$n$ -Dimensional Discrete Cat Map Generation Using Laplace Expansions
提出一种基于拉普拉斯展开的n维猫映射高效生成方法，避免了传统矩阵乘法的高计算复杂度，并引入新参数控制空间配置，适用于需要高随机性和低相关性的伪随机数生成场景。
Different from existing methods that use matrix multiplications and have high computation complexity, this paper proposes an efficient generation method of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${n}$ </tex-math></inline-formula> -dimensional ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${n}\text{D}$ </tex-math></inline-formula> ) Cat maps using Laplace expansions. New parameters are also introduced to control the spatial configurations of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${n}\text{D}$ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${O(n^{4})}$ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${n}\text{D}$ </tex-math></inline-formula> Cat matrices with a lower inner correlation, and thus yields more random and unpredictable outputs of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${n}\text{D}$ </tex-math></inline-formula> Cat maps.