The distinct properties of chaos, especially its extreme sensitivity to tiny variations of initial conditions and system parameters, have granted it to be a good candidate for cryptographic algorithms. In fact, chaos-based algorithm (CBA) has shown some exceptionally good properties in many aspects regarding security, complexity, speed, computing power and computational overhead. The huge potential of chaotic maps in various applications has also ignited great demands of new chaotic maps with more complicate and nicer dynamical nature. By referring to the dimensionality of the maps, they are generally categorized as one-dimensional, two-dimensional and so on (some examples of chaotic maps are given in Appendices). The designs of low-dimensional chaotic maps are basically relied on the non-linear or piecewise-linear transformations, so that the necessity expansion and contraction can be obtained in the same invariant set. In contrast, the formation of high-dimensional chaotic maps is rather ad hoc. Therefore, it is desirable if some systematic ways can be obtained for the derivation of chaotic maps with any desired dimension. We are interested in extending the low-dimensional chaotic. High dimensional chaotic map can be obtained by replacing the scalar values of the transformation matrix of a conventional Cat map by symmetric m-matrices with natural entries. A more general approach, known as multi-dimensional generalization, is to be presented with the proof of some important properties, such as areapreservative and positive Lyapunov exponent. Another possible way to form a chaotic map with any dimension is known as spatial extension. mapping function to high-dimensional space. Cellular Automata (CA), formally introduced by John von Neumann in 1951, are a stroked="f" filled="f" path="m@4@5l@4@11@9@11@9@5xe" o:preferrelative="t" o:spt="75" coordsize="21600,21600" It is shown that there exists one Bernoulli-measure global attractor of rule 119, which is also the nonwandering set of the rule. Moreover, it is demonstrated that rule 119 is topologically mixing on the global attractor and possesses the positive topological entropy. Therefore, rule 119 is chaotic in the sense of both Li-Yorke and Devaney on the global attractor.