FPGA-Based Cosimulation of S-Box Constitution from Fractional Order Liu System

Öz

Substitution box(S-box) has an important duty in encryption. In recent years plenty of S-box designs by chaotic systems have been studied. In this paper a high speed S-box constitution from fractional order Liu system on FPGA is proposed. The design was implemented by using Xilinx System Generator(XSG) toolbox in MATLAB/Simulink software in single precision floating point numbers format and a low cost Basys 3 FPGA trainer card was used. In addition, numerical fractional order equations was solved by using Grünwald-Letnikov(GL) method with 1024 elements length. The S-box table was constructed about 0.137s. After all performance analyzes as nonlinearity, strict avalanche criterion, differential probability, bits independent criterion, and equiprobable input/output XOR distribution, were performed. The performance analyzes shown that the design was satisfactory.

Anahtar Kelimeler

• S-box
• fractional order systems
• Liu chaotic system
• FPGA

References

1. [1] Kocarev, L. chaos-based cryptography: a brief overview. IEEE Circuits and Systems Magazine 2001;1(3), 6-21.
2. [2] Kiani-B A, Fallahi K, Pariz N, Leung H. A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter. Communications in Nonlinear Science and Numerical Simulation 2009;14:863–79. https://doi.org/https://doi.org/10.1016/j.cnsns.2007.11.011.
3. [3] Zhang X. Relationship between integer order systems and fractional order system and its two applications. IEEE/CAA Journal of Automatica Sinica 2018;5:639–43. https://doi.org/10.1109/JAS.2016.7510205.
4. [4] Petráš I. Fractional-order nonlinear systems: modeling, analysis and simulation. Springer Science & Business Media; 2011.
5. [5] Tolba MF, AbdelAty AM, Soliman NS, Said LA, Madian AH, Azar AT, et al. FPGA implementation of two fractional order chaotic systems. AEU - International Journal of Electronics and Communications 2017;78:162–72. https://doi.org/https://doi.org/10.1016/j.aeue.2017.04.028.
6. [6] Ávalos-Ruiz LF, Zúñiga-Aguilar CJ, Gómez-Aguilar JF, Escobar-Jiménez RF, Romero-Ugalde HM. FPGA implementation and control of chaotic systems involving the variable-order fractional operator with Mittag–Leffler law. Chaos, Solitons & Fractals 2018;115:177–89. https://doi.org/https://doi.org/10.1016/j.chaos.2018.08.021.
7. [7] Tlelo-Cuautle E, Rangel-Magdaleno JJ, Pano-Azucena AD, Obeso-Rodelo PJ, Nunez-Perez JC. FPGA realization of multi-scroll chaotic oscillators. Communications in Nonlinear Science and Numerical Simulation 2015;27:66–80. https://doi.org/https://doi.org/10.1016/j.cnsns.2015.03.003.
8. [8] Pano-Azucena AD, Ovilla-Martinez B, Tlelo-Cuautle E, Manuel Muñoz-Pacheco J, de la Fraga LG. FPGA-based implementation of different families of fractional-order chaotic oscillators applying Grünwald–Letnikov method. Communications in Nonlinear Science and Numerical Simulation 2019;72:516–27. https://doi.org/https://doi.org/10.1016/j.cnsns.2019.01.014.

9. [9] Karakaya B, Gülten A, Frasca M. A true random bit generator based on a memristive chaotic circuit: Analysis, design and FPGA implementation. Chaos, Solitons & Fractals 2019;119:143–9. https://doi.org/https://doi.org/10.1016/j.chaos.2018.12.021.
10. [10] Shannon CE. Communication Theory of Secrecy Systems. Bell System Technical Journal 1949; 28(4):656–715. DOI: 10.1002/j.1538-7305.1949.tb00928.x.
11. [11] Saber M, Hagras EAA. Parallel multi-layer selector S-Box based on lorenz chaotic system with FPGA implementation. Indonesian Journal of Electrical Engineering and Computer Science 2020;19:784–92. https://doi.org/10.11591/ijeecs.v19.i2.pp784-792.
12. [12] Elsafty AH, Tolba MF, Said LA, Madian AH, Radwan AG. FPGA Speech Encryption Realization Based on Variable S-Box and Memristor Chaotic Circuit. 2018 30th International Conference on Microelectronics (ICM), 2018, p. 152–5. https://doi.org/10.1109/ICM.2018.8704019.
13. [13] Yassin HM, Mohamed AT, Abdel-Gawad AH, Tolba MF, Saleh HI, Madian AH, et al. Speech Encryption on FPGA Using a Chaotic Generator and S-Box Table. 2019 Fourth International Conference on Advances in Computational Tools for Engineering Applications (ACTEA), 2019, p. 1–4. https://doi.org/10.1109/ACTEA.2019.8851086.
14. [14] ElSafty AH, Tolba MF, Said LA, Madian AH, Radwan AG. Hardware realization of a secure and enhanced s-box based speech encryption engine. Analog Integrated Circuits and Signal Processing 2021;106:385–97. https://doi.org/10.1007/s10470-020-01614-z.
15. [15] Abd El-Maksoud AJ, Abd El-Kader AA, Hassan BG, Rihan NG, Tolba MF, Said LA, et al. FPGA implementation of sound encryption system based on fractional-order chaotic systems. Microelectronics Journal 2019;90:323–35. https://doi.org/https://doi.org/10.1016/j.mejo.2019.05.005.
16. [16] Maazouz M, Toubal A, Bengherbia B, Houhou O, Batel N. FPGA implementation of a chaos-based image encryption algorithm. Journal of King Saud University - Computer and Information Sciences 2022. https://doi.org/https://doi.org/10.1016/j.jksuci.2021.12.022.
17. [17] Hafsa A, Gafsi M, Malek J, MacHhout M. FPGA implementation of improved security approach for medical image encryption and decryption. Scientific Programming 2021;2021. https://doi.org/10.1155/2021/6610655.
18. [18] Yang CH, Chien YS. FPGA implementation and design of a hybrid chaos-aes color image encryption algorithm. Symmetry 2020;12. https://doi.org/10.3390/sym12020189.
19. [19] Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier; 1998.
20. [20] Aydin, Y., Garipcan, A. M., & Özkaynak, F. (2025). A novel secure S-box design methodology based on FPGA and SHA-256 hash algorithm for block cipher algorithms. Arabian Journal for Science and Engineering, 50(2), 1247-1260.
21. [21] Garipcan, A. M., Aydın, Y., & Özkaynak, F. (2025). A novel s-box generation method based on metastable inducing over FPGA for block ciphers. Knowledge-Based Systems, 310, 112968.
22. [22] Malal, A., & Tezcan, C. (2024). FPGA-friendly compact and efficient AES-like 8× 8 S-box. Microprocessors and Microsystems, 105, 105007.
23. [23] Hong, R., Zhang, L., Pan, Z., Xiao, C., & Wang, J. (2025). Research and implementation of large-scale S-box for MK-3 algorithm based on polynomial basis: in FPGA. Journal of Cryptographic Engineering, 15(1), 1-13.
24. [24] Elrefai, H. M., Sayed, W. S., & Said, L. A. (2024). Hardware implementation of a 2D chaotic map-based audio encryption system using s-box. Electronics, 13(21), 4254.
25. [25] Liu C, Liu L, Liu T. A novel three-dimensional autonomous chaos system. Chaos, Solitons & Fractals 2009;39:1950–8. https://doi.org/https://doi.org/10.1016/j.chaos.2007.06.079.
26. [26] https://digilent.com/reference/programmable-logic/basys-3/reference-manual
27. [27] Rukhin A, Soto J, Nechvatal J, Smid M, Barker E, Leigh S, et al. NIST Special Publication 800-22 (with revisions dated 2001;22.
28. [28] Özkaynak F. An Analysis and Generation Toolbox for Chaotic Substitution Boxes: A Case Study Based on Chaotic Labyrinth Rene Thomas System. Iranian Journal of Science and Technology - Transactions of Electrical Engineering 2020;44:89–98. https://doi.org/10.1007/s40998-019-00230-6.
29. [29] Fumy W. On the F-function of FEAL. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 1988; 293 LNCS(2):434–437. DOI: 10.1007/3-540-48184-2_38.
30. [30] Adams C, Tavares S. The structured design of cryptographically good s-boxes. Journal of Cryptology 1990; 3(1):27–41. DOI: 10.1007/BF00203967.
31. [31] Webster, A.F.; Tavares, S.E. On the design of s-boxes. In Conference on the Theory and Application of Cryptographic Techniques; Williams, H.C., Ed.; Springer: Berlin/Heidelberg, Germany, 1986; pp. 523–534.
32. [32] Biham E, Shamir A. Differential cryptanalysis of feal and N-hash. Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 1991;547 LNCS:1–16. https://doi.org/10.1007/3-540-46416-6_1.
33. [33] Wang X, Teng L, Qin X. A novel colour image encryption algorithm based on chaos. Signal Processing 2012;92:1101–8. https://doi.org/https://doi.org/10.1016/j.sigpro.2011.10.023.
34. [34] Zhang YQ, Wang XY. Analysis and improvement of a chaos-based symmetric image encryption scheme using a bit-level permutation. Nonlinear Dynamics 2014;77:687–98. https://doi.org/10.1007/s11071-014-1331-3.
35. [35] Özkaynak F. A novel method to improve the performance of chaos based evolutionary algorithms. Optik 2015;126:5434–8. https://doi.org/https://doi.org/10.1016/j.ijleo.2015.09.098.
36. [36] Farah T, Rhouma R, Belghith S. A novel method for designing S-box based on chaotic map and Teaching–Learning-Based Optimization. Nonlinear Dynamics 2017;88:1059–74. https://doi.org/10.1007/s11071-016-3295-y.
37. [37] Khan M, Shah T. A construction of novel chaos base nonlinear component of block cipher. Nonlinear Dynamics 2014;76:377–82. https://doi.org/10.1007/s11071-013-1132-0.
38. [38] Özkaynak F, Özer AB. A method for designing strong S-Boxes based on chaotic Lorenz system. Physics Letters A 2010;374:3733–8. https://doi.org/https://doi.org/10.1016/j.physleta.2010.07.019.
39. [39] Özkaynak F. On the effect of chaotic system in performance characteristics of chaos based s-box designs. Physica A: Statistical Mechanics and Its Applications 2020;550:124072. https://doi.org/https://doi.org/10.1016/j.physa.2019.124072.
40. [40] Wang Y, Zhang Z, Zhang LY, Feng J, Gao J, Lei P. A genetic algorithm for constructing bijective substitution boxes with high nonlinearity. Information Sciences 2020;523:152–66. https://doi.org/https://doi.org/10.1016/j.ins.2020.03.025.
41. [41] Liu L. Designing a random S-box with the mixed spatiotemporal chaos. Journal of Physics: Conference Series 2021; 1983:0–7. https://doi.org/10.1088/1742-6596/1983/1/012040.
42. [42] Cassal-Quiroga BB, Campos-Cantón E. Generation of Dynamical S-Boxes for Block Ciphers via Extended Logistic Map. Mathematical Problems in Engineering 2020;2020. https://doi.org/10.1155/2020/2702653.
43. [43] Türk Ö. FPGA simulation of chaotic tent map-based S-Box design. International Journal of Circuit Theory and Applications n.d.;n/a. https://doi.org/https://doi.org/10.1002/cta.3242.