Synchronizing Hyperchaotic Maps to Encode/Decode Information

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.

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Appendix

Proposition 2.1 Let the nonlinear chaotic system, x_k+1 = f(x_k ), y_k = h(x_k) be locally observable, and suppose that corresponding to the constant value, y_e, there exists a unique state vector equilibrium value, x_e. Then, the system is constructible, i.e. there exists a map φ n such that the state x_k of the system can be exactly reconstructed, from time k = 0, on, in terms of the output y_k and a finite string of previously obtained outputs, in the form: x_k = φ (y_k , y_k__x,..., y_k–(n–1)), k > 0 provided the string of outputs, {y_k} 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 x_k for all k > n–1.