Advertisement

Secure Communication Systems Based on the Synchronization of Chaotic Systems

  • Samir Bendoukha
  • Salem AbdelmalekEmail author
  • Adel Ouannas
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 200)

Abstract

Over the last three decades, chaotic dynamical systems have found many applications in science and engineering particularly in the field of secure communications and data encryption. The vast majority of such applications start from the synchronization control of two chaotic systems where one system’s states are forced to follow the exact same trajectory set out by another system with different initial conditions. The general theme seems to be that a master system is placed at the transmitter and a slave at the receiver. Once the pair is synchronized, the states can be used to secure the communication channel in one of four ways: chaotic modulation schemes, chaotic multi–carrier schemes, chaotic multiple access schemes, and chaos–based encryption schemes. This chapter aims to give an overview of secure communications and chaos and summarize the latest advancements in the field of chaos based communications. In addition, a case study is selected assuming antipodal chaos shift keying (ACSK) modulation and the complete communication system is described. Simulation results are presented to highlight the performance of chaotic modulation systems.

Keywords

Chaotic dynamical systems Secure communications Data encryption Modulation Multi–carrier modulation Multiple access 

References

  1. 1.
    Pecora, L.M., Carrol, T.L.: Synchronization in chaotic systems. Phys. Rev. A 64, 821–824 (1990)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Paar, C., Pelzl, J.: Understanding Cryptography. Springer, Berlin Heidelberg (2010)zbMATHCrossRefGoogle Scholar
  3. 3.
    Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton, Florida, USA (1997)zbMATHGoogle Scholar
  4. 4.
    van Tilborg, H.C.A.: Encyclopedia of Cryptography and Security. Springer, Berlin (2005)zbMATHCrossRefGoogle Scholar
  5. 5.
    Shanon, C.E.: Communication theory of secrecy systems. Bell Sys. Tech. J. 28, 656–715 (1949)MathSciNetCrossRefGoogle Scholar
  6. 6.
    De Cannière, C., Preneel, B.: Trivium specifications, eSTREAM submitted papers (2006)Google Scholar
  7. 7.
    Rivest, R.: The RC4 Encryption Algorithm, http://www.rsasecurity.com (1992)
  8. 8.
    Diffie, W., Hellman, M.: New directions in cryptography. IEEE Trans. Inf. Theor. 22(6), 644–654 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kocarev, L., Lian, S.: Chaos-based Cryptography: Theory, Algorithms and Applications. Springer-Verlag, Berlin Heidelberg (2011)zbMATHCrossRefGoogle Scholar
  11. 11.
    Alvarez, G., Li, S.: Some basic cryptographic requirements for chaos based cryptosystems. Int. J. Bifurcat. Chaos 16, 2129–2151 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Yamada, T., Fujisaca, H.: Stability theory of synchronized motion in coupled-oscillator systems. II. Prog. Theor. Phys. 70(5), 1240–1248 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Yamada, T., Fujisaca, H.: Stability theory of synchronized motion in coupled-oscillator systems. III. Prog. Theor. Phys. 72(5), 885–894 (1984)CrossRefGoogle Scholar
  14. 14.
    Afraimovich, V.S., Verochev, N.N., Robinovich, M.I.: Stochastic synchronization of oscillations in dissipative systems. Radio. Phys. Quantum Electron. 29, 795–803 (1983)CrossRefGoogle Scholar
  15. 15.
    Yu, Y., Zhang, S.: The synchronization of linearly bidirectional coupled chaotic systems. Chaos. Solitons Fractals 22, 189–197 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Chen, G., Yu, X.: Chaos Control: Theory and Applications. Springer, Berlin, Germany (2003)zbMATHCrossRefGoogle Scholar
  17. 17.
    Ott, E., Grebogi, C., Yorke, J.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Smaoui, N., Karouma, A.: Secure communications based on the synchronization of the hyperchaotic Chen and the unified chaotic systems. Commun. Nonlin. Sci. Num. Simul. 16(8), 3279–3293 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Smaoui, N., Karouma, A., Zribi, M.: Synchronization of the hyperchaotic Lüsystems using a sliding mode controller. Kuwait J. Sci. Eng. 38(2A), 69–91 (2011)MathSciNetGoogle Scholar
  20. 20.
    Smaoui, N., Karouma, A., Zribi, M.: Adaptive synchronization of hyperchaotic Chen systems with application to secure communication. Int. J Innov. Comput. Info. Cont. 9(3), 1127–1144 (2013)Google Scholar
  21. 21.
    Grassi, Giuseppe: Observer-based hyperchaos synchronization in cascaded discrete-time systems. Chaos, Solitons Fractals 40(2), 1029–1039 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Grassi, G.: Arbitrary full-state hybrid projective synchronization for chaotic discrete-time systems via a scalar signal. Chin. Phys. B 21(6), 060504 (2012)CrossRefGoogle Scholar
  23. 23.
    Boccaletti, S., Kurths, J., Osipov, G., Valladares, D., Zhou, C.: The synchronization of chaotic systems. Phys. Rep. 366, 1–101 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Li, X.F., Leung, A., Han, X.P., Liu, X.J., Chu, Y.D.: Complete (anti-) synchronization of chaotic systems with fully uncertain parameters by adaptive control. Nonlinear Dyn. 63, 263–275 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Zhang, X., Zhu, H.: Anti-synchronization of two different hyperchaotic systems via active and adaptive control. Int. J. Nonlinear Sci. 6, 216–223 (2008)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Mahmoud, G., Mahmoud, E.: Phase and antiphase synchronization of two identical hyperchaotic complex nonlinear systems. Nonlinear Dyn. 61, 141–152 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Qiang, J.: Projective synchronization of a new hyperchaotic Lorenz system chaotic systems. Phys. Lett. A 370, 40–45 (2007)zbMATHCrossRefGoogle Scholar
  28. 28.
    Li, C., Yan, J.: The synchronization of three fractional differential systems. Chaos Solitons Fractals 22(3), 751–757 (2007)CrossRefGoogle Scholar
  29. 29.
    Cai, G., Hu, P., Li, Y.: Modified function lag projective synchronization of a financial hyperchaotic system. Nonlinear Dyn. 69, 1457–1464 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Zhanguo, L., Wei, X.: Hybrid function projective synchronization of chaotic systems with fully unknown parameters. Stud. Math. Sci. 2(1), 80–87 (2011)Google Scholar
  31. 31.
    Wang, J., Xiong, X., Zhang, Y.: Extending synchronization scheme to chaotic fractional-order Chen systems. Phys. A 370(2), 279–85 (2006)CrossRefGoogle Scholar
  32. 32.
    Zhang, G., Liu, Z., Ma, Z.: Generalized synchronization of different dimensional chaotic dynamical systems. Chaos Soliton Fractals 32, 773–779 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Qun, L., Hai-Peng, P., Ling-Yu, X., Xian, Y.: Lag synchronization of coupled multidelay systems. Math. Prob. Eng. (2012)Google Scholar
  34. 34.
    Ouannas, A., Azar, A.T., Vaidyanathan, S.: On a simple approach for Q-S synchronization of chaotic dynamical systems in continuous-time. Int. J. Comp. Sci. Math. 8(1), 20–27 (2009)CrossRefGoogle Scholar
  35. 35.
    Ouannas, A., Al-sawalha, M.M.: Synchronization between different dimensional chaotic systems using two scaling matrices. Opt.-Int. J. Light Electron Opt. 127, 959–963 (2016)CrossRefGoogle Scholar
  36. 36.
    Ouannas, A., Al-sawalha, M.M.: On \(\Lambda \)-\(\Phi \) generalized synchronization of chaotic dynamical systems in continuous-time. Eur. Phys. J. Spec. Top. 225(1), 187–196 (2016)zbMATHCrossRefGoogle Scholar
  37. 37.
    Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421–428 (1992)CrossRefGoogle Scholar
  38. 38.
    Feng, C.F.: Projective synchronization between two different time-delayed chaotic systems using active control approach. Nonlinear Dyn. 62, 453–459 (2010)zbMATHCrossRefGoogle Scholar
  39. 39.
    Ho, M.C., Hung, Y.C.: Synchronization of two different chaotic systems using generalized active control. Phys. Lett. A 301, 424–428 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Chen, S., Lu, J.: Synchronization of an uncertain unified systems via adaptive control. Chaos Soliton Fractals 14, 643–647 (2002)zbMATHCrossRefGoogle Scholar
  41. 41.
    Zhao, J., Lu, J.: Parameter identification and backstepping control of uncertain Lu system. Chaos, Soliton Fractals 17, 721–729 (2003)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Yau, H.T.: Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos, Soliton Fractals 22, 341–347 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Lu, J., Wu, X., Han, X., Lu, J.: Adaptive feedback synchronization of unified chaotic systems. Phys. Lett. A 329, 327–333 (2004)zbMATHCrossRefGoogle Scholar
  44. 44.
    Li, X.: Generalized projective synchronization using nonlinear control method. Int. J. Nonlinear Sci. 8, 79–85 (2009)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Yan, Z.: Q-S synchronization in 3D Hènon-like map and generalized Hènon map via a scalar controller. Phys. Lett. A 342(4), 309–317 (2005)zbMATHCrossRefGoogle Scholar
  46. 46.
    Yan, Z.Y.: Q-S (complete or anticipated) synchronization backstepping scheme in a class of discrete-time chaotic (hyperchaotic) systems: a symbolic-numeric computation approach. Chaos 16(1), 013119 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Grassia, G., Miller, D.A.: Dead-beat full state hybrid projective synchronization for chaotic maps using a scalar synchronizing signal. Nonlinear Sci. Numer. Simul. 17(4), 1824–1830 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Aguilar-Bustos, A.Y., Cruz Hernandez, Y.C.: Synchronization of discrete-time hyperchaotic systems: an application in communications. Chaos Solitons Fractals 41(3), 1301–1310 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Liu, W., Wang, Z.M., Zhang, W.D.: Controlled synchronization of discrete-time chaotic systems under communication constraints. Nonlinear Dyn. 69(1–2), 223–230 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Filali, R.L., Benrejeb, M., Borne, P.: On observer-based secure communication design using discrete-time hyperchaotic systems. Commun. Nonlinear Sci. Numer. Simul. 19(5), 1424–1432 (2014)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Ouannas, A.: A new Q-S synchronization scheme for discrete chaotic systems. Far East J. Appl. Math. 84(2), 89–94 (2013)zbMATHGoogle Scholar
  52. 52.
    Ouannas, A.: Co-existence of complete synchronization and anti-synchronization in a class of discrete rational chaotic systems. Far East J. Dyn. Syst. 23(1–2), 41–48 (2013)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Ouannas, A.: Chaos synchronization approach for coupled of arbitrary 3-D quadratic dynamical systems in discrete-time. Far East J. Appl. Math. 86(3), 225–232 (2014)zbMATHGoogle Scholar
  54. 54.
    Ouannas, A., Aljazaery, I.: A new method to generate a discrete chaotic dynamical systems using synchronization technique. Far East J. Dyn. Syst. 24(1–2), 15–24 (2014)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Ouannas, A.: A new chaos synchronization criterion for discrete dynamical systems. Appl. Math. Sci. 8(41), 2025–2034 (2014)MathSciNetGoogle Scholar
  56. 56.
    Ouannas, A.: Nonlinear control method for chaos synchronization of arbitrary 2D quadratic dynamical systems in discrete-time. J. Math. Analy. 8(41), 2025–2034 (2014)MathSciNetGoogle Scholar
  57. 57.
    Ouannas, A.: On full state hybrid projective synchronization of general discrete chaotic systems. J. Nonlinear Dyn. 2014, 1–6 (2014)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Ouannas, A.: Some synchronization criteria for N-dimensional chaotic systems in discrete-Time. J. Adv. Res. Appl. Math. 6(4), 1–10 (2014)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Ouannas, A.: A synchronization criterion for a class of sinusoidal chaotic maps via linear controller. Int. J. Contemp. Math. Sci. 9(14), 677–683 (2014)CrossRefGoogle Scholar
  60. 60.
    Ouannas, A.: Synchronization and inverse synchronization of different dimensional discrete chaotic systems via scaling matrix. Int. J. Chaos Control Model. Simul. 3(4), 1–12 (2014)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Ouannas, A.: Synchronization criterion for a class of N-dimensional discrete chaotic systems. J. Adv. Res. Dyn. Control Syst. 7(1), 82–89 (2015)MathSciNetGoogle Scholar
  62. 62.
    Ouannas, A.: A new synchronization scheme for general 3D quadratic chaotic systems in discrete-time. Nonlinear Dyn. Syst. Theor. 15(2), 163–170 (2015)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Ouannas, A., Odibat, Z.: Generalized synchronization of different dimensional chaotic dynamical systems in discrete-time. Nonlinear Dyn. 81(1), 765–771 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Ouannas, A.: A new generalized-type of synchronization for discrete chaotic dynamical systems. J. Comput. Nonlinear Dyn. 10(6), 061019 (2015)CrossRefGoogle Scholar
  65. 65.
    Ouannas, A., Odibat, Z., Shawagfeh, N.: A new Q–S synchronization result for discrete chaotic systems. In: Differential Equations and Dynamical Systems, pp. 1–10Google Scholar
  66. 66.
    Ouannas, A., Odibat, Z., Shawagfeh, N.: Universal chaos synchronization control laws for general quadratic discrete systems. Appl. Math. Model. 45, 636–641Google Scholar
  67. 67.
    Martinez-Guerra, R., Mata-Machuca, J.L.: Fractional generalized synchronization in a class of nonlinear fractional order systems. Nonlinear Dyn. 77, 1237–1244 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Mahmoud, G.M., Abed-Elhameed, T.M., Ahmed, M.E.: Generalization of combination-combination synchronization of chaotic \(n\)-dimensional fractional-order dynamical systems. Nonlinear Dyn. 83(4), 1885–93 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Maheri, M., Arifin, N.: Synchronization of two different fractional-order chaotic systems with unknown parameters using a robust adaptive nonlinear controller. Nonlinear Dyn. 85(2), 825–38 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Ouannas, A., Al-sawalha, M.M., Ziar, T.: Fractional chaos synchronization schemes for different dimensional systems with non-identical fractional-orders via two scaling matrices. Optik 127(20), 8410–8418 (2016)CrossRefGoogle Scholar
  71. 71.
    Sheu, L.J., Chen, H.K., Chen, J.H., Tam, L.M.: Chaos in a new system with fractional order. Chaos Solitons Fractals 31(5), 1203–12 (2007)CrossRefGoogle Scholar
  72. 72.
    Yan, J., Li, C.: On chaos synchronization of fractional differential equations. Chaos Solitons Fractals 32(2), 725–35 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Xin, L., Yong, C.: Function projective synchronization of two identical new hyperchaotic systems. Commun. Theor. Phys., Beijing, China 48(5), 864–870 (2007)Google Scholar
  74. 74.
    Odibat, Z.: Adaptive feedback control and synchronization of non-identical chaotic fractional order systems. Nonlinear Dyn. 60(4), 479–87 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    Odibat, Z., Corson, N., Aziz-Alaoui, M.A., Bertelle, C.: Synchronization of chaotic fractional-order systems via linear control. Int. J. Bifur. Chaos 20(1), 81–97 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Odibat, Z.: A note on phase synchronization in coupled chaotic fractional order systems. Nonlinear Anal. RWA 13(2), 779–89 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    Odibat, Z., Corson, N., Aziz–Alaoui, M.A., Alsaedi, A.: Chaos in fractional order cubic Chua system and synchronization. Int. J. Bifur. Chaos 27(10), 1750161–13 (2017)Google Scholar
  78. 78.
    Wang, Y., Guan, Z.: Generalized synchronization of continuous chaotic systems. Chaos Soliton Fractals 27, 97–101 (2006)zbMATHCrossRefGoogle Scholar
  79. 79.
    Li, C.P., Deng, W.H., Xu, D.: Chaos synchronization of the chua system with a fractional order. Phys. A 360(2), 171–85 (2006)MathSciNetCrossRefGoogle Scholar
  80. 80.
    Zhu, H., Zhou, S., Zhang, J.: Chaos and synchronization of the fractional-order Chua’s system. Chaos Solitons Fractals 39(4), 1595–603 (2009)zbMATHCrossRefGoogle Scholar
  81. 81.
    Ouannas, A., Abu-Saris, R.: A robust control method for Q-S synchronization between different dimensional integer-order and fractional-order chaotic systems. J. Control Sci. Eng. 2015 (2015)Google Scholar
  82. 82.
    Ouannas, A., Karouma, A.: Different generalized synchronization schemes between integer-order and fractional-order chaotic systems with different dimensions. In: Differential Equations and Dynamical Systems, pp. 1–13 (2016)Google Scholar
  83. 83.
    Ouannas, A., Wang, X., Pham, V.-T., Ziar, T.: Dynamic analysis of complex synchronization schemes between integer order and fractional order chaotic systems with different dimensions. Complexity 2017 (2017)Google Scholar
  84. 84.
    Ouannas, A., Azar, A.T., Radwan, A.G.: On inverse problem of generalized synchronization between different dimensional integer-order and fractional-order chaotic systems. In: IEEE Conference ICM (2016)Google Scholar
  85. 85.
    Ouannas, A., Zehrour, O., Laadjal, Z.: Nonlinear methods to control synchronization between fractional-order and integer-order chaotic systems. Nonlinear Stud. 25(1), 1–13 (2018)MathSciNetzbMATHGoogle Scholar
  86. 86.
    Gasri, A., Ouannas, A.: A general control method for inverse hybrid function projective synchronization of class of chaotic systems. Inter. J. Math. Analy. 9(9), 429–436 (2015)zbMATHGoogle Scholar
  87. 87.
    Ouannas, A., Abu-Saris, R.: On matrix projective synchronization and inverse matrix projective synchronization for different and identical dimensional discrete-time chaotic systems. J. Chaos 2015, 1–7 (2015)zbMATHGoogle Scholar
  88. 88.
    Ouannas, A., Mahmoud, E.E.: Inverse matrix projective synchronization for discrete chaotic systems with different dimensions. J. Comput. Intell. Electron. Syst. 3(3), 188–192 (2014)CrossRefGoogle Scholar
  89. 89.
    Ouannas, A., Grassi, G.: Inverse full state hybrid projective synchronization for chaotic maps with different dimensions. Chin. Phys. B 25(9), 090503–6 (2016)CrossRefGoogle Scholar
  90. 90.
    Ouannas, A., Grassi, G., Ziar, T., Odibat, Z.: On a function projective synchronization scheme between non-identical fractional-order chaotic (hyperchaotic) systems with different dimensions and orders. Optik 136, 513–523 (2017)CrossRefGoogle Scholar
  91. 91.
    Ouannas, A., Azar, A.T., Ziar, T.: On inverse full state hybrid function projective synchronization for continuous-time chaotic dynamical systems with arbitrary dimensions. In: Differential Equations and Dynamical Systems, pp. 1–14 (2016)Google Scholar
  92. 92.
    Ouannas, A.: On inverse generalized synchronization of continuous chaotic dynamical systems. Int. J. Appl. Comp. Math. 2(1), 1–11 (2016)MathSciNetCrossRefGoogle Scholar
  93. 93.
    Ouannas, A., Azar, A.T., Abu-Saris, R.: A new type of hybrid synchronization between arbitrary hyperchaotic maps. Int. J. Mach. Learn. Cybern. 8(6), 1–8 (2016)Google Scholar
  94. 94.
    Ouannas, A.: Co-existence of various synchronization-types in hyperchaotic maps. Nonlinear Dyn. Syst. Theor. 16(3), 312–321 (2016)zbMATHGoogle Scholar
  95. 95.
    Ouannas, A., Grassi, G.: A new approach to study co-existence of some synchronization types between chaotic maps with different dimensions. Nonlinear Dyn. 86(2), 1319–1328 (2016)zbMATHCrossRefGoogle Scholar
  96. 96.
    Ouannas, A., Azar, A.T., Vaidyanathan, S.: New hybrid synchronization schemes based on coexistence of various types of synchronization between master-slave hyperchaotic systems. Int. J. Comp. App. Tech. 55(2), 112–120 (2017)CrossRefGoogle Scholar
  97. 97.
    Ouannas, A., Azar, A.T., Vaidyanathan, S.: A new fractional hybrid Chaos synchronization. Int. J. Model. Ident. Control 27(4), 314–323 (2017)zbMATHCrossRefGoogle Scholar
  98. 98.
    Ouannas, A., Azar, A.T., Vaidyanathan, S.: A robust method for New fractional hybrid chaos synchronization. Math. Meth. Appl. Sci. 40, 1804–1812 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  99. 99.
    Ouannas, A., Abdelmalek, S., Bendoukha, S.: Coexistence of some chaos synchronization types in fractional-order differential equations. Electron. J. Diff. Equ. 2017(128), 1–15 (2017)MathSciNetzbMATHGoogle Scholar
  100. 100.
    Ouannas, A., Odibat, Z.: Fractional analysis of co-existence of some types of chaos synchronization. Chaos, Solution Fractal 105, 215–223 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    Ouannas, A., Wang, X., Pham, V.T., Grassi, G., Ziar, T.: Co-existence of some synchronization types between non-identical commensurate and incommensurate fractional-order chaotic systems with different dimensions. In: Advances in Difference Equations, vol. 21 (2018)Google Scholar
  102. 102.
    Kocarev, L.: Chaos-based cryptography: a brief overview. IEEE Circ. Syst. Mag. 1(3), 6–21 (2001)CrossRefGoogle Scholar
  103. 103.
    Dachselt, F., Schwarz, W.: Chaos and cryptography. IEEE Trans. Circuits Syst. I Fund. Theor. Appl. 48(12), 1498–1509 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  104. 104.
    Kolumban, G., Kennedy, M.P., Kis, G., Jako, Z.: FM-DCSK: a novel method for chaotic communications. In: Proceedings of the 1998 IEEE International Symposium on Circuits and Systems, ISCAS 1998, vol. 4, pp. 477–480 (1998)Google Scholar
  105. 105.
    Sushchik, M., Rulkov, N., Larson, L., Tsimring, L., Abarbanel, H., Yao, K., Volkovskii, A.: Chaotic pulse position modulation: a robust method of communicating with chaos. IEEE Commun. Lett. 4(4), 128–130 (2000)CrossRefGoogle Scholar
  106. 106.
    Chien, T., Liao, T.: Design of secure digital communication systems using chaotic modulation, cryptography and chaotic synchronization. Chaos, Solitons Fractals 24, 241–255 (2005)zbMATHCrossRefGoogle Scholar
  107. 107.
    Masuda, N., Aihara, K.: Cryptosystems with discretized chaotic maps. IEEE Trans. Circuits Syst. I Fund. Theor. Appl. 49(1), 28–40 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  108. 108.
    Lawande, Q.V., Ivan, B.R., Dhodapkar, S.D.: Chaos Based Cryptography: A New Approach to Secure Communications. BARC Newsletter, Bombay (2005)Google Scholar
  109. 109.
    Masuda, N., Jakimoski, G., Aihara, K., Kocarev, L.: Chaotic block ciphers: from theory to practical algorithms. IEEE Trans. Circuits Syst. I Regul. Pap. 53, 1341–1352 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  110. 110.
    Matthews, R.A.J.: On the derivation of a ‘chaotic’ encryption algorithm. Cryptologia 13, 29–42 (1989)MathSciNetCrossRefGoogle Scholar
  111. 111.
    Wheeler, D.D.: Problems with chaotic cryptosystems. Cryptologia 13, 243–250 (1989)CrossRefGoogle Scholar
  112. 112.
    Wheeler, D.D., Matthews, R.A.J.: Supercomputer investigations of a chaotic encryption algorithm. Cryptologia 15(2), 140–152 (1991)CrossRefGoogle Scholar
  113. 113.
    Gerosa, A., Bernardini, R., Pietri, S.: A fully integrated chaotic system for the generation of truly random numbers. IEEE Trans. Circuits Syst. I Fund. Theor. Appl. 49(7), 993–1000 (2002)CrossRefGoogle Scholar
  114. 114.
    Bergamo, P., D’Arco, P., De Santis, A., Kocarev, L.: Security of public-key cryptosystems based on Chebyshev polynomials. IEEE Trans. Circuits Syst. I Regul. Pap. 52(7), 1382–1393 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  115. 115.
    Kocarev, L., Makraduli, J.: Public-key encryption based on Chebyshev polynomials. Circuits Syst. Sign. Process. 24(5), 497–517 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  116. 116.
    Cuomo, K.M., Oppenheim, A.V., Strogatz, S.H.: Robustness and signal recovery in a synchronized chaotic system. Int. J. Bifurcat. Chaos 3(6), 1629–1638 (1993)zbMATHCrossRefGoogle Scholar
  117. 117.
    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. SP 40(10), 634–642 (1993)CrossRefGoogle Scholar
  118. 118.
    Xu, W., Tan, Y., Lau, F.C.M., Kolumban, G.: Design and optimization of differential chaos shift keying scheme with code index modulation. IEEE Trans. Commun. PP(99), 1–1 (2018)Google Scholar
  119. 119.
    Hu, W., Wang, L., Kaddoum, G.: Design and performance analysis of a differentially spatial modulated chaos shift keying modulation system. IEEE Trans. Circuits Syst. II Express Briefs 64(11), 1302–1306 (2017)CrossRefGoogle Scholar
  120. 120.
    Huang, T., Wang, L., Xu, W., Chen, G.: A multi-carrier \(M\)-ary differential chaos shift keying system with low PAPR. IEEE Access 5, 18793–18803 (2017)CrossRefGoogle Scholar
  121. 121.
    Jako, Z.: Performance Improvement of Differential Chaos Shift Keying Modulation Scheme. Budapest University of Technology and Economics, Hungary (2003). PhD thesisGoogle Scholar
  122. 122.
    Parlitz, U., Chua, L.O., Kocarev, L., Halle, K., Shang, A.: Transmission of digital signals by chaotic synchronization. Int. J. Bifurcat. Chaos 2(4), 973–977 (1992)zbMATHCrossRefGoogle Scholar
  123. 123.
    Cruz, C., Nijmeijer, H.: Synchronization through filtering. Int. J. Bifurcat. Chaos 10(4), 763–775 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  124. 124.
    Millerioux, G., Daafouz, J.: Unknown input observers for message embedded chaos synchronization of discrete time systems. Int. J. Bifurcat. Chaos 14(4), 1357–1368 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  125. 125.
    Brand, A., Aghvami, H.: Multiple Access Protocols for Mobile Communications: GPRS, UMTS, and Beyond. Wiley & Sons Ltd., (2002)Google Scholar
  126. 126.
    He, D., Leung, H.: Quasi-orthogonal chaotic CDMA multi-user detection using optimal chaos synchronization. IEEE Trans. Circuits Syst. II Express Briefs 52(11), 739–743 (2005)CrossRefGoogle Scholar
  127. 127.
    Ihan Martoyo, P., Susanto, A., Wijanto, E., Kanalebe, H., Gandi, K.: Chaos codes vs. orthogonal codes for CDMA. In: IEEE 11th International Symposium Spread Spectrum Techniques and Applications, pp. 189–193 (2010)Google Scholar
  128. 128.
    Mansingka, A.S., Zidan, M.A., Radwan, A.G., Salama, K.N.: Secure DS-CDMA spreading codes using fully digital multidimensional multiscroll chaos. In: IEEE 56th International Midwest Symposium Circuits & Systems (MWSCAS), pp. 1334–1338 (2013)Google Scholar
  129. 129.
    Tam, W., Lao, F., Tse, C. : Digital Communications with Chaos: Multiple Access Techniques and Performance. Elsevier Ltd. (2007)Google Scholar
  130. 130.
    Engels, M.: Wireless OFDM systems. In: International Series Engineering & Computer Science (SECS), vol. 692. Springer, Berlin (2002)Google Scholar
  131. 131.
    Yang, X., Hu, X., Shen, Z., He, H., Hu, W., Bai, C.: Physical layer signal encryption using digital chaos in OFDM-PON. In: 10th International Conference on Information Communications and Signal Processing (ICICS), Singapore, pp. 1–6 (2015)Google Scholar
  132. 132.
    Zhang, W., Zhang, C., Jin, W., Chen, C., Jiang, N., Qiu, K.: Chaos coding-based QAM IQ-encryption for improved security in OFDMA-PON. IEEE Photonics Tech. Lett. 26(19), 1964–1967 (2014)CrossRefGoogle Scholar
  133. 133.
    Shen, Z., Yang, X., He, H., Hu, W.: Secure transmission of optical DFT-S-OFDM data encrypted by digital chaos. IEEE Photonics J. 8(3), 1–9 (2016)CrossRefGoogle Scholar
  134. 134.
    Zhang, Jian-Zhong, Wang, An-Bang, Wang, Juan-Fen, Wang, Yun-Cai: Wavelength division multiplexing of chaotic secure and fiber-optic communications. Opt. Express 17, 6357–6367 (2009)CrossRefGoogle Scholar
  135. 135.
    Chen, B., Zhang, L., Lu, H.: High security differential chaos-based modulation with channel scrambling for WDM-aided VLC system. IEEE Photonics J. 8(5), 1–13 (2016)Google Scholar
  136. 136.
    Jiang, N., Xue, C., Zhang, C., Qiu, K.: Physical-enhanced secure communication based on wavelength division multiplexing chaos synchronization of multimode semiconductor lasers. In: IEEE/CIC International Conference on Communications in China (ICCC), pp. 1–5 (2016)Google Scholar
  137. 137.
    Yang, T., Chua, L.: Impulsive stabilization for control and synchronization of chaotic systems, theory and applications to secure communications. IEEE Trans. Circuits Syst. I 44(10), 976–988 (1997)MathSciNetCrossRefGoogle Scholar
  138. 138.
    Millerioux, G., Mira, C.: Coding scheme based on chaos sychronization from noninvertible maps. Int. J. Bifurcat. Chaos 8(10), 2019–2029 (1998)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Samir Bendoukha
    • 1
  • Salem Abdelmalek
    • 2
    Email author
  • Adel Ouannas
    • 3
  1. 1.Department of Electrical EngineeringTaibah UniversityYanbuSaudi Arabia
  2. 2.Department of MathematicsUniversity of TebessaTebessaAlgeria
  3. 3.Laboratory of Mathematics, Informatics and Systems (LAMIS)University of Larbi TebessiTebessaAlgeria

Personalised recommendations