This paper and its companion (Part II) are devoted to the analysis of the application of a chaotic piecewise-linear onedimensional (PL1D) map as random number generator (RNG). Piecewise linearity of the map enables us to mathematically find parameter values for which a generating partition is Markov and the RNG behaves as a Markov information source, and then to mathematically analyze the information generation process and the RNG. In the companion paper we discuss practical aspects of our chaos-based RNGs.

This paper and its companion (Part I) are devoted to to the analysis of the application of a chaotic piecewise-linear one-dimensional (PL1D) map as Random Number Generator (RNG). In Part I, we have mathematically analyzed the information generation process of a class of PL1D maps. In this paper, we find optimum parameters that give an RNG with lowest redundancy and maximum margin against parasitic attractors. Further, the map is implemented in a 0.8 m standard CMOS process utilizing switched current techniques. Post-layout circuit simulations of the RNG indicate no periodic attractors over variations in temperature, power supply and process conditions, and maximum redundancy of 0.4%. We esimate that the output bit rate of our RNG is 1 Mbit/s, which is substantially higher than the output bit rate of RNGs available on the market.

We discuss a general approach for chaotic synchronization of dynamical systems that is based on an active-passive decomposition ͑APD͒ of given dynamical systems. It is shown how this approach can be used to construct high-dimensional synchronizing systems in a systematic way using low-dimensional systems as building blocks. Furthermore, two methods for encoding messages are considered that are both based on synchronization. Using these methods the quality of the reconstructed information signal is higher and the encoding is more secure compared to other encryption methods based on synchronization. The main ideas are illustrated using experimental and numerical examples based on continuous and discrete dynamical systems.

For a class of dynamical systems driven by chaotic impulses we give conditions for the occurrence of chaos locking. It is shown how the concept of time-discontinuous coupling of two chaotic systems may lead to generalized synchronization. The method of synchronization is interpreted as a nonlinear analog of the sampling theorem. Furthermore, we examine the effect of amplitude quantization of the driving signal on their synchronization. Even though two time-discontinuously coupled dynamical systems are no more exactly synchronized when the driving signal is digitized, their trajectories are close enough to allow correct transmission of digital information signals between them. ͓S1063-651X͑96͒07708-2͔PACS number͑s͒: 42.81.ϪiOver the past few decades, there has been considerable interest in the studies of chaos and its ubiquitous nature. This is due to two facts. First, the study of chaotic behavior in almost all fields of science is essential for an appropriate description and modeling of various phenomena in nature ͓1͔. Second, nonlinear phenomena may lead to new applications in engineering. For example, recently there has been considerable interest in potential applications of synchronized chaotic systems in the area of analog communication ͓2-4͔. However, almost exclusively today communications are digital. Motivated by this challenge-digital communications-we discuss in this paper the question of the exchange of digital information signals between two synchronized continuous chaotic systems.The paper is organized as follows. First we develop a general theory of driving ͑chaotic͒ systems by chaotic impulses. As a consequence, a criterion for the occurrence of generalized synchronization in unidirectionally systems coupled at discrete times is given. Then we address some questions related to synchronizing two identical systems by chaotic impulses and finally we discuss the relevance of our results to digital communication using chaos synchronization.Consider a driven N-dimensional chaotic dynamical system whose behavior is governed by ẋϭF͑x,s T ͒, ͑1͒