On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps

When chaotic systems are realized with finite precisions in digital computers, their dynamical properties are often found to be entirely different from the original versions in the continuous setting. In the literature, there does not seem to be much work on quantitative analysis of such degradation of digitized chaos and how to reduce its negative influence on chaos-based digital systems. Focusing on 1D piecewise linear chaotic maps (PWLCM), this paper reports some findings on a new series of dynamical indicators, which can quantitatively reflect the degradation effects on a digital PWLCM realized with a fixed-point finite precision. On top of that, the paper introduces a new method for studying digital chaos from an algorithmic point of view. In addition, the theoretical results obtained in this paper should be very helpful for the consideration of reducing negative influence of dynamical degradation in real design of various digital chaotic systems. As typical examples, the proposed dynamical indicators are applied to the performance comparison of different remedies for improving dynamical degradation, cryptanalysis of digital chaotic ciphers based on 1D PWLCM, and the design of chaotic pseudo-random number generators with desired characteristics.