$n$ -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.