Designing pair of nonlinear components of a block cipher over quaternion integers

Muhammad Sajjad; Tariq Shah; Huda Alsaud; Maha Alammari; Muhammad Sajjad; Tariq Shah; Huda Alsaud; Maha Alammari

doi:10.3934/math.20231074

AIMS Mathematics

2023, Volume 8, Issue 9: 21089-21105. doi: 10.3934/math.20231074

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Designing pair of nonlinear components of a block cipher over quaternion integers

1.
Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan
2.
Department of Mathematics, College of Science, King Saud University, P.O. Box 22452 Riyadh 11495, Saudi Arabia

Received: 19 May 2023 Revised: 23 June 2023 Accepted: 23 June 2023 Published: 03 July 2023
MSC : 68P25, 68U15

In the field of cryptography, block ciphers are widely used to provide confidentiality and integrity of data. One of the key components of a block cipher is its nonlinear substitution function. In this paper, we propose a new design methodology for the nonlinear substitution function of a block cipher, based on the use of Quaternion integers (QI). Quaternions are an extension of complex numbers that allow for more complex arithmetic operations, which can enhance the security of the cipher. We demonstrate the effectiveness of our proposed design by implementing it in a block cipher and conducting extensive security analysis. Quaternion integers give pair of substitution boxes (S-boxes) after fixing parameters but other structures give only one S-box after fixing parameters. Our results show that the proposed design provides superior security compared to existing designs, two making on a promising approach for future cryptographic applications.
- quaternion integers,
- residue class,
- block cipher,
- security analysis
Citation: Muhammad Sajjad, Tariq Shah, Huda Alsaud, Maha Alammari. Designing pair of nonlinear components of a block cipher over quaternion integers[J]. AIMS Mathematics, 2023, 8(9): 21089-21105. doi: 10.3934/math.20231074

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Abstract

In the field of cryptography, block ciphers are widely used to provide confidentiality and integrity of data. One of the key components of a block cipher is its nonlinear substitution function. In this paper, we propose a new design methodology for the nonlinear substitution function of a block cipher, based on the use of Quaternion integers (QI). Quaternions are an extension of complex numbers that allow for more complex arithmetic operations, which can enhance the security of the cipher. We demonstrate the effectiveness of our proposed design by implementing it in a block cipher and conducting extensive security analysis. Quaternion integers give pair of substitution boxes (S-boxes) after fixing parameters but other structures give only one S-box after fixing parameters. Our results show that the proposed design provides superior security compared to existing designs, two making on a promising approach for future cryptographic applications.

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