Batch generated strongly nonlinear S-Boxes using enhanced quadratic maps

Mohammad Mazyad Hazzazi; Farooq E Azam; Rashad Ali; Muhammad Kamran Jamil; Sameer Abdullah Nooh; Fahad Alblehai; Mohammad Mazyad Hazzazi; Farooq E Azam; Rashad Ali; Muhammad Kamran Jamil; Sameer Abdullah Nooh; Fahad Alblehai

doi:10.3934/math.2025262

AIMS Mathematics

2025, Volume 10, Issue 3: 5671-5695. doi: 10.3934/math.2025262

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Research article

Batch generated strongly nonlinear S-Boxes using enhanced quadratic maps

1.
Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia
2.
Department of Mathematics, Riphah International University, 50390 Lahore, Pakistan
3.
Department of Mathematics, University of Trento, 38122 Trento, Italy
4.
Department of Mathematics, Riphah International University, 54660 Lahore, Pakistan
5.
Faculty of Computing and Information Technology King AbdulAziz University Jeddah 80200, Saudi Arabia
6.
Computer Science Department, Community College, King Saud University, Riyadh 11437, Saudia Arabia

Received: 11 December 2024 Revised: 24 January 2025 Accepted: 20 February 2025 Published: 14 March 2025
MSC : 94A60, 68P25

One of the most crucial elements in the design of a block cipher is the substitution box or S-box. Its cipher strength directly impacts the cipher algorithm's security, and the block cipher algorithm requires a good S-box. According to the cryptanalysis result of the S-box construction in AES: (1) the number of irreducible polynomials can be increased to 30; (2) the affinity transformation constant c can be chosen from all elements if the existence of fixed points and reverse fixed points in an S-box is ignored; and (3) the S-box in AES is fixed, which poses possible security risks to the AES algorithm. The study above led us to build a non-degenerate 2D enhanced quadratic map (2D-EQM) with unpredictability and ergodicity. From there, we generated affine transformation constants and affine transformation matrices, which were then applied to seed S-boxes to create a batch of strongly nonlinear S-boxes. Finally, we assessed the performance of suggested S-boxes using six criteria. Security and statistical research showed that the suggested S-box batch generation procedure was practical and effective.
- 2D Enhanced Quadratic Map (2D-EQM),
- affine transformation,
- cryptographic algorithms,
- nonlinearity,
- S-Box design
Citation: Mohammad Mazyad Hazzazi, Farooq E Azam, Rashad Ali, Muhammad Kamran Jamil, Sameer Abdullah Nooh, Fahad Alblehai. Batch generated strongly nonlinear S-Boxes using enhanced quadratic maps[J]. AIMS Mathematics, 2025, 10(3): 5671-5695. doi: 10.3934/math.2025262

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Abstract

One of the most crucial elements in the design of a block cipher is the substitution box or S-box. Its cipher strength directly impacts the cipher algorithm's security, and the block cipher algorithm requires a good S-box. According to the cryptanalysis result of the S-box construction in AES: (1) the number of irreducible polynomials can be increased to 30; (2) the affinity transformation constant c can be chosen from all elements if the existence of fixed points and reverse fixed points in an S-box is ignored; and (3) the S-box in AES is fixed, which poses possible security risks to the AES algorithm. The study above led us to build a non-degenerate 2D enhanced quadratic map (2D-EQM) with unpredictability and ergodicity. From there, we generated affine transformation constants and affine transformation matrices, which were then applied to seed S-boxes to create a batch of strongly nonlinear S-boxes. Finally, we assessed the performance of suggested S-boxes using six criteria. Security and statistical research showed that the suggested S-box batch generation procedure was practical and effective.

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