Sincronización entre Mapas Hipercaóticos para Codificar y Decodificar Información
Carlos Aguilar–Ibáñez, Miguel S. Suárez–Castañón, Humberto Sossa–Azuela and Ricardo Barrón–Fernández
Centro de Investigación en Computación del IPN Av. Juan de Dios Bátiz s/n Esq. con Manuel Othón de Mendizabal Col. San Pedro Zacatenco, A.P. 75476 07700 México, D.F., México e–mails: caguilar@pollux.cic.ipn.mx ; sasuarez@prodigy.net.mx Phone: 52–5–7296000, x–56568
Article received on May 06, 2004
Accepted on August 23, 2004
Abstract
In this work we propose to use hyperchaotic maps synchronization to encode and decode information. The information to be encode is input to the transmitter as an external perturbation. The transmitted signal is used for synchronization and as the encode information carrier. Once the receiver is synchronized with the transmitter, the former decode the information by reconstruct the external perturbation. Roughly speaking, we design a simple schema to encode and decode data, as a simple inverse problem approach. The schema performance shows to be quite satisfactory, as assess from the numerical implementation. We use the results to build an application to establish secure on–line communication over Internet.
Keywords: Information Encoding, Information Decoding, Cryptography, Hyperchaotic, Map Synchronization.
Resumen
En este trabajo se propone el uso de sincronización entre mapas hipercaóticos para codificar y decodificar información. La información a ser codificada es introducida al transmisor como una perturbación externa. La señal transmitida es empleada tanto para la sincronización y como portadora de la información codificada. Una vez que el receptor esta sincronizado con el transmisor, el primero decodifica la información mediante la reconstrucción de la perturbación externa. En términos generales, se diseñó un esquema sencillo para codificar y decodificar datos, enfocado como un problema inverso. El desempeño del esquema mostró ser muy satisfactorio, como se comprobó en la implantación numérica. Los resultados obtenidos se usaron para construir una aplicación para comunicación segura en línea sobre internet.
Palabras Clave: Información Codificada, información decodificada, criptografía, hipercaótico, Sincronización de Mapas.
DESCARGAR ARTÍCULO EN FORMATO PDF
]]>References
1. Schneier, B., (1995). Applied Cryptography, Wiley. [ Links ]
2. (Stinson, D., (2002). Cryptography: Theory and Practice, CRC Press. [ Links ]
3. Goldreich, O., (2000). Foundations of Cryptography: Basic Tools, Cambridge University Press. [ Links ]
4. (Sira H., Aguilar C., Suárez M., (2002). Exact State Reconstructors in the Recovery of Messages Encrypted by the States of Nonlinear Discrete–time Chaotic Systems, International Journal of Bifurcation and Chaos, Vol. 12, num. 1. [ Links ]
5. Cuomo, G., Oppenheim, A. V., Strogatz, S. H. (1997). Synchronization of Lorenz–Based Chaotic Circuits with Applications to Communications, IEEE Trans. Circuits Syst–II: Analog and Digital Signal Processing, 40, 626–633. [ Links ]
6. Carrol, T. L., Pecora, L. M., (1998). Synchronization Hyperchaotic Volume–Preserving Maps and Circuits, IEEE Trans, on Circuits and Systems–I: Fundamental Theory and Applications, Vol. 45, num. 6. [ Links ]
7. Special Issue on Chaos synchronization and control: Theory and Applications, (1997). IEEE Trans. Circuits Syst. I: Fundamental Th. and Appl. 44 (10). [ Links ]
8. Parlitz, U., Junge, L., Kocarev, L., (1999). Chaos Synchronization, Lecture Notes in Control and Information Sciences 244, New Directions in Nonlinear Observer Design, H. Nijmeier and T.I. Fossen (Eds.), Springer Verlag, pp. 511–525. [ Links ]
9. Parlitz, U., Junge, L., (1999). Synchronization of chaotic systems, Proceedings of the European Control Conference ECC'99, Paper F1056–5, 31.Aug.–3.Sept. 1999, Karsruhe, Germany. [ Links ]
10. Packard, N. H., Crutchfield, J. P., Farmer, J. D., Shaw, R. S., (1980). Geometry From a Time Series, Phys. Rev. Lett. 45, pp. 712–716. [ Links ]
11. Sauer T., Yorke J. & Casdagli M.(1991). Embedology, J. Stat. Phys. 65, pp 579–616. [ Links ]
12. Takens F., (1981). Detecting strange attractors in turbulence, Lecture Notes in Mathematics, 898, pp. 366–387. [ Links ]
13. (Parlitz U., Zöller R., Holzfuss J and Lauterborn W., (1994). Reconstructing Physical Variables and Parameters From Dynamical Systems, International Journal of Bifurcation and Chaos, vol. 4, pp.1715–1719. [ Links ]
14. Itoh Makoto, Wah Wu Chai, Chua Leon O., (1997). Communication Systems Via Chaotic Signals From a Reconstruction Viewpoint, International Journal of Bifurcation and Chaos, vol. 7, pp.275–286. [ Links ]
15. Isidori, A., (1989). NonLinear Control Systems, 2nd. ed. Berlin, (Springer–Verlag, Germany). [ Links ]
16. Pecora L.M. and Carroll T.L.,(1990). Synchronization in chaotic systems, Phys. Rev. Lett., vol 64, pp. 821–824. [ Links ]
17. Huijberts Henri, Nijmeijer H, and Willems Rob, (2000). System Identification in Communication with Chaotic Systems, IEEE Transactions on Circuits and Systems –I: Fundamental Theory and Applications, vol. 47, pp. 800–808. [ Links ]
18. Lichtenberg, A. J. and Lieberman, M. A., (1983). Regular and Stochastic Motion, New York, Springer–Verlag. [ Links ]
19. Pecora, L. M. and Carroll, T. L., Johnson, G., Mar, D. J., (1997). Volume–Preserving and Volume Expanding Synchronized Chaotic Systems, Phys. Rev. E, vol. 56, pp. 5090–5100. [ Links ]
20. Kotta, Ü., (1995). Inversion Method in the Discrete–time Nonlinear Control Systems Synthesis Problems, Lecture Notes in Control and Information Sciences 205, Springer, Berlin, 152 pp. [ Links ]
Appendix
Proposition 2.1 Let the nonlinear chaotic system, xk+1 = f(xk ), yk = h(xk) be locally observable, and suppose that corresponding to the constant value, ye, there exists a unique state vector equilibrium value, xe. Then, the system is constructible, i.e. there exists a map φ 
n such that the state xk of the system can be exactly reconstructed, from time k = 0, on, in terms of the output yk and a finite string of previously obtained outputs, in the form: xk = φ (yk , yk_x,..., yk–(n–1)), k > 0 provided the string of outputs, {yk} for y–n + 1 <k<0, is completely known. Moreover, an initializations of (3.5) with arbitrarily chosen values, y–i ,i = 1,2,...,n — 1, and the actual y0 , still results in an exact reconstruction of xk for all k > n–1.