A Novel Algorithm to Improve Digital Chaotic Sequence Complexity through CCEMD and PE

Abstract

In this paper, a three-dimensional chaotic system with a hidden attractor is introduced. The complex dynamic behaviors of the system are analyzed with a Poincaré cross section, and the equilibria and initial value sensitivity are analyzed by the method of numerical simulation. Further, we designed a new algorithm based on complementary ensemble empirical mode decomposition (CEEMD) and permutation entropy (PE) that can effectively enhance digital chaotic sequence complexity. In addition, an image encryption experiment was performed with post-processing of the chaotic binary sequences by the new algorithm. The experimental results show good performance of the chaotic binary sequence.Empirical mode decomposition (EMD) in digital signal processing has been extensively applied in nonlinear signal processing [15][16][17][18]. EMD was first proposed by Huang et al. [19][20][21]. It is an effective tool for analyzing nonlinear and non-stationary signals. The EMD method is closely related to the corresponding Hilbert transform method. Through the decomposition of nonlinear and non-stationary signals, a series of intrinsic mode functions (IMFs) are obtained, which makes each IMF a stable signal for narrowband [22]. The IMFs play a crucial role in the analysis of non-stationary or nonlinear signals. However, there are some problems with the EMD method, of which the main one is mode mixing. Complementary ensemble empirical mode decomposition (CEEMD) can effectively restrain the mode mixing of EMD at a certain level [23][24][25]. Based on the above considerations, we proposed a new algorithm which combines CEEMD with permutation entropy (PE) [26] to effectively improve the complexity of the digital chaotic sequence.The rest of this paper is organized as follows: Section 2 describes a hidden chaos attractor with no equilibria. The dynamic characteristics of a complex chaotic system are studied by means of numerical simulation and theoretical analysis. Section 3 proposes a new algorithm to improve the complexity of the digital chaotic sequence. Section 4 considers image encryption with post-processing of the chaotic binary sequences by the algorithm outlined in Section 3. The security of the encrypted image is analyzed through key sensitivity, information entropy, and histogram analysis. Section 5 summarizes the discussions of this paper. The Characteristic Analysis of a Chaotic SystemIn this section, a system can be expressed as the following set of differential equations:where a, b, c are real parameters. When a = 2, b = 0.35, c = 1 and the initial value is (−1.6, 0.82, 1.9), the system displays a single-scroll chaotic system [27]. Different projections of the chaotic attractor for this system are shown in Figure 1.

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