has the ability and is good enough to withstand linear and differential attacks. From different analyses, it is ensured that the proposed S-boxes are better than as compared to the existing S-boxes. Further these S-boxes can be utilized in the security of the image data and the text data.


Cite This Article

A. Mahboob, M. Asif, R. M. Zulqarnain, I. Saddique, H. Ahmad et al., "A mathematical approach for generating a highly non-linear substitution box using quadratic fractional transformation," Computers, Materials & Continua, vol. 77, no.2, pp. 2565–2578, 2023.

cc This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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