Formation of High-Dimensional Chaotic Maps and Their Uses in Cryptography

Abstract

Being a particular class of nonlinearity, chaos nowadays becomes one of the most well known and potentially useful dynamics. Although a chaotic system is only governed by some simple and low order deterministic rules, it possesses many distinct characteristics, such as deterministic but random-like complex temporal behavior, high sensitivity to initial conditions and system parameters, fractal structure, long-term unpredictability and so on. These properties have been widely explored for the last few decades and found to be useful for many engineering problems such as cryptographic designs, digital communications, network behaviour modeling, to name a few. The increasing interests in chaos-based applications have also ignited tremendous demand for new chaos generators with complicate dynamics but simple designs. In this chapter, two different approaches are described for the formation of high-dimensional chaotic maps and their dynamical characteristics are studied. As reflected by the statistical results, strong mixing nature is acquired and these high-dimensional chaotic maps are ready for various cryptographic usages. Firstly, it is used as a simple but effective post-processing function which outperforms other common post-processing functions. Based on such a chaos-based post-processing function, two types of pseudo random number generators (PRNGs) are described. The first one is a 32-bit PRNG providing a fast and effective solution for random number generation. The second one is an 8-bit PRNG system design, meeting the challenge of low bit-precision system environment. The framework can then be easily extended for some practical applications, such as for image encryption. Detailed analyses on these applications are carried out and the effectiveness of the high-dimensional chaotic map in cryptographic applications is confirmed.