A Simple Chaotic Flow with Hyperbolic Sinusoidal Function and Its Application to Voice Encryption
Abstract
:1. Introduction
2. Chaotic Flow with a Hyperbolic Sinusoidal Function
2.1. Scenario A: Line of Equilibria
2.2. Scenario B: No Equilibrium Point
2.3. Scenario C: SelfExcited Attractor
3. Circuit Design of the Proposed Chaotic Flow
4. Voice Encryption Algorithm Design and Its Analysis
4.1. RNG Algorithm Design and NIST 80022 Test Results
4.2. Voice Encryption Algorithm Design and Its Application
5. Conclusions and Discussion
Author Contributions
Funding
Conflicts of Interest
Date Availability
Appendix A
Algorithm 1 RNG Design Algorithm Pseudo Code 

References
 Hua, Z.; Zhou, B.; Zhou, Y. SineTransformBased Chaotic System With FPGA Implementation. IEEE Trans. Ind. Electron. 2018, 65, 2557–2566. [Google Scholar] [CrossRef]
 Thoai, V.P.; Kahkeshi, M.S.; Huynh, V.V.; Ouannas, A.; Pham, V.T. A Nonlinear FiveTerm System: Symmetry, Chaos, and Prediction. Symmetry 2020, 12, 865. [Google Scholar] [CrossRef]
 Chain, K.; Kuo, W.C. A new digital signature scheme based on chaotic maps. Nonlinear Dyn. 2013, 74, 1003–1012. [Google Scholar] [CrossRef]
 Muthukumar, P.; Balasubramaniam, P.; Ratnavelu, K. Sliding mode control design for synchronization of fractional order chaotic systems and its application to a new cryptosystem. Int. J. Dyn. Control 2017, 5, 115–123. [Google Scholar] [CrossRef]
 Deng, Y.; Hu, H.; Liu, L. Feedback control of digital chaotic systems with application to pseudorandom number generator. Int. J. Mod. Phys. C 2015, 26, 1550022. [Google Scholar] [CrossRef]
 CastroRamírez, J.; MartínezGuerra, R.; CruzVictoria, J.C. A new reducedorder observer for the synchronization of nonlinear chaotic systems: An application to secure communications. Chaos Interdiscip. J. Nonlinear Sci. 2015, 25, 103128. [Google Scholar] [CrossRef]
 Wang, G.; Chen, D.; Lin, J.; Chen, X. The application of chaotic oscillators to weak signal detection. IEEE Trans. Ind. Electron. 1999, 46, 440–444. [Google Scholar] [CrossRef]
 Sakthivel, R.; Santra, S.; Anthoni, S.M.; Kuppili, V. Synchronisation and antisynchronisation of chaotic systems with application to DC–DC boost converter. IET Gener. Transm. Distrib. 2017, 11, 959–967. [Google Scholar] [CrossRef]
 Chen, E.; Min, L.; Chen, G. Discrete Chaotic Systems with OneLine Equilibria and Their Application to Image Encryption. Int. J. Bifurc. Chaos 2017, 27, 1750046. [Google Scholar] [CrossRef]
 Glushkov, A.V.; Khetselius, O.; Brusentseva, S.V.; Zaichko, P.A.; Ternovsky, V.B. Studying interaction dynamics of chaotic systems within a nonlinear prediction method: Application to neurophysiology. Adv. Neural Netw. Fuzzy Syst. Artif. Intell. 2014, 21, 69–75. [Google Scholar]
 AguilarLópez, R.; MartínezGuerra, R.; PerezPinacho, C.A. Nonlinear observer for synchronization of chaotic systems with application to secure data transmission. Eur. Phys. J. Spec. Top. 2014, 223, 1541–1548. [Google Scholar] [CrossRef]
 Radwan, A.; Moaddy, K.; Salama, K.N.; Momani, S.; Hashim, I. Control and switching synchronization of fractional order chaotic systems using active control technique. J. Adv. Res. 2014, 5, 125–132. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Boulkroune, A.; Bouzeriba, A.; Hamel, S.; Bouden, T. Adaptive fuzzy controlbased projective synchronization of uncertain nonaffine chaotic systems. Complexity 2015, 21, 180–192. [Google Scholar] [CrossRef]
 Mobayen, S. Finitetime stabilization of a class of chaotic systems with matched and unmatched uncertainties: An LMI approach. Complexity 2016, 21, 14–19. [Google Scholar] [CrossRef]
 Ma, D.; Sun, Q.; Li, X. Synchronization of masterslave chaotic system with coupling timevarying delay based on sampleddata control. In Proceedings of the Control and Decision Conference (CCDC), 2015 27th Chinese, Qingdao, China, 23–25 May 2015; pp. 6545–6550. [Google Scholar]
 Xiong, W.; Huang, J. Finitetime control and synchronization for memristorbased chaotic system via impulsive adaptive strategy. Adv. Differ. Equ. 2016, 2016, 101. [Google Scholar] [CrossRef] [Green Version]
 Song, Q.; Huang, T. Stabilization and synchronization of chaotic systems with mixed timevarying delays via intermittent control with nonfixed both control period and control width. Neurocomputing 2015, 154, 61–69. [Google Scholar] [CrossRef]
 Mobayen, S.; Baleanu, D.; Tchier, F. Secondorder fast terminal sliding mode control design based on LMI for a class of nonlinear uncertain systems and its application to chaotic systems. J. Vib. Control 2017, 23, 2912–2925. [Google Scholar] [CrossRef]
 Wei, Z. Dynamical behaviors of a chaotic system with no equilibria. Phys. Lett. A 2011, 376, 102–108. [Google Scholar] [CrossRef]
 Jafari, S.; Sprott, J.C.; Golpayegani, S. Elementary quadratic chaotic flows with no equilibria. Phys. Lett. A 2013, 377, 699–702. [Google Scholar] [CrossRef]
 Wang, X.; Chen, G. A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 2012, 17, 1264–1272. [Google Scholar] [CrossRef] [Green Version]
 Molaie, M.; Jafari, S.; Sprott, J.C.; Golpayegani, S. Coexisting hidden attractors in a 4D simplified Lorenz system. Int. J. Bifurc. Chaos 2013, 23, 1350188. [Google Scholar] [CrossRef]
 Shilnikov, L.P. A case of the existence of a denumerable set of periodic motions. Sov. Math. 1965, 24, 163–166. [Google Scholar]
 Leonov, G.A.; Kuznetsov, N.V.; Kuznetsova, O.A.; Seledzhi, S.M.; Vagaitsev, V.I. Hidden oscillations in dynamical systems. Trans. Syst. Contr. 2011, 6, 54–67. [Google Scholar]
 Leonov, G.A.; Kuznetsov, N.V.; Vagaitsev, V.I. Hidden attractor in smooth chua systems. Phys. D Nonlinear Phenom. 2012, 241, 1482–1486. [Google Scholar] [CrossRef]
 Leonov, G.A.; Kuznetsov, N.V. Hidden attractors in dynamical systems. From hidden oscillations in Hilbert Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int. J. Bifurc. Chaos 2013, 23, 1330002. [Google Scholar] [CrossRef] [Green Version]
 Wang, Z.; Volos, C.; Kingni, S.T.; Azar, A.T.; Pham, V.T. Fourwing attractors in a novel chaotic system with hyperbolic sine nonlinearity. Opt. Int. J. Light Electron Opt. 2017, 131, 1071–1078. [Google Scholar] [CrossRef]
 Pham, V.T.; Volos, C.; Kingni, S.T.; Kapitaniak, T.; Jafari, S. Bistable Hidden Attractors in a Novel Chaotic System with Hyperbolic Sine Equilibrium. Circuitssyst. Signal Process. 2017, 37, 1028–1043. [Google Scholar] [CrossRef]
 Kuznetsov, A.; Kuznetsov, S.; Mosekilde, E.; Stankevich, N. Coexisting hidden attractors in a radiophysical oscillator system. J. Phys. A Math. 2015, 48, 125101. [Google Scholar] [CrossRef]
 Zhusubaliyev, Z.T.; Mosekilde, E. Multistability and hidden attractors in a multilevel DC/DC converter. Math. Comput. Simul. 2015, 109, 32–45. [Google Scholar] [CrossRef]
 Kiseleva, M.A.; Kuznetsov, N.V.; Leonov, G.A. Hidden attractors in electromechanical systems with and without equilibria. IFAC Pap. 2016, 49, 51–55. [Google Scholar] [CrossRef]
 Zhusubaliyev, Z.T.; Mosekilde, E.; Rubanov, V.G.; Nabokov, R.A. Multistability and hidden attractors in a relay system with hysteresis. Phys. D Nonlinear Phenom. 2015, 306, 6–15. [Google Scholar] [CrossRef]
 Yu, M.; Sun, K.; Liu, W.; He, S. A hyperchaotic map with grid sinusoidal cavity. Chaossolitons Fractals 2018, 106, 107–117. [Google Scholar] [CrossRef]
 Zhang, X.; Li, C.; Lei, T.; Liu, Z.; Tao, C. A symmetric controllable hyperchaotic hidden attractor. Symmetry 2020, 12, 550. [Google Scholar] [CrossRef] [Green Version]
 Sadkhan, S.B.; Ali, H. A proposed speech scrambling based on hybrid chaotic key generators. In Proceedings of the 2016 AlSadeq IEEE International Conference on Multidisciplinary in IT and Communication Science and Applications (AICMITCSA), AlNajaf, Iraq, 9–10 May 2016; pp. 1–6. [Google Scholar]
 Mobayen, S.; Vaidyanathan, S.; Sambas, A.; Kacar, S.; Çavuşoğlu, Ü. A novel chaotic system with boomerangshaped equilibrium, its circuit implementation and application to sound encryption. Iran. J. Sci. Technol. Trans. Electr. Eng. 2019, 43, 1–12. [Google Scholar]
 Raheema, A.M.; Sadkhan, S.B.; Sattar, S.M.A. Design and implementation of speech encryption based on hybrid chaotic maps. In Proceedings of the 2018 IEEE International Conference on Engineering Technology and Their Applications (IICETA), AlNajaf, Iraq, 8–9 May 2018; pp. 112–117. [Google Scholar]
 Nosrati, K.; Volos, C. Bifurcation Analysis and Chaotic Behaviors of FractionalOrder Singular Biological Systems. In Nonlinear Dynamical Systems with SelfExcited and Hidden Attractors; Springer: Berlin/Heidelberg, Germany, 2018; pp. 3–44. [Google Scholar]
 Wu, F.; Ma, J. The chaos dynamic of multiproduct Cournot duopoly game with managerial delegation. Discret. Dyn. Nat. Soc. 2014, 2014. [Google Scholar] [CrossRef]
 Barrera, J.; Flores, J.J.; FuerteEsquivel, C. Generating complete bifurcation diagrams using a dynamic environment particle swarm optimization algorithm. J. Artif. Evol. Appl. 2007, 2008. [Google Scholar] [CrossRef] [Green Version]
 Ouannas, A.; Khennaoui, A.A.; Wang, X.; Pham, V.T.; Boulaaras, S.; Momani, S. Bifurcation and chaos in the fractional form of HénonLozi type map. Eur. Phys. J. Spec. Top. 2020, 229, 2261–2273. [Google Scholar] [CrossRef]
 Zhu, X.; Du, W.S. New chaotic systems with two closed curve equilibrium passing the same point: Chaotic behavior, bifurcations, and synchronization. Symmetry 2019, 11, 951. [Google Scholar] [CrossRef] [Green Version]
 Awrejcewicz, J.; Krysko, A.V.; Erofeev, N.P.; Dobriyan, V.; Barulina, M.A.; Krysko, V.A. Quantifying chaos by various computational methods. Part 1: Simple systems. Entropy 2018, 20, 175. [Google Scholar] [CrossRef] [Green Version]
 Kong, G.; Zhang, Y.; Khalaf, A.J.M.; Panahi, S.; Hussain, I. Parameter estimation in a new chaotic memristive system using ions motion optimization. Eur. Phys. J. Spec. Top. 2019, 228, 2133–2145. [Google Scholar] [CrossRef]
 Huang, W.; Kamenski, L.; Lang, J. Conditioning of implicit Runge–Kutta integration for finite element approximation of linear diffusion equations on anisotropic meshes. J. Comput. Appl. Math. 2019. [Google Scholar] [CrossRef] [Green Version]
 Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A. Determining Lyapunov exponents from a time series. Phys. D Nonlinear Phenom. 1985, 16, 285–317. [Google Scholar] [CrossRef] [Green Version]
 Bouali, S.; Buscarino, A.; Fortuna, L.; Frasca, M.; Gambuzza, L. Emulating complex business cycles by using an electronic analogue. Nonlinear Anal. Real World Appl. 2012, 13, 2459–2465. [Google Scholar] [CrossRef]
 Banerjee, T.; Biswas, D. Theory and experiment of a firstorder chaotic delay dynamical system. Int. J. Bifurc. Chaos 2013, 23, 1330020. [Google Scholar] [CrossRef]
 Zhou, W.J.; Wang, Z.P.; Wu, M.W.; Zheng, W.H.; Weng, J.F. Dynamics analysis and circuit implementation of a new threedimensional chaotic system. Opt. Int. J. Light Electron Opt. 2015, 126, 765–768. [Google Scholar] [CrossRef]
 Lai, Q.; Wang, L. Chaos, bifurcation, coexisting attractors and circuit design of a threedimensional continuous autonomous system. Opt. Int. J. Light Electron Opt. 2016, 127, 5400–5406. [Google Scholar] [CrossRef]
 Gokyildirim, A.; Uyaroglu, Y.; Pehlivan, I. A novel chaotic attractor and its weak signal detection application. Opt. Int. J. Light Electron Opt. 2016, 127, 7889–7895. [Google Scholar] [CrossRef]
 Hajipour, A.; Tavakoli, H. Analysis and circuit simulation of a novel nonlinear fractional incommensurate order financial system. Opt. Int. J. Light Electron Opt. 2016, 127, 10643–10652. [Google Scholar] [CrossRef]
 Wang, Y.; Wong, K.W.; Liao, X.; Chen, G. A new chaosbased fast image encryption algorithm. Appl. Soft Comput. 2011, 11, 514–522. [Google Scholar] [CrossRef]
 Çavuşoğlu, Ü.; Kaçar, S.; Pehlivan, I.; Zengin, A. Secure image encryption algorithm design using a novel chaos based SBox. Chaossolitons Fractals 2017, 95, 92–101. [Google Scholar] [CrossRef]
 Bakhache, B.; Ghazal, J.M.; El Assad, S. Improvement of the security of zigbee by a new chaotic algorithm. IEEE Syst. J. 2014, 8, 1024–1033. [Google Scholar] [CrossRef]
 Khan, M. A novel image encryption scheme based on multiple chaotic Sboxes. Nonlinear Dyn. 2015, 82, 527–533. [Google Scholar] [CrossRef]
 Çavuşoğlu, Ü.; Zengin, A.; Pehlivan, I.; Kaçar, S. A novel approach for strong SBox generation algorithm design based on chaotic scaled Zhongtang system. Nonlinear Dyn. 2017, 87, 1081–1094. [Google Scholar] [CrossRef]
 Hua, Z.; Zhou, Y.; Pun, C.M.; Chen, C.P. 2D Sine Logistic modulation map for image encryption. Inf. Sci. 2015, 297, 80–94. [Google Scholar] [CrossRef]
 Rukhin, A.; Soto, J.; Nechvatal, J.; Smid, M.; Barker, E. A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications; BoozAllen and Hamilton Inc Mclean Va: McLean, VA, USA, 2001. [Google Scholar]
NIST Statistical Tests  pValue (x⊕y)  pValue (x⊕z)  pValue (y⊕z)  pValue (x⊕y⊕z)  Results 

Frequency (Monobit) Test  0.32708  0.70840  0.07409  0.83679  Passed 
BlockFrequency Test  0.05028  0.44384  0.87530  0.24483  Passed 
CumulativeSums Test  0.49997  0.79399  0.12548  0.88754  Passed 
Runs Test  0.28235  0.05285  0.01530  0.34010  Passed 
LongestRun Test  0.91108  0.88963  0.46730  0.057827  Passed 
Binary Matrix Rank Test  0.17994  0.15263  0.55596  0.39136  Passed 
Discrete Fourier Transform Test  0.17441  0.92688  0.52063  0.20211  Passed 
Overlapping Templates Test  0.63213  0.12006  0.96148  0.29966  Passed 
Maurer’s Universal Statistical Test  0.59708  0.81350  0.40059  0.48723  Passed 
Approximate Entropy Test  0.95048  0.38285  0.27585  0.52635  Passed 
RandomExcursions Test (x = −4)  0.82604  0.57997  0.40488  0.34822  Passed 
RandomExcursions Variant Test (x = −4)  0.74935  0.63538  0.19136  0.46211  Passed 
Serial Test1  0.53650  0.89087  0.74965  0.92028  Passed 
Serial Test2  0.13577  0.48589  0.27236  0.75602  Passed 
LinearComplexity Test  0.72956  0.94527  0.31945  0.78612  Passed 
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Mobayen, S.; Volos, C.; Çavuşoğlu, Ü.; S. Kaçar, S. A Simple Chaotic Flow with Hyperbolic Sinusoidal Function and Its Application to Voice Encryption. Symmetry 2020, 12, 2047. https://doi.org/10.3390/sym12122047
Mobayen S, Volos C, Çavuşoğlu Ü, S. Kaçar S. A Simple Chaotic Flow with Hyperbolic Sinusoidal Function and Its Application to Voice Encryption. Symmetry. 2020; 12(12):2047. https://doi.org/10.3390/sym12122047
Chicago/Turabian StyleMobayen, Saleh, Christos Volos, Ünal Çavuşoğlu, and Sezgin S. Kaçar. 2020. "A Simple Chaotic Flow with Hyperbolic Sinusoidal Function and Its Application to Voice Encryption" Symmetry 12, no. 12: 2047. https://doi.org/10.3390/sym12122047