#
A Novel Chaotic System with a Line Equilibrium: Analysis and Its Applications to Secure Communication and Random Bit Generation^{ †}

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Proposed Chaotic System

## 3. Application to Secure Communications

#### 3.1. Symmetric Chaos Shift Keying Modulation

#### 3.2. Bit Error Rate Performance

## 4. Application to Random Bit Generation

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Lorenz, E.N. Deterministic nonperiodic flow. J. Atmos. Sci.
**1963**, 20, 130–141. [Google Scholar] [CrossRef] [Green Version] - Guan, Z.H.; Huang, F.; Guan, W. Chaos-based image encryption algorithm. Phys. Lett. A
**2005**, 346, 153–157. [Google Scholar] [CrossRef] - Karmakar, J.; Nandi, D.; Mandal, M. A novel hyper-chaotic image encryption with sparse-representation based compression. Multimed. Tools Appl.
**2020**, 79, 28277–28300. [Google Scholar] [CrossRef] - Jithin, K.; Sankar, S. Colour image encryption algorithm combining Arnold map, DNA sequence operation, and a Mandelbrot set. J. Inf. Secur. Appl.
**2020**, 50, 102428. [Google Scholar] [CrossRef] - Zaher, A.A.; Abu-Rezq, A. On the design of chaos-based secure communication systems. Commun. Nonlinear Sci. Numer. Simul.
**2011**, 16, 3721–3737. [Google Scholar] [CrossRef] - Dedieu, H.; Kennedy, M.P.; Hasler, M. Chaos shift keying: Modulation and demodulation of a chaotic carrier using self-synchronizing Chua’s circuits. IEEE Trans. Circuits Syst. II Analog Digit. Signal Process.
**1993**, 40, 634–642. [Google Scholar] [CrossRef] - Rajagopal, K.; Cicek, S.; Akgul, A.; Jafari, S.; Karthikeyan, A. Chaotic cuttlesh: King of camouage with self-excited and hidden flows, its fractional-order form and communication designs with fractional form. Discret. Contin. Dyn. Syst.-B
**2019**, 25, 1001. [Google Scholar] [CrossRef] [Green Version] - Stavroulakis, P. Chaos Applications in Telecommunications; CRC Press: Boca Raton, FL, USA, 2005. [Google Scholar]
- Souza, C.E.; Chaves, D.P.; Pimentel, C. Digital communication systems based on three-dimensional chaotic attractors. IEEE Access
**2019**, 7, 10523–10532. [Google Scholar] [CrossRef] - Patidar, V.; Sud, K.K.; Pareek, N.K. A pseudo random bit generator based on chaotic logistic map and its statistical testing. Informatica
**2009**, 33, 441–452. [Google Scholar] - Strogatz, S.H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Phys. Today
**2015**, 68, 54. [Google Scholar] - Hassan, S.S.; Reddy, M.P.; Rout, R.K. Dynamics of the modified n-degree Lorenz system. Appl. Math. Nonlinear Sci.
**2019**, 4, 315–330. [Google Scholar] [CrossRef] [Green Version] - Wang, X.; Wang, M. A hyperchaos generated from Lorenz system. Phys. A Stat. Mech. Its Appl.
**2008**, 387, 3751–3758. [Google Scholar] [CrossRef] - Han, C. An image encryption algorithm based on modified logistic chaotic map. Optik
**2019**, 181, 779–785. [Google Scholar] [CrossRef] - Xiong, L.; Zhang, S.; Zeng, Y.; Liu, B. Dynamics of a new composite four–Scroll chaotic system. Chin. J. Phys.
**2018**, 56, 2381–2394. [Google Scholar] [CrossRef] - Wang, Z.; Volos, C.; Kingni, S.T.; Azar, A.T.; Pham, V.T. Four-wing attractors in a novel chaotic system with hyperbolic sine nonlinearity. Optik
**2017**, 131, 1071–1078. [Google Scholar] [CrossRef] - Pham, V.T.; Vaidyanathan, S.; Volos, C.; Jafari, S. Hidden attractors in a chaotic system with an exponential nonlinear term. Eur. Phys. J. Spec. Top.
**2015**, 224, 1507–1517. [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. Circuits Syst. Signal Process.
**2018**, 37, 1028–1043. [Google Scholar] [CrossRef] - Dalkiran, F.Y.; Sprott, J.C. Simple chaotic hyperjerk system. Int. J. Bifurc. Chaos
**2016**, 26, 1650189. [Google Scholar] [CrossRef] [Green Version] - Zhou, C.; Yang, C.; Xu, D.; Chen, C. Dynamic analysis and synchronisation control of a novel chaotic system with coexisting attractors. Pramana
**2020**, 94, 19. [Google Scholar] [CrossRef] - Dudkowski, D.; Jafari, S.; Kapitaniak, T.; Kuznetsov, N.V.; Leonov, G.A.; Prasad, A. Hidden attractors in dynamical systems. Phys. Rep.
**2016**, 637, 1–50. [Google Scholar] [CrossRef] - Pham, V.T.; Volos, C.; Jafari, S.; Kapitaniak, T. A novel cubic–equilibrium chaotic system with coexisting hidden attractors: Analysis, and circuit implementation. J. Circuits Syst. Comput.
**2018**, 27, 1850066. [Google Scholar] [CrossRef] - Jafari, S.; Sprott, J. Simple chaotic flows with a line equilibrium. Chaos Solitons Fractals
**2013**, 57, 79–84. [Google Scholar] [CrossRef] - Gotthans, T.; Sprott, J.C.; Petrzela, J. Simple chaotic flow with circle and square equilibrium. Int. J. Bifurc. Chaos
**2016**, 26, 1650137. [Google Scholar] [CrossRef] - Azar, A.T.; Serrano, F.E. Stabilization of port Hamiltonian chaotic systems with hidden attractors by adaptive terminal sliding mode control. Entropy
**2020**, 22, 122. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Nag Chowdhury, S.; Ghosh, D. Hidden attractors: A new chaotic system without equilibria. Eur. Phys. J. Spec. Top.
**2020**, 229, 1299–1308. [Google Scholar] [CrossRef] - Sushchik, M.; Tsimring, L.S.; Volkovskii, A.R. Performance analysis of correlation-based communication schemes utilizing chaos. IEEE Trans. Circuits Syst. I Fundam. Theory Appl.
**2000**, 47, 1684–1691. [Google Scholar] [CrossRef] - Wang, X.; Akgul, A.; Cicek, S.; Pham, V.T.; Hoang, D.V. A chaotic system with two stable equilibrium points: Dynamics, circuit realization and communication application. Int. J. Bifurc. Chaos
**2017**, 27, 1750130. [Google Scholar] [CrossRef] - Moysis, L.; Volos, C.; Stouboulos, I.; Goudos, S.; Ciçek, S.; Pham, V.T.; Mishra, V.K. A Novel Chaotic System with Application to Secure Communications. In Proceedings of the 2020 9th International Conference on Modern Circuits and Systems Technologies (MOCAST), Bremen, Germany, 7–9 September 2020; pp. 1–4. [Google Scholar]
- 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] - Singh, J.P.; Roy, B. Coexistence of asymmetric hidden chaotic attractors in a new simple 4-D chaotic system with curve of equilibria. Optik
**2017**, 145, 209–217. [Google Scholar] [CrossRef] - 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] - Çiçek, S.; Kocamaz, U.E.; Uyaroğlu, Y. Secure communication with a chaotic system owning logic element. AEU-Int. J. Electron. Commun.
**2018**, 88, 52–62. [Google Scholar] [CrossRef] - Çiçek, S.; Ferikoğlu, A.; Pehlivan, I. A new 3D chaotic system: Dynamical analysis, electronic circuit design, active control synchronization and chaotic masking communication application. Optik
**2016**, 127, 4024–4030. [Google Scholar] [CrossRef] - Kocamaz, U.E.; Çiçek, S.; Uyaroğlu, Y. Secure communication with chaos and electronic circuit design using passivity-based synchronization. J. Circuits, Syst. Comput.
**2018**, 27, 1850057. [Google Scholar] [CrossRef] - Pone, J.R.M.; Çiçek, S.; Kingni, S.T.; Tiedeu, A.; Kom, M. Passive–active integrators chaotic oscillator with anti-parallel diodes: Analysis and its chaos-based encryption application to protect electrocardiogram signals. Analog. Integr. Circuits Signal Process.
**2019**, 1–15. [Google Scholar] [CrossRef] - Kingni, S.T.; Rajagopal, K.; Çiçek, S.; Srinivasan, A.; Karthikeyan, A. Dynamic analysis, FPGA implementation, and cryptographic application of an autonomous 5D chaotic system with offset boosting. Front. Inf. Technol. Electron. Eng.
**2020**, 21, 950–961. [Google Scholar] [CrossRef] - Rajagopal, K.; Çiçek, S.; Khalaf, A.J.M.; Pham, V.T.; Jafari, S.; Karthikeyan, A.; Duraisamy, P. A novel class of chaotic flows with infinite equilibriums and their application in chaos-based communication design using DCSK. Z. Für Naturforschung A
**2018**, 73, 609–617. [Google Scholar] [CrossRef] - Rajagopal, K.; Pham, V.T.; Çiçek, S.; Jafari, S.; Karthikeyan, A.; Arun, S. A chaotic jerk system with different types of Equilibria and its application in communication system. Teh. Vjesn.
**2020**, 27, 681–686. [Google Scholar] - Alvarez, G.; Li, S. Some basic cryptographic requirements for chaos-based cryptosystems. Int. J. Bifurc. Chaos
**2006**, 16, 2129–2151. [Google Scholar] [CrossRef] [Green Version] - Huang, X.; Liu, L.; Li, X.; Yu, M.; Wu, Z. A New Pseudorandom Bit Generator Based on Mixing Three-Dimensional Chen Chaotic System with a Chaotic Tactics. Complexity
**2019**, 2019, 1–9. [Google Scholar] [CrossRef] [Green Version] - Hu, H.; Liu, L.; Ding, N. Pseudorandom sequence generator based on the Chen chaotic system. Comput. Phys. Commun.
**2013**, 184, 765–768. [Google Scholar] [CrossRef] - Tuna, M. A novel secure chaos-based pseudo random number generator based on ANN-based chaotic and ring oscillator: Design and its FPGA implementation. Analog Integr. Circuits Signal Process.
**2020**, 105, 167–181. [Google Scholar] [CrossRef] - Moysis, L.; Volos, C.; Jafari, S.; Munoz-Pacheco, J.M.; Kengne, J.; Rajagopal, K.; Stouboulos, I. Modification of the Logistic Map Using Fuzzy Numbers with Application to Pseudorandom Number Generation and Image Encryption. Entropy
**2020**, 22, 474. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Demir, K.; Ergün, S. An analysis of deterministic chaos as an entropy source for random number generators. Entropy
**2018**, 20, 957. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhao, Y.; Gao, C.; Liu, J.; Dong, S. A self-perturbed pseudo-random sequence generator based on hyperchaos. Chaos Solitons Fractals X
**2019**, 4, 100023. [Google Scholar] [CrossRef] - Datcu, O.; Macovei, C.; Hobincu, R. Chaos Based Cryptographic Pseudo-Random Number Generator Template with Dynamic State Change. Appl. Sci.
**2020**, 10, 451. [Google Scholar] [CrossRef] [Green Version] - Rukhin, A.; Soto, J.; Nechvatal, J.; Smid, M.; Barker, E. A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications; Technical Report; Booz-Allen and Hamilton Inc.: Mclean, VA, USA, 2001. [Google Scholar]
- Lynnyk, V.; Sakamoto, N.; Čelikovskỳ, S. Pseudo random number generator based on the generalized Lorenz chaotic system. IFAC-PapersOnLine
**2015**, 48, 257–261. [Google Scholar] [CrossRef] - Hamza, R. A novel pseudo random sequence generator for image-cryptographic applications. J. Inf. Secur. Appl.
**2017**, 35, 119–127. [Google Scholar] [CrossRef] - Moysis, L.; Tutueva, A.; Volos, C.; Butusov, D.; Munoz-Pacheco, J.M.; Nistazakis, H. A Two-Parameter Modified Logistic Map and Its Application to Random Bit Generation. Symmetry
**2020**, 12, 829. [Google Scholar] [CrossRef] - Nazaré, T.E.; Nepomuceno, E.G.; Martins, S.A.; Butusov, D.N. A Note on the Reproducibility of Chaos Simulation. Entropy
**2020**, 22, 953. [Google Scholar] [CrossRef] - Sayed, W.S.; Radwan, A.G.; Fahmy, H.A.; El-Sedeek, A. Software and Hardware Implementation Sensitivity of Chaotic Systems and Impact on Encryption Applications. Circuits Syst. Signal Process.
**2020**, 39, 5638–5655. [Google Scholar] [CrossRef] - Liu, B.; Xiang, H.; Liu, L. Reducing the Dynamical Degradation of Digital Chaotic Maps with Time-Delay Linear Feedback and Parameter Perturbation. Math. Probl. Eng.
**2020**, 2020, 4926937. [Google Scholar] [CrossRef] [Green Version] - Kaddoum, G. Wireless chaos-based communication systems: A comprehensive survey. IEEE Access
**2016**, 4, 2621–2648. [Google Scholar] [CrossRef]

**Figure 1.**Phase portraits of system (1), for $a=0.65$ and $b=0.1$ and $x\left(0\right)=(0,0.1,0.1)$, for 1500 s.

**Figure 2.**3D chaotic attractor of system (1), for $a=0.65$ and $b=0.1$ and $x\left(0\right)=(0,0.1,0.1)$, for 1500 s.

**Figure 3.**State trajectories of system (1), for $a=0.65$ and $b=0.1$ and $x\left(0\right)=(0,0.1,0.1)$, for 1500 s.

**Figure 4.**Phase portraits of system (1) for each state versus its derivative, for $a=0.65$ and $b=0.1$ and $x\left(0\right)=(0,0.1,0.1)$, for 1500 s.

**Figure 10.**Simulation results of the SCSK secure communications design, for chaotic signal $100\xb7x$.

**Figure 11.**Simulation results of the SCSK secure communications design, for chaotic signal $100\xb7sin(x+y+z)$.

**Figure 12.**Simulation results of the SCSK secure communications design, for chaotic signal $100\xb7sin3(x+y+z)$.

**Figure 13.**Simulation results of the SCSK secure communications design, for chaotic signal $100\xb7sin5(x+y+z)$.

If $\mathit{p}\ge \mathit{\alpha}$, the Test Is Successful | ||||
---|---|---|---|---|

No. | Statistical Test | p-Value | Proportion | Result |

1 | Frequency | 0.023545 | 50/50 | Success |

2 | Block Frequency | 0.191687 | 49/50 | Success |

3 | Cumulative Sums | 0.935716 | 49/50 | Success |

4 | Runs | 0.171867 | 49/50 | Success |

5 | Longest Run | 0.350485 | 49/50 | Success |

6 | Rank | 0.935716 | 49/50 | Success |

7 | FFT | 0.779188 | 50/50 | Success |

8 | Non-Overlapping Template | 0.319084 | 50/50 | Success |

9 | Overlapping Template | 0.137282 | 48/50 | Success |

10 | Universal | 0.191687 | 50/50 | Success |

11 | Approximate Entropy | 0.085587 | 50/50 | Success |

12 | Random Excursions | 0.010606 | 29/29 | Success |

13 | Random Excursions Variant | 0.186566 | 29/29 | Success |

14 | Serial | 0.574903 | 50/50 | Success |

15 | Linear Complexity | 0.262249 | 50/50 | Success |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Moysis, L.; Volos, C.; Stouboulos, I.; Goudos, S.; Çiçek, S.; Pham, V.-T.; Mishra, V.K.
A Novel Chaotic System with a Line Equilibrium: Analysis and Its Applications to Secure Communication and Random Bit Generation. *Telecom* **2020**, *1*, 283-296.
https://doi.org/10.3390/telecom1030019

**AMA Style**

Moysis L, Volos C, Stouboulos I, Goudos S, Çiçek S, Pham V-T, Mishra VK.
A Novel Chaotic System with a Line Equilibrium: Analysis and Its Applications to Secure Communication and Random Bit Generation. *Telecom*. 2020; 1(3):283-296.
https://doi.org/10.3390/telecom1030019

**Chicago/Turabian Style**

Moysis, Lazaros, Christos Volos, Ioannis Stouboulos, Sotirios Goudos, Serdar Çiçek, Viet-Thanh Pham, and Vikas K. Mishra.
2020. "A Novel Chaotic System with a Line Equilibrium: Analysis and Its Applications to Secure Communication and Random Bit Generation" *Telecom* 1, no. 3: 283-296.
https://doi.org/10.3390/telecom1030019