Generating Hyperchaotic Attractor via the Sprott B System and Its Application in Highly Secure Voice Encryption

Haneche Nabil; Tayeb Hamaizia

doi:10.32323/ujma.1684079

Generating Hyperchaotic Attractor via the Sprott B System and Its Application in Highly Secure Voice Encryption

Abstract

This paper presents a new 4D system constructed based on the well-known Sprott B system and characterized by high complexity. This system can display chaotic, hyperchaotic, periodic, and quasi-periodic behaviors for some parameter values. The findings demonstrate that the novel system can exhibit multistability and different coexisting attractors. Additionally, the study used Lyapunov exponents, bifurcation diagrams, stability of equilibrium points, dissipativity, phase plots, dynamical maps, and Poincare sections to analyze the dynamical characteristics of this system. Since the novel 4D system is very advantageous for chaos-based applications due to its hyperchaotic dynamics and multistability, an efficient voice encryption scheme that uses the proposed hyperchaotic system in secure voice transmission is presented. Experiments and tests are carried out to evaluate the performance and security of the proposed encryption scheme against various attacks. The simulation results demonstrate the robustness of our voice cryptosystem against cryptographic attacks.

Keywords

Hyperchaos, Lyapunov exponents, Multistability, Sprott B system, Voice encryption

References

[1] H. Jahanshahi, K. Rajagopal, A. Akgul, N. M. Sari, H. Namazi, S. Jafari, Complete analysis and engineering applications of a megastable nonlinear oscillator, Int. J. Non-Linear Mech., 107 (2018), 126-136. https://doi.org/10.1016/j.ijnonlinmec.2018.08.020
[2] K. Bukht Mehdi, A. A. Mousa, S. A. Baloch, U. Demirbilek, A. Ghallab, I. Siddique, R. M. Zulqarnain, Exploration of soliton dynamics and chaos in the Landau-Ginzburg-Higgs equation through extended analytical approaches, J. Nonlinear Math. Phys., 32(1) (2025), Article ID 22. https://doi.org/10.1007/s44198-025-00272-x
[3] U. Demirbilek, A. H. Tedjani, A. R. Seadawy, Analytical solutions of the combined Kairat-II-X equation: A dynamical perspective on bifurcation, chaos, energy, and sensitivity, AIMS Math., 10(6) (2025), 13664-13691. https://doi.org/10.3934/math.2025615
[4] X. Wang, Y. Chen, A new chaotic image encryption algorithm based on L-shaped method of dynamic block, Sens. Imaging, 22(1) (2021), Article ID 31. https://doi.org/10.1007/s11220-021-00357-z
[5] U. Demirbilek, S. A. Baloch, I. Siddique, H. Bulut, T. Radwan, Exploring lump solutions, bifurcation, sensitivity, and chaotic dynamics in the extended KP-Boussinesq equation, J. Nonlinear Math. Phys., 32 (2025), Article ID 49. https://doi.org/10.1007/s44198-025-00304-6
[6] E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20(2) (1963), 130-141. https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[7] O. E. Rössler, The chaotic hierarchy, Z. Naturforsch. A, 38a(7) (1983), 788-801. https://doi.org/10.1515/zna-1983-0714
[8] T. Matsumoto, A chaotic attractor from Chua’s circuit, IEEE Trans. Circuits Syst., 31(12) (1984), 1055-1058. https://doi.org/10.1109/TCS.1984.1085459
[9] J. Lü, G. Chen, A new chaotic attractor coined, Int. J. Bifur. Chaos, 12(03) (2002), 659-661. http://dx.doi.org/10.1142/S0218127402004620
[10] G. Chen, T. Ueta, Yet another chaotic attractor, Int. J. Bifur. Chaos, 9(07) (1999), 1465-1466. http://dx.doi.org/10.1142/S0218127499001024

[11] J. C. Sprott, Simple chaotic systems and circuits, Am. J. Phys., 68(8) (2000), 758-763. https://doi.org/10.1119/1.19538
[12] O. E. Rössler, An equation for hyperchaos, Phys. Lett. A, 71(2-3) (1979), 155-157. https://doi.org/10.1016/0375-9601(79)90150-6
[13] X. Cheng, H. Zhu, L. Liu, K. Mao, J. Liu, Dynamic analysis of a novel hyperchaotic system based on STM32 and application in image encryption, Sci. Rep., 14(1) (2024), Article ID 20452. http://dx.doi.org/10.1038/s41598-024-71338-x
[14] S. Zhu, C. Zhu, Secure image encryption algorithm based on hyperchaos and dynamic DNA coding, Entropy, 22(7) (2020), Article ID 772. https://doi.org/10.3390/e22070772
[15] J. Fang, J. Wang, K. Zhao, Y. Jiang, W. Liang, A new dual memristor hyperchaotic system: Dynamic properties, electronic circuit, and image encryption, Analog Integr. Circuits Signal Process., 122(2) (2025), Article ID 20. https://doi.org/10.1007/s10470-025-02322-2
[16] H. Lin, C. Wang, L. Cui, Y. Sun, X. Zhang, W. Yao, Hyperchaotic memristive ring neural network and application in medical image encryption, Nonlinear Dyn., 110(1) (2022), 841-855. https://doi.org/10.1007/s11071-022-07630-0
[17] S. Kaouache, Projective synchronization of the modified fractional-order hyperchaotic Rössler system and its application in secure communication, Univ. J. Math. Appl., 4(2) (2021), 50-58. https://doi.org/10.32323/ujma.739649
[18] N. Haneche, T. Hamaizia, A secure communication scheme based on generalized modified projective synchronization of a new 4-D fractional-order hyperchaotic system, Phys. Scr., 99(9) (2024), Article ID 095203. http://dx.doi.org/10.1088/1402-4896/ad6515
[19] X. Wang, M. Wang, A hyperchaos generated from Lorenz system, Phys. A: Stat. Mech. Appl., 387(14) (2008), 3751–3758. https://doi.org/10.1016/j.physa.2008.02.020
[20] S. Pang, Y. Liu, A new hyperchaotic system from the Lü system and its control, J. Comput. Appl. Math., 235(8) (2011), 2775-2789. https://doi.org/10.1016/j.cam.2010.11.029
[21] Y. Li, W. K. Tang, G. Chen, Generating hyperchaos via state feedback control, Int. J. Bifur. Chaos, 15(10) (2005), 3367-3375. https://doi.org/10.1142/S0218127405013988
[22] J. Wang, Z. Chen, G. Chen, Z. Yuan, A novel hyperchaotic system and its complex dynamics, Int. J. Bifur. Chaos, 18(11) (2008), 3309-3324. https://doi.org/10.1142/S0218127408022391
[23] L. Stenflo, Generalized Lorenz equations for acoustic-gravity waves in the atmosphere, Phys. Scr., 53(1) (1996), Article ID 83. https://doi.org/10.1088/0031-8949/53/1/015
[24] M. Yan, H. Xu, The multi-scroll hyper-chaotic coexistence attractors and its application, Signal Process. Image Commun., 95 (2021), Article ID 116210. http://dx.doi.org/10.1016/j.image.2021.116210
[25] Z. Jiang, X. Liu, Image encryption algorithm based on discrete quantum Baker map and Chen hyperchaotic system, Int. J. Theor. Phys., 62(2) (2023), Article ID 22. https://doi.org/10.1007/s10773-023-05277-0
[26] S. Fu, X. Cheng, J. Liu, Dynamics, circuit design, feedback control of a new hyperchaotic system and its application in audio encryption, Sci. Rep., 13(1) (2023), Article ID 19385. https://doi.org/10.1038/s41598-023-46161-5
[27] S. Zhou, Y. Yin, U. Erkan, A. Toktas, Y. Zhang, Novel hyperchaotic system: Implementation to audio encryption, Chaos Solitions Fractals, 193 (2025), Article ID 116088. https://doi.org/10.1016/j.chaos.2025.116088
[28] X. Liu, X. Tong, M. Zhang, Z. Wang, A highly secure image encryption algorithm based on conservative hyperchaotic system and dynamic biogenetic gene algorithms, Chaos Solitions Fractals, 171 (2023), Article ID 113450. https://doi.org/10.1016/j.chaos.2023.113450
[29] X. Jiao, E. Dong, Z. Wang, Dynamic analysis and FPGA implementation of a Kolmogorov-like hyperchaotic system, Int. J. Bifur. Chaos, 31(04) (2021), Article ID 2150052. https://doi.org/10.1142/S0218127421500528
[30] T. Haridas, S. D. Upasana, G. Vyshnavi, M. S. Krishnan, S. S. Muni, Chaos-based audio encryption: Efficacy of 2D and 3D hyperchaotic systems, Franklin Open, 8 (2024), Article ID 100158. https://doi.org/10.1016/j.fraope.2024.100158
[31] T. Alaa-Eldeen, G. M. Alzabeedy, W. Albalawi, K. S. Nisar, A. H. Abdel-Aty, M. El-Kady, M. Abdelhakem, Spectral tau explicit form for approximating solutions to real-life IBVPs using Chebyshev derivatives, Int. J. Geom. Methods Mod. Phys., 22(03) (2025), Article ID 2450324. https://doi.org/10.1142/S0219887824503249
[32] M. Gamal, M. El-Kady, M. Abdelhakem, Solving real-life BVPs via the second derivative Chebyshev pseudo-Galerkin method, Int. J. Mod. Phys. C, 35(07) (2024), Article ID 2450089. https://doi.org/10.1142/S012918312450089X
[33] M. Gamal, M. A. Zaky, M. El-Kady, M. Abdelhakem, Chebyshev polynomial derivative-based spectral tau approach for solving high-order differential equations, Comput. Appl. Math., 43(7) (2024), Article ID 412. https://doi.org/10.1007/s40314-024-02908-y
[34] A. J. Ali, A. F. Abbas, M. Abdelhakem, Comparative analysis of Adams-Bashforth-Moulton and Runge-Kutta methods for solving ordinary differential equations using MATLAB, Math. Model. Eng. Probl., 11(3) (2024), 641-647. https://doi.org/10.18280/mmep.110307
[35] J. C. Sprott, Some simple chaotic flows, Phys. Rev. E, 50(2) (1994), Article ID R647. https://doi.org/10.1103/PhysRevE.50.R647
[36] J. L. Kaplan, J. A. Yorke, Chaotic behavior of multidimensional difference equations, In H. O. Peitgen, H. O. Walther (Eds.), Functional Differential Equations and Approximation of Fixed Points, Springer, Berlin, Heidelberg, 1979, pp. 204-227. https://doi.org/10.1007/BFb0064319
[37] A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, Determining Lyapunov exponents from a time series, Phys. D: Nonlinear Phenom., 16(3) (1985), 285-317. https://doi.org/10.1016/0167-2789(85)90011-9
[38] E. Ozpolat, A. Gulten, Synchronization and application of a novel hyperchaotic system based on adaptive observers, Appl. Sci., 14(3) (2024), Article ID 1311. https://doi.org/10.3390/app14031311
[39] A. K. Khashman, S. F. Al-Azzawi, A new simple no-equilibrium 4D hyperchaotic system with an exponential nonlinear term, Proceedings of the 2022 International Conference on Computer Science and Software Engineering (CSASE), (2022), 283-288. https://doi.org/10.1109/CSASE51777.2022.9759724
[40] S. Li, Y. Wu, X. Zhang, Analysis and synchronization of a new hyperchaotic system with exponential term, Mathematics, 9(24) (2021), Article ID 3281. https://doi.org/10.3390/math9243281
[41] M. D. Johansyah, S. M. Hamidzadeh, K. Benkouider, S. Vaidyanathan, A. Sambas, M. A. Mohamed, A. A. Aziz, A novel hyperchaotic financial system with sinusoidal hyperbolic nonlinearity: From theoretical analysis to adaptive neural fuzzy controller method, Chaos Theory Appl., 6(1) (2024), 26-40. https://doi.org/10.51537/chaos.1336838
[42] P. K. Naskar, S. Paul, D. Nandy, A. Chaudhuri, DNA encoding and channel shuffling for secured encryption of audio data, Multimed. Tools Appl., 78(17) (2019), 25019-25042. https://doi.org/10.1007/s11042-019-7696-z
[43] W. Dai, X. Xu, X. Song, G. Li, Audio encryption algorithm based on Chen memristor chaotic system, Symmetry, 14(1) (2022), Article ID 17. https://doi.org/10.3390/sym14010017
[44] R. I. Abdelfatah, Audio encryption scheme using self-adaptive bit scrambling and two multi chaotic-based dynamic DNA computations, IEEE Access, 8 (2020), 69894-69907. https://doi.org/10.1109/ACCESS.2020.2987197
[45] T. Bonny, W. Al Nassan, Optimizing security and cost efficiency in N-level cascaded chaotic-based secure communication system, Appl. Syst. Innov., 7(6) (2024), Article ID 107. https://doi.org/10.3390/asi7060107
[46] N. R. Babu, M. Kalpana, P. Balasubramaniam, A novel audio encryption approach via finite-time synchronization of fractional order hyperchaotic system, Multimed. Tools Appl., 80(12) (2021), 18043-18067. https://doi.org/10.1007/s11042-020-10288-8
[47] P. K. Naskar, S. Bhattacharyya, A. Chaudhuri, An audio encryption based on distinct key blocks along with PWLCM and ECA, Nonlinear Dyn., 103(2) (2021), 2019-2042. https://doi.org/10.1007/s11071-020-06164-7

Details

Primary Language

English

Subjects

Numerical and Computational Mathematics (Other)

Journal Section

Research Article

Authors

Haneche Nabil ^*
0009-0005-2866-6853
Algeria

Tayeb Hamaizia
0000-0001-8507-572X
Algeria

Early Pub Date

September 5, 2025

Publication Date

September 17, 2025

Submission Date

April 25, 2025

Acceptance Date

August 7, 2025

Published in Issue

Year 2025 Volume: 8 Number: 3

DOI

https://doi.org/10.32323/ujma.1684079

IZ

https://izlik.org/JA62AJ97EN

APA

Nabil, H., & Hamaizia, T. (2025). Generating Hyperchaotic Attractor via the Sprott B System and Its Application in Highly Secure Voice Encryption. Universal Journal of Mathematics and Applications, 8(3), 126-140. https://doi.org/10.32323/ujma.1684079

AMA

1.Nabil H, Hamaizia T. Generating Hyperchaotic Attractor via the Sprott B System and Its Application in Highly Secure Voice Encryption. Univ. J. Math. Appl. 2025;8(3):126-140. doi:10.32323/ujma.1684079

Chicago

Nabil, Haneche, and Tayeb Hamaizia. 2025. “Generating Hyperchaotic Attractor via the Sprott B System and Its Application in Highly Secure Voice Encryption”. Universal Journal of Mathematics and Applications 8 (3): 126-40. https://doi.org/10.32323/ujma.1684079.

EndNote

Nabil H, Hamaizia T (September 1, 2025) Generating Hyperchaotic Attractor via the Sprott B System and Its Application in Highly Secure Voice Encryption. Universal Journal of Mathematics and Applications 8 3 126–140.

IEEE

[1]H. Nabil and T. Hamaizia, “Generating Hyperchaotic Attractor via the Sprott B System and Its Application in Highly Secure Voice Encryption”, Univ. J. Math. Appl., vol. 8, no. 3, pp. 126–140, Sept. 2025, doi: 10.32323/ujma.1684079.

ISNAD

Nabil, Haneche - Hamaizia, Tayeb. “Generating Hyperchaotic Attractor via the Sprott B System and Its Application in Highly Secure Voice Encryption”. Universal Journal of Mathematics and Applications 8/3 (September 1, 2025): 126-140. https://doi.org/10.32323/ujma.1684079.

JAMA

1.Nabil H, Hamaizia T. Generating Hyperchaotic Attractor via the Sprott B System and Its Application in Highly Secure Voice Encryption. Univ. J. Math. Appl. 2025;8:126–140.

MLA

Nabil, Haneche, and Tayeb Hamaizia. “Generating Hyperchaotic Attractor via the Sprott B System and Its Application in Highly Secure Voice Encryption”. Universal Journal of Mathematics and Applications, vol. 8, no. 3, Sept. 2025, pp. 126-40, doi:10.32323/ujma.1684079.

Vancouver

1.Haneche Nabil, Tayeb Hamaizia. Generating Hyperchaotic Attractor via the Sprott B System and Its Application in Highly Secure Voice Encryption. Univ. J. Math. Appl. 2025 Sep. 1;8(3):126-40. doi:10.32323/ujma.1684079