$n$ -Dimensional Discrete Cat Map Generation Using Laplace Expansions

Different from existing methods that use matrix multiplications and have high computation complexity, this paper proposes an efficient generation method of ${n}$ -dimensional ( ${n}\text{D}$ ) Cat maps using Laplace expansions. New parameters are also introduced to control the spatial configurations of the ${n}\text{D}$ Cat matrix. Thus, the proposed method provides an efficient way to mix dynamics of all dimensions at one time. To investigate its implementations and applications, we further introduce a fast implementation algorithm of the proposed method with time complexity ${O(n^{4})}$ and a pseudorandom number generator using the Cat map generated by the proposed method. The experimental results show that, compared with existing generation methods, the proposed method has a larger parameter space and simpler algorithm complexity, generates ${n}\text{D}$ Cat matrices with a lower inner correlation, and thus yields more random and unpredictable outputs of ${n}\text{D}$ Cat maps.