Image encryption based on hyper-chaotic multi-attractors

Abstract

Linear time-delay feedback method makes the stable system generate the infinite-dimensional hyper-chaos, which possesses more than one positive Lyapunov exponent, infinite-dimensional, a wider chaotic parameter range, and multi-attractors, including the single-scroll attractor, the double-scroll attractor, and the composite multi-scroll attractor. The infinite-dimensional hyper-chaotic multi-attractors Chen system generated by linear time-delay feedback (HCMACS) is used for image encryption. Firstly, the state time sequences of the infinite-dimensional HCMACS are preprocessed to achieve the ideal statistical property. Secondly, two random matrices generated by random number generators are used to expand the image size. Finally, the preprocessed chaotic sequences are used to confuse and diffuse the expanded digital image to obtain the encrypted image. The special feature of the proposed method is the theoretically infinite-dimensional secret key space. Results of analysis and computer simulation indicate that the encryption algorithm has good encryption performance, which can effectively resist various attacks.

Appendix: The sensitivity of the initial conditions of HCMACS

Fig. 14

Time domain waveform of x(t) with different initial conditions

Fig. 15

The cross-correlation of two sequences x(t) generated initial condition cases (1) and (2)

Since time delay is introduced, HCMACS generates hyper-chaos with theoretically infinite dimension in the sense that a continuum of initial conditions over the interval $-\tau \le t < 0$ is required to specify the dynamical behavior, that is, when $t > 0$, the behavior of the system is not only dependent on the states at time 0, but also related to the state on [$-\tau$, 0]. So the initial state space of the system is infinite dimensional.

HCMACS is sensitively dependent on the state on [$-0.3, 0$]. To show this point, assume $\tau = 0.3$, $a = 35$, $b = 3$, $c = 18.5$, $k = 3.8$, $x(0) = 0.1$, $y(0) = 0.1$, and $z (0) = 0.1$, the state of z(t) for $-\tau \le t < 0$ is given as two slightly different cases:

• (1) $z(t) = 0$ for $-\tau \le t < 0$.

• (2) $z(t) = 0$ for $-\tau \le t < -0.0195$ and $-0.0195< t < 0$, while, $z(-0.0195) = 2^{-32}$.

The time domain waveforms of the state x of the system for slightly different initial condition are given in Fig. 14. From Fig. 14, we can see that two time sequences are quite different. The cross-correlation of the time sequences with different initial conditions is given in Fig. 15, from which, we know that the cross-correlation is near zero. This indicates that the system is sensitive on the initial condition in the time-delay continuum. Therefore, the time-delay continuum can be used as key to expand the key space.