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A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter. (English) Zbl 1221.94049
Summary: In recent years chaotic secure communication and chaos synchronization have received ever increasing attention. In this paper, for the first time, a fractional chaotic communication method using an extended fractional Kalman filter is presented. The chaotic synchronization is implemented by the EFKF design in the presence of channel additive noise and processing noise. Encoding chaotic communication achieves a satisfactory, typical secure communication scheme. In the proposed system, security is enhanced based on spreading the signal in frequency and encrypting it in time domain. In this paper, the main advantages of using fractional order systems, increasing nonlinearity and spreading the power spectrum are highlighted. To illustrate the effectiveness of the proposed scheme, a numerical example based on the fractional Lorenz dynamical system is presented and the results are compared to the integer Lorenz system.

37D45Strange attractors, chaotic dynamics
93E11Filtering in stochastic control
94A12Signal theory (characterization, reconstruction, filtering, etc.)
Full Text: DOI
[1] Hifler, R.: Application of fractional calculus in physics, (2000)
[2] Riewe, F.: Mechanics with fractional derivatives, Phys rev, No. 55, 3581-3592 (1997)
[3] Podlubny, I.: Geometric and physical interpretation of fractional integral and fractional derivatives, J fract calc 5, No. 4, 367-386 (2002) · Zbl 1042.26003
[4] Ben Adda, F.: Geometric interpretation of the fractional derivative, J fract calc, 21-52 (1997) · Zbl 0907.26005
[5] Bullock, G. L.: A geometric interpretation of the Riemann -- Stieltjes integral, Am math mon 95, No. 5, 448-455 (1988) · Zbl 0651.26009 · doi:10.2307/2322483
[6] Bologna, M.; Grigolini, P.: Physics of fractal operators, (2003)
[7] Vinagre BM, Monje CA, Calderson AJ. Fractional order system and fractional order control actions, FC application in automatic control. Las Vegas, USA; December 9, 2002. p. 3690 -- 700.
[8] Loverro, A.: Fractional calculus: history, definitions and applications for engineer, (2004)
[9] Debnath, L.: Fractional calculus fundamentals, (2002)
[10] Kilbas; Srivastava; Trujillo: Theory and application of fractional differential equations, North holland mathematics studies 207 (2006) · Zbl 1092.45003
[11] Heidari-Bateni, G.; Mcgillem, C. D.: A chaotic direct-sequence spread spectrum communication system, IEEE trans commun 42, No. 2 -- 4, 1524-1527 (1994)
[12] Satish, K.; Jayakar, T.; Tobin, C.; Madhavi, K.; Murali, K.: Chaos based spread spectrum image steganography, Chaos based spread spectrum image steganography, 587-590 (2005)
[13] Kolumban, G.; Kennedy, M. P.; Chua, L. O.: The role of synchronization in digital communications using chaos --- part I. Fundamentals of digital communications, IEEE trans circ syst I 44, 927-936 (1997)
[14] Kolumban, G.; Kennedy, M. P.; Chua, L. O.: The role of synchronization in digital communications using chaos --- part II. Chaotic modulation and chaotic synchronization, IEEE trans circ syst I 45, 1129-1140 (1998) · Zbl 0991.93097 · doi:10.1109/81.735435
[15] Kolumban, G.; Kennedy, M. P.: The role of synchronization in digital communications using chaos --- part III. Performance bounds for correlation receivers, IEEE trans circ syst I 47, No. 3, 1673-1683 (2002) · Zbl 0990.94002 · doi:10.1109/81.899919
[16] Dachselt, F.; Schwatrz, W.: Chaos and cryptography, IEEE trans circ syst I 48, 1498-1509 (2001) · Zbl 0999.94030 · doi:10.1109/TCSI.2001.972857
[17] Azemi A, Raoufi R, Fallahi K, Hosseini-Khayat S. A sliding-mode adaptive observer chaotic communication scheme. In: 13th Iranian conference on electrical engineering (ICEE2005), vol. 2, Zanjan, Iran; May 10 -- 12, 2005.
[18] Fallahi K, Khoshbin H. A new communication scheme using one-dimensional chaotic maps. In: 15th Iranian conference on electrical engineering (ICEE2007), Iran; May, 2007.
[19] Ruan H, Zhai T, Yaz EE. A chaotic secure chaotic communication scheme with extended Kalman filter based parameter estimation. In: Proceeding of IEEE conference on control applications, vol. 1; June 2003. p. 404 -- 8.
[20] Murali, K.: Digital signal transmission with cascaded heterogeneous chaotic systems, Phys rev E 63, 016217-016223 (2001)
[21] Alvarez, G.; Montoya, F.; Pastor, G.; Romera, M.: Chaotic cryptosystems, Int J bifurcat chaos, 332-338 (1999)
[22] Short, K. M.: Unmasking a modulated chaotic communication scheme, Int J bifurcat chaos 6, 367-375 (1996) · Zbl 0870.94004 · doi:10.1142/S0218127496000114
[23] Perez, G.; Cerdeira, H. A.: Extracting messages masked by chaos, Phys rev lett 74, 1970-1973 (1995)
[24] Yang, T.; Wu, C. W.; Chua, L. O.: Cryptography based on chaotic systems, IEEE trans circ syst --- I: Fundamen appl 44, No. 5, 469-472 (1997) · Zbl 0884.94021 · doi:10.1109/81.572346
[25] Fallahi, K.; Raoufi, R.; Khoshbin, H.: An application of Chen system for secure chaotic communication based on extended Kalman filter and multi-shift cipher algorithm, Commun nonlinear sci numer simul 13, 763-781 (2008) · Zbl 1221.94046 · doi:10.1016/j.cnsns.2006.07.006
[26] Oldham, K. B.; Spanier, J.: The fractional calculus, (1974) · Zbl 0292.26011
[27] Loverro, A.: Fractional calculus: history, definitions and applications for engineer, (2004)
[28] Li, C. G.; Chen, G.: Chaos in the fractional order Chen system and its control, Chaos solitons fract 22, 549-554 (2003) · Zbl 1069.37025 · doi:10.1016/j.chaos.2004.02.035
[29] Diethelm, K.; Ford, N. J.; Freed, A. D.: Detailed error analysis for a fractional Adams method, Numer algorithms 36, 31-52 (2004) · Zbl 1055.65098 · doi:10.1023/B:NUMA.0000027736.85078.be
[30] Li, C.; Peng, G.: Chaos in Chen’s system with a fractional order, Chaos solitons fract 22, 443-450 (2004) · Zbl 1060.37026 · doi:10.1016/j.chaos.2004.02.013
[31] Diethelm, K.; Ford, N. J.; Freed, A. D.: A predictor -- corrector approach for the numerical solution of fractional differential equations, Nonlinear dyn 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[32] Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order, Elec trans numer anal 5, 1-6 (1997) · Zbl 0890.65071 · emis:journals/ETNA/vol.5.1997/pp1-6.dir/pp1-6.html
[33] Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations, Int J math anal appl 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194
[34] Sierociuk, D.; Dzielin’ski, A.: Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation, Int J appl math comput sci 16, No. 1, 129-140 (2006)
[35] Tavazoei, M. S.; Haeri, M.: A necessary condition for double scroll attractor existence in fractional order systems, Phys lett A (2007) · Zbl 1209.37037
[36] Ostalczyk P. Fractional-order backward difference equivalent forms. Part I --- Horner’s form. In: Proceedings of the first IFAC workshop fractional differentiation and its applications, FDA’04, Enseirb, Bordeaux, France. p. 342 -- 7.