Algebraic construction of secure S-boxes using finite fields
*Deepak KumarCorresponding authordeepak.kumar1@bmu.edu.inDepartment of Computer Science and EngineeringSchool of Engineering and TechnologyBML Munjal UniversitySidhrawali, Gurugram, Haryana, 122413, IndiaView full profile → , Kothakonda Sairamkothakondasairam.ks@proton.meCentre for Multidisciplinary ResearchAnurag UniversityHyderabad, Telangana, 500088, IndiaView full profile → , V. S. N. Kumarvsnkumar@gmail.comDepartment of Computer ScienceAditya Degree & PG Colleges (Autonomous)Kakinada, Andhra Pradesh, 533003, IndiaView full profile → , D. Balajibalaji@avit.ac.inDepartment of Humanities and Sciences (Mathematics Division)Aarupadai Veedu Institute of TechnologyVinayaka Mission’s Research Foundation (Deemed to be University)Paiyanoor, Chennai, Tamil Nadu, 603104, IndiaView full profile → , Lokesh Vermalokesh.verma.orp@chitkara.edu.inCentre for Research Impact and OutcomeChitkara UniversityRajpura, Punjab, 140417, IndiaView full profile → , Pooja Dilip Moreypooja.morey@vit.eduDepartment of Artificial Intelligence & Data scienceVishwakarma Institute of TechnologyBibwewadi, Pune, Maharashtra, 411037, IndiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Apr 2026
- Published Online:
- 31 Oct 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-3027
- Pages:
- 4009–4019
Abstract
S-Boxes are important devices in the current cryptographic systems, which guarantee confusion and antagonism against attacks. This article extends a mathematical model of the building of resilient S-Boxes with the use of finite fields GF(2 n). The method takes advantage of multiplicative inverses, affine transformations, and permutation polynomials to provide high nonlinearity and low differential uniformity. Irreducible structures and polynomial representations are also used to come up with algebraic normal forms that increase the cryptographic strength. The experimental analysis shows an enhancement of performance, with a nonlinearity of above 112 and a differential uniformity of 4 surpassing traditional designs by a factor of about 12. The suggested technique guarantees a balanced output, great avalanche effect, as well as resistance to linear and differential cryptanalysis. It has the scalability that enables a flexible generation of S-Boxes, allowing lightweight and high-security applications, and adds a systematic and efficient design strategy. Future work examines the optimization of hardware and its resistance to side-channel attacks, and how to integrate both into more advanced encryption architecture that can be implemented in practice.
Keywords
Subject Classifications
References
[1] D. C. Ventura, L. M. X. Rodríguez-Henríquez, and S. E. P. Hernández, “Understanding S-box security assessment: A practical guide,” Mathematical and Computational Applications, vol. 31, no. 1, Art. no. 27 (2026), doi: 10.3390/mca31010027.
[2] A. Prathiba and V. S. K. Bhaaskaran, “Lightweight S-box architecture for secure Internet of Things,” Information, vol. 9, no. 1, Art. no. 13 (2018), doi: 10.3390/info9010013.
[3] P.-P. Duong and C.-K. Pham, “Constructing 8 × 8 S-boxes with optimal Boolean function nonlinearity,” Cryptography, vol. 9, no. 4, Art. no. 67 (2025), doi: 10.3390/cryptography9040067.
[4] P.-P. Duong, T.-K. Dang, T.-T. Hoang, and C.-K. Pham, “Compact 8-bit S-boxes based on multiplication in a Galois field GF(24),” Cryptography, vol. 9, no. 2, Art. no. 21 (2025), doi: 10.3390/cryptography9020021.
[5] M. I. Huilgol and S. Asok, “PBIB-designs with constant block sums arising from different graph parameters,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 28, no. 7, pp. 2697–2721 (2025), doi: 10.47974/JDMSC-2064.
[6] M. Dimitrov and T. Baicheva, “On the pentanomial power mapping classification of 8-bit to 8-bit S-boxes,” Mathematics, vol. 12, no. 14, Art. no. 2154 (2024), doi: 10.3390/math12142154.
[7] L. C. Nizam Chew and E. S. Ismail, “S-box construction based on linear fractional transformation and permutation function,” Symmetry, vol. 12, no. 5, Art. no. 826 (2020), doi: 10.3390/sym12050826.
[8] R. C. Poonia, T. Singh, C. K. Rajashri, D. Mondal, and S. M. Patil, “Optimizing communication system security through comprehensive testing and intelligent network configurations,” Journal of Information and Optimization Sciences, vol. 47, no. 5-A, pp. 1709–1717 (2026), doi: 10.47974/JIOS-2261.
[9] J. Ni, J. Zhang, G. Wang, R. Li, and Y. Shen, “Algebraic attacks against Grendel: An arithmetization-oriented primitive with the Legendre symbol,” Symmetry, vol. 15, no. 8, Art. no. 1563 (2023), doi: 10.3390/sym15081563.
[10] Z. Jiang and Q. Ding, “Construction of an S-box based on chaotic and bent functions,” Symmetry, vol. 13, no. 4, Art. no. 671 (2021), doi: 10.3390/sym13040671.
[11] A. Kuznetsov, L. Wieclaw, N. Poluyanenko, L. Hamera, S. Kandiy, and Y. Lohachova, “Optimization of a simulated annealing algorithm for S-boxes generating,” Sensors, vol. 22, no. 16, Art. no. 6073 (2022), doi: 10.3390/s22166073.
[12] W. N. Anandpwar, S. M. Barhate, and M. P. Dhore, “Designing an efficient machine learning-based intrusion detection system for enhanced cyber security in modern networks,” International Journal on Advanced Computer Theory and Engineering, vol. 14, no. 1, pp. 70–75 (2025), doi: 10.65521/ijacte.v14i1.216.
[13] R. C. Poonia, A. Almakki, A. K. J. Saudagar, A. Altameem, and M. Albathan, “An optimization approach for real-time object detection in IoT devices through edge computing and deep learning,” Journal of Information and Optimization Sciences, vol. 45, no. 5, pp. 1465–1475 (2024), doi: 10.47974/JIOS-1768.
[14] E. Özpolat, V. Çelik, and A. Gülten, “Hyperchaotic system-based PRNG and S-box design for a novel secure image encryption,” Entropy, vol. 27, no. 3, Art. no. 299 (2025), doi: 10.3390/e27030299.
