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A Novel Scheme for Designing Secure Substitution Boxes (S-Boxes) Based on Mobius Group and Finite Field

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Abstract

Data encryption with substitution-box (S-box) is now widely regarded as the most secure method of communicating in order to prevent unauthorized access to sensitive and confidential information. An S-box is the basic component in a cryptosystem, which provides confusion in data. An S-box that holds a high degree of nonlinearity, as well as a low level of linear and differential approximation probability, is considered cryptographically secure. In this regard, cryptography plays an important role in the encryption of private data and information. The field of graph theory is important in a variety of domains. Graph theory, in particular, is commonly utilized as an encryption technique, as graphs can be transformed into matrices. In this study, we use graph theory and adjacency matrices to generate highly nonlinear S-boxes. We use an adjacency matrix corresponding to the coset diagram for the action of the Mobius group \(M = \left\langle {x,y:x^{2} = y^{6} = 1} \right\rangle\) on \(PL\left( {F_{7} } \right)\). We then use this matrix to the Galois field \(GF\left( {2^{8} } \right)\) to obtain new S-boxes. The proposed S-boxes are then analyzed using algebraic and statistical methods. The analyses and findings are encouraging, demonstrating that the proposed S-boxes are well protected. Also, as a means of confusion, the proposed S-boxes have a high level of security and enhance resistance against the potential attacks.

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Availability of data and material (data transparency)

The data that support the findings of this study are available from the corresponding author, Mr. Bilal Arshad, upon reasonable request.

Code availability

Due to the nature of this research, participants of this study did not agree for their code to be shared publicly, so supporting code is not available.

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Authors and Affiliations

  1. Department of Basic Sciences, University of Engineering and Technology, Taxila, Pakistan

    Bilal Arshad & Nasir Siddiqui

  2. Department of Mathematics, University of Wah, Wah Cantt, Pakistan

    Zamir Hussain

  3. Department of Creative Technologies, Air University, Islamabad, Pakistan

    Muhammad Ehatisham-ul-Haq

Authors
  1. Bilal Arshad
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  2. Nasir Siddiqui
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  3. Zamir Hussain
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  4. Muhammad Ehatisham-ul-Haq
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All authors have made equally contributions to the writing of this research paper.

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Correspondence to Bilal Arshad.

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Arshad, B., Siddiqui, N., Hussain, Z. et al. A Novel Scheme for Designing Secure Substitution Boxes (S-Boxes) Based on Mobius Group and Finite Field. Wireless Pers Commun 124, 3527–3548 (2022). https://doi.org/10.1007/s11277-022-09524-1

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