Avoiding Dynamical Degradation in Computer Simulation of Chaotic Systems Using Semi-Explicit Integration: Rössler Oscillator Case

Dynamical degradation is a known problem in the computer simulation of chaotic systems. Data type limitations, sampling, and rounding errors give rise to the periodic behavior. In applications of chaotic systems in secure communication and cryptography systems, such effects can reduce data storage security and operation. In this study, we considered a possible solution to this problem by using semi-explicit integration. The key idea is to perturb the chaotic trajectory by switching between two integrators, which possess close but still different numerical solutions. Compared with the traditional approach based on the perturbation of the bifurcation parameter, this technique does not significantly change the nonlinear properties of the system. We verify the efficiency of the proposed perturbation method through several numerical experiments using the well-known Rössler oscillator.

Hybrid Control of Digital Baker Map with Application to Pseudo-Random Number Generator

Dynamical degradation occurs when chaotic systems are implemented on digital devices, which seriously threatens the security of chaos-based cryptosystems. The existing solutions mainly focus on the compensation of dynamical properties rather than on the elimination of the inherent biases of chaotic systems. In this paper, a unidirectional hybrid control method is proposed to improve the dynamical properties and to eliminate the biases of digital chaotic maps. A continuous chaotic system is introduced to provide external feedback control of the given digital chaotic map. Three different control modes are investigated, and the influence of control parameter on the properties of the controlled system is discussed. The experimental results show that the proposed method can not only improve the dynamical degradation of the digital chaotic map but also make the controlled digital system produce outputs with desirable performances. Finally, a pseudorandom number generator (PRNG) is proposed. Statistical analysis shows that the PRNG has good randomness and almost ideal entropy values.

A Spatiotemporal Chaotic System Based on Pseudo-Random Coupled Map Lattices and Elementary Cellular Automata

The coupled map lattices (CML) is a spatiotemporal chaotic system with complex dynamic behavior. In this paper, we propose a spatiotemporal chaotic system with a novel pseudo-random coupling method based on the elementary cellular automata (ECA), and add different perturbations to lattices in each iteration according to ECA. We investigate the spatiotemporal dynamic properties and chaotic behaviors of the proposed system such as bifurcation diagrams, Kolmogorov-Sinai entropy density, and universality. Moreover, the correlation between any two lattices is discussed. Theory analysis and simulation test indicate that the new system has better performance in complexity, ergodic and unpredictability than conventional CML systems such as adjacent CML and mixed linear-nonlinear CML. Furthermore, the correlation coefficient between any two lattices in proposed system is significantly lower than other systems, and another advantage of the proposed system is utilizing the output of ECA to perturb the chaotic system which can effectively alleviate the dynamical degradation in digital system. The excellent performance of proposed system demonstrates that it has great potential for crypto-system.

An Iteration-Time Combination Method to Reduce the Dynamic Degradation of Digital Chaotic Maps

A new and improved method based on the number of iterations is proposed to reduce the dynamical degradation of the digital chaotic map in this study. We construct a control function by introducing iteration time instead of external systems, thereby replacing the control parameters in the original chaotic map. Experimental results show that the chaotic map based on the iteration-time combination method is more complicated and effective. The period is extended without completely destroying the phase space, which indicates that our method is effective and can compete with other proposed techniques. A type of pseudorandom bit generator based on the iteration-time combination method is proposed to demonstrate its simple application.

Reducing the Dynamical Degradation of Digital Chaotic Maps with Time-Delay Linear Feedback and Parameter Perturbation

Digital chaotic maps are not secure enough for cryptographic applications due to their dynamical degradation. In order to improve their dynamics, in this paper, a novel method with time-delay linear feedback and parameter perturbation is proposed. The delayed state variable is used to construct the linear feedback function and parameter perturbation function. This method is universal for all different digital chaotic maps. Here, two examples are presented: one is 1D logistic map and the other is 2D Baker map. To show the effectiveness of this method, we take some numerical experiments, including trajectory and phase space analysis, correlation analysis, period analysis, and complexity analysis. All the numerical results prove that the method can greatly improve the dynamics of digital chaotic maps and is quite competitive with other proposed methods. Furthermore, a simple pseudorandom bit generator (PRBG) based on digital Baker map is proposed to show its potential application. The proposed PRBG is completely constructed by the digital chaotic map, without any other complex operations. Several numerical results indicate that this PRBG has good randomness and high complexity level.

Counteracting the Dynamical Degradation of Digital Chaos by Applying Stochastic Jump of Chaotic Orbits

The performance of digital chaotic systems in secure communications and cryptographic applications is significantly degraded due to the limited precision of digital computational devices that are used to implement digital chaotic systems. However, regardless of the initial value, the digital chaotic map employed in these applications converges to several periodic orbits. Thus, a periodic orbit search algorithm (POSA) can be designed to accurately locate fixed points and limit cycles. In addition, an effective method based on stochastic jumps of chaotic orbits is proposed to counteract the performance degradation of digital chaos. Numerical simulations were performed to assess the effectiveness and feasibility of the proposed method in terms of autocorrelation, phase space inspection, approximate entropy and periodicity analysis. Furthermore, comparison with other existing methods is presented to show the competitiveness of the proposed scheme. Finally, a pseudorandom bit generator (PRBG), with desirable statistical properties, is constructed for chaos cryptography and other applications.