A two-stage algebraic framework for designing cryptographically strong S-boxes for secure image encryption
4 affiliations
- 1Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa, Saudi Arabia
- 2Department of Mathematics, Division of Science and Technology, University of Education, Lahore, Pakistan
- 3Department of Mathematics, College of Science & Arts, King Abdul Aziz University, Rabigh, Saudi Arabia
- 4Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
Abstract
The security of modern block ciphers heavily relies on the cryptographic strength of Substitution boxes (S-boxes), which introduce non-linearity and confusion. While numerous methods for S-box generation exist, there is a continuous pursuit of designs that offer a superior and balanced profile against various cryptanalytic attacks. This paper presents a new, two-step approach to the construction of highly nonlinear S-boxes, with group-theoretic underpinnings. During the first step, we use a parameterized permutation representation of the triangle group Δ(2,3,43) on the projective line over the finite field F257, denoted by P1(F257), to produce an initial S-box with a nonlinearity of 106. The second stage is the main innovation, in which we use a refinement process via the action of the permutation group C7406532×C3 on this original S-box. The refinement increases the average nonlinearity to 112 and yields differential uniformity 4, linear approximation probability 0.0625, strong bit-independence behavior, and strict avalanche characteristics close to the ideal value. The proposed S-box is then incorporated into a multi-round grayscale image-encryption scheme based on fixed S-box substitution, key- and nonce-dependent position permutation, SHA-256-based keystream whitening, and a final bidirectional nonlinear diffusion layer. Experiments on Moon Surface, Airplane, and Clock images produce ciphertext entropies of 7.9968–7.9975, near-zero pixel correlations, NPCR values of 99.7040%–99.7559%, and UACI values of 33.4902%–33.6390%. Cropping and noise experiments further show exact recovery without attack and useful attack-dependent reconstruction under ciphertext degradation, particularly under mild cropping and low-density salt-and-pepper noise. These results demonstrate strong statistical security, differential sensitivity, and practical applicability for grayscale image protection.
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BibTeX
@article{razzaque2026twostage,
title = {{A two-stage algebraic framework for designing cryptographically strong S-boxes for secure image encryption}},
author = {Asima Razzaque and Hafiza Zara Mustafa and Dilshad Alghazzawi and Ghaliah Alhamzi},
journal = {PLOS One},
year = {2026},
month = sep,
doi = {10.1371/journal.pone.0357107},
url = {https://doi.org/10.1371/journal.pone.0357107},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A two-stage algebraic framework for designing cryptographically strong S-boxes for secure image encryption
AU - Asima Razzaque
AU - Hafiza Zara Mustafa
AU - Dilshad Alghazzawi
AU - Ghaliah Alhamzi
PY - 2026
DA - 2026/09/15
JO - PLOS One
DO - 10.1371/journal.pone.0357107
UR - https://doi.org/10.1371/journal.pone.0357107
AB - The security of modern block ciphers heavily relies on the cryptographic strength of Substitution boxes (S-boxes), which introduce non-linearity and confusion. While numerous methods for S-box generation exist, there is a continuous pursuit of designs that offer a superior and balanced profile against various cryptanalytic attacks. This paper presents a new, two-step approach to the construction of highly nonlinear S-boxes, with group-theoretic underpinnings. During the first step, we use a parameterized permutation representation of the triangle group Δ(2,3,43) on the projective line over the finite field F257, denoted by P1(F257), to produce an initial S-box with a nonlinearity of 106. The second stage is the main innovation, in which we use a refinement process via the action of the permutation group C7406532×C3 on this original S-box. The refinement increases the average nonlinearity to 112 and yields differential uniformity 4, linear approximation probability 0.0625, strong bit-independence behavior, and strict avalanche characteristics close to the ideal value. The proposed S-box is then incorporated into a multi-round grayscale image-encryption scheme based on fixed S-box substitution, key- and nonce-dependent position permutation, SHA-256-based keystream whitening, and a final bidirectional nonlinear diffusion layer. Experiments on Moon Surface, Airplane, and Clock images produce ciphertext entropies of 7.9968–7.9975, near-zero pixel correlations, NPCR values of 99.7040%–99.7559%, and UACI values of 33.4902%–33.6390%. Cropping and noise experiments further show exact recovery without attack and useful attack-dependent reconstruction under ciphertext degradation, particularly under mild cropping and low-density salt-and-pepper noise. These results demonstrate strong statistical security, differential sensitivity, and practical applicability for grayscale image protection.
SN - 1932-6203
ER -