n-dimensional Discrete Cat Map Generation using Laplace Expansions

Introduction

Different from existing methods that use matrix multiplications and have high computation complexity, this paper proposes an efficient generation method of n-dimensional (nD) Cat maps using Laplace expansions. New parameters are also introduced to control the spatial configurations of the nD 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(n4) and a pseudo-random 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 nD Cat matrices with a lower inner correlation, and thus yields more random and unpredictable outputs of nD Cat maps.

Proposed nD Cat map generation method

Algorithm 1 describes the proposed nD Cat map generation method. The parameters are setting as follows

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Performance of proposed approach

Shannon entropy can be used to test the randomness of Cat maps generated by different methods. Bigger entropy value means the corresponding Cat map can generate better random outputs. Figure 1 shows the test results. Obviously, the proposed method can generate nD Cat map with better randomness.

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Figure 1. The Shannon entropy results of nD Cat maps generated by different methods with (a) different dimension n and (b) different number of states N.