Abstract
Synchronizing chaotic oscillators has been a challenge to guarantee successful applications in secure communications. That way, three synchronization techniques are applied herein to twenty two chaotic oscillators, three of them based on piecewise-linear functions and nineteen proposed by Julien C. Sprott. These chaotic oscillators are simulated to generate chaotic time series that are used to evaluate their Lyapunov exponents and Kaplan-Yorke dimension to rank their unpredictability. The oscillators with the high positive Lyapunov exponent are implemented into a field-programmable gate array (FPGA), and afterwards they are synchronized in a master-slave topology applying three techniques: the seminal work introduced by Pecora-Carroll, Hamiltonian forms and observer approach, and open-plus-closed-loop (OPCL). These techniques are compared with respect to their synchronization error and latency that is associated to the FPGA implementation. Finally, the chaotic oscillators providing the high positive Lyapunov exponent are synchronized and applied to a communication system with chaotic masking to perform a secure image transmission. Correlation analysis is performed among the original image, the chaotic channel and the recovered image for the three synchronization schemes. The experimental results show that both Hamiltonian forms and OPCL can recover the original image and its correlation with the chaotic channel is as low as 0.00002, demonstrating the advantage of synchronizing chaotic oscillators with high positive Lyapunov exponent to guarantee high security in data transmission.
Highlights
Secure communication systems have been developed since the introduction of the first synchronization approach between two chaotic oscillators by Pecora and Carroll [1, 2]
It consists of three state variables x1, x2, x3, four coefficients a, b, c, d1 and the saturated nonlinear function (SNLF) f(x1) that can be approached by the piecewise-linear functions (PWL) function described by (2) and shown in Fig 1(a) to generate 2-scrolls
We selected the six chaotic oscillators providing high positive Lyapunov exponent values, they are: the chaotic oscillator based on SNLF series, and Sprott’s cases G and L from Table 4, and the chaotic oscillator based on Negative Slopes, and Sprott’s cases B and S from Table 5
Summary
Secure communication systems have been developed since the introduction of the first synchronization approach between two chaotic oscillators by Pecora and Carroll [1, 2]. The main objective of synchronizing two chaotic oscillators is oriented to develop secure communication systems to preserve privacy, provide security and be robust to attacks These issues can be accomplished using chaotic oscillators because they have the property of high sensitivity to the initial conditions, which can be quantified by evaluating and maximizing the positive Lyapunov exponent. The evaluation of the fractal dimension provides characteristics to rank the randomness and unpredictability of chaotic oscillators In this manner, this work shows that the master-slave synchronization of two chaotic oscillators having high positive Lyapunov exponents guarantees high security, and if the synchronization error is very low the original information can be recovered without loss of data.
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