1. Introduction
Since Lorenz [
1] discovered the first three-dimensional chaos model, chaos theory has grown with the development of computer science. Chaos is an unpredictable and random motion in deterministic dynamical systems due to its sensitivity to initial values. The certainty of a dynamic system is a concept defined in mathematics, which means that the state of the system at any time can be determined by the initial state of the system. Although the motion state of the deterministic dynamic system at any time can be calculated according to the initial state and motion law, the measurement of the initial state and data cannot be completely accurate. Even a slight difference will lead to a very large error in the predicted results, to an unpredictable degree. In recent years, as chaotic systems have many advantages in encryption, such as ergodicity, unpredictability, pseudo-randomicity, and high sensitivity to parameters and initial values [
2], image encryption based on chaos has become a research hotspot. Aside from image encryption based on chaotic systems, there are many representative methods such as: based on one-time keys, bit-level permutation, DNA rule, matrix, and semi-tensor product theory [
3,
4].
At present, research on 1D chaos, such as Logistic mapping [
5,
6,
7]; 2D chaos, such as Henon mapping [
8,
9,
10]; and 3D chaos, such as Rossler chaotic attractor [
11,
12,
13], Chua [
14,
15,
16], and Chen [
17,
18,
19], have been very extensive and mature. With the development of chaos theory, many people began to study high-dimensional chaotic attractors, such as 4D chaotic attractor subsystems [
20,
21,
22,
23], 5D chaotic attractor subsystems [
24,
25,
26,
27], and 6D chaotic attractor subsystems [
28]. In recent years, fractional-order chaotic systems [
29,
30,
31], hidden attractors [
32,
33,
34], and chaotic systems with co-existing attractors [
35,
36] have also been extensively studied. In ordinary three-dimensional chaotic attractors, linear or nonlinear state feedback controllers can generate different kinds of four-dimensional chaotic systems. The 4D hyperchaotic system has better computational complexity and two or more positive Lyapunov exponents [
37,
38].
Recently, many scholars have generated many new chaotic systems on the basis of studying the Lorenz chaotic system, which are collectively referred to as Lorenz type hyperchaotic systems [
39,
40]. These new systems are applied to many aspects, such as chaotic synchronization [
41], image encryption [
8,
12,
13,
19], stream cryptography [
42,
43], and so on.
The chaos-based image encryption systems are usually applied to generate chaotic stream ciphers for exchanging the positions or values of the pixels in the original images. A 2D chaotic Arnold cat map was used to generate a 3D cat map, which then was used in image encryption [
44]. The results show that the scheme is fast and safe. The authors of [
45] applied Henon mapping to the image encryption scheme, and proved that the encryption method could resist selective plaintext attack, etc. The authors of [
46] proposed an image encryption scheme based on Logistic mapping, and the authors of [
47] proposed an image encryption scheme based on the 3D chaotic system. The above image encryption methods using chaotic systems are based on low-dimensional chaotic systems with at most one positive Lyapunov exponent, which have many advantages, such as simple format, few control parameters, and ease of implementation. However, low-dimensional chaotic systems are vulnerable to attack. If low-dimensional chaotic systems are changed into high-dimensional chaotic systems, the encryption will be more effective. Lyapunov exponent (LE) is an effective method to measure chaotic systems. If a chaotic system has two or more positive LEs, it can be called a hyperchaotic system, which usually has a larger key space and much higher security in encryption schemes [
37,
38]. As the chaotic systems with four dimensions or more have two or more Lyapunov exponents and better dynamic characteristics, the application to image encryption will have better practical effects [
37,
38]. The authors of [
48] presented a novel approach that uses a hyperchaotic system, Pixel-level, and DNA-level diffusion. The authors of [
49] proposed a new image encryption method based on matrix semi-tensor product theory and hyperchaotic Lorenz. The research above shows that the application of hyperchaotic system encryption has become an important trend.
The main contributions of this paper are shown as follows: (1) A new 4D hyperchaotic system is generated, and the dynamic properties of the attractor such as phase space, local stability, poincare section, periodic attractor, quasi-periodic attractor, chaotic attractor, bifurcation diagram, and Lyapunov index are analyzed; (2) Then the new hyperchaotic system is normalized and binary serialized, and the binary hyperchaotic stream generated by the system is statistically tested and entropy analyzed; (3) The hyperchaotic binary stream is applied to the gray image encryption; (4) The histogram, correlation coefficient, entropy test, and security analysis show that the hyperchaotic system has good random characteristics and can be applied to the gray image encryption.
The main advantages of this paper are shown as follows: (1) A new 4D hyperchaotic system based on Lorenz is proposed and analyzed; (2) The hyperchaotic system with two positive LEs is much more random, which is then used to generate sequences for the encryption operations; (3) The new hyperchaotic system in this paper is obtained by adding a new variable, , and a feedback controller, , to the classical Lorenz chaotic attractor system. In this way, an equilibrium point curve exists in the system which is a new phenomenon in the system.
In this paper, a new 4D hyperchaotic system is proposed by studying Lorenz-type hyperchaotic system, and the corresponding dynamic properties, such as Lyapunov exponent, phase space diagram, poincare section diagram, and local stability are studied. The method of normalization and binarization is applied to the encryption of gray image. Finally, the entropy test and security analysis of image encryption are carried out.
The rest of this paper is organized as follows:
Section 2 introduces a new 4D hyperchaotic system based on Lorenz system with two positive LEs. In
Section 3, analyses of the dynamic properties are done, such as judgment of local stability, Poincare section diagram, periodic attractor, etc. In
Section 4, normalization and quantization are done. Furthermore, NIST tests, permutation entropy, and approximate entropy are completed to test the time series of the hyperchaotic system. In
Section 5, the hyperchaotic system is used in image encryption. Then, the encryption effect and security are tested by correlation coefficient analysis, information entropy, differential attack, etc. Finally, the paper is summarized in
Section 6.
2. A New Hyperchaotic System
In 1963, a representative Lorenz equation in chaotic attractors was proposed [
1]. The differential expression of this equation is shown as follows Equation (1):
The equation set is a third-order system of ordinary differential equations, and each variable in the equations does not obviously contain time
, so the equation set is called an autonomous system. Its parameters,
,
, and
, are all constants greater than zero. When the parameters of this equation are taken as
,
, and
, the system presents chaotic attractor state, namely the classical Lorenz attractor, and its phase space is shown in
Figure 1. The numeric computation method used to compute the chaotic system or hyperchaotic system is the 4th order Runge–Kutta method.
The new hyperchaotic system in this paper is obtained by adding a new variable,
, and a feedback controller,
, to the classical Lorenz chaotic attractor system. The new hyperchaotic system is expressed as follows in Equation (2):
Where
,
,
, and
are all constants greater than zero. Let
. There are two positive LEs over a wide range of parameters, which implies that the system here is hyperchaotic, as shown in Figure 9b. Fix the parameter
, then set the parameters of the system with
,
,
,
, and initial condition
. The system can present the state of a hyperchaotic system, as shown in
Figure 2. The Lyapunov exponent corresponding to the hyperchaotic system is shown as follows:
The divergence of the hyperchaotic system can be expressed as Equation (3):
According to Equation (3), when , the hyperchaotic system is a dissipative system.
Figure 2 shows the new hyperchaotic attractors and the phase diagrams with parameters
,
,
, and
. (a) shows the hyperchaotic attractor, (b) shows the hyperchaotic attractor on y–w plane, (c) shows the hyperchaotic attractor on x–y plane, (c) shows the hyperchaotic attractor on y–z plane. The time series diagrams of the phases
,
,
, and
of the hyperchaotic system is shown in
Figure 3.
Figure 3 shows the time series diagrams of the hyperchaotic system, and it can be seen that the sequences have good randomness.
4. Normalization and Quantization
In order to put the hyperchaotic system into use, normalization and quantization are done. The time series after normalization and quantization are tested.
4.1. Normalization Treatment
In order to facilitate the data processing of the hyperchaotic system, the normalization is carried out first. In this paper, the time series data of four output signals,
,
,
, and
, are mapped to the interval
and then quantified. The stream of the normalized hyperchaotic system is shown in
Figure 10.
Figure 10 shows the time series are normalized into the interval
.
4.2. Quantization
For the above hyperchaotic system, it must be converted into binary stream. Here, the quantization function expression is set as
, and the definition is shown as follows:
Here
,
is the quantized binary stream. The conversion value falls within the corresponding interval of the quantization function and gets 0 or 1, respectively. As chaotic signals
have good random statistical properties, the quantized stream (
) should have excellent statistical properties of equilibrium 0-1 ratio in theory. The streams after quantization of the time series
,
,
, and
are shown in
Figure 11.
Figure 11 shows the time series of the hyperchaotic system are quantized into 0-1 sequences.
4.3. NIST Test
The NIST SP 800-22 [
50] random test package for stream cryptography (NIST random test) was provided by the National Institute of Standards and Technology. In order to verify the statistical performance of the quantized streams of the hyperchaotic system, NIST tests are carried out by using the test programs. The test package includes frequency test within a block, binary matrix rank test, non-overlapping template matching test, etc. These tests can be used to test binary sequences of an arbitrary length, generated by the pseudo-random number generator, which can be used to determine the non-randomness hidden in the stream. All of the test results are determined by
. If
, then the stream is not random. If
, then the stream is considered random. In order to make the system get better randomness, this paper carries out NIST tests to prove that the random streams generated by the system can be used in the encryption application.
Table 1 shows the test results. It can be seen that the quantized streams have good statistical characteristics and have passed the tests.
Table 1 shows that the sequences generated by the new hypersystem have passed all the tests in statistical NIST tests.
4.4. Permutation Entropy
The permutation entropy can be used to measure the complexity of time series. Permutation entropy is obtained by adding the permutation idea into the calculation of the complexity of sub-sequences. The algorithm is described as follows:
1. Define a time series , , …,, is the embedded dimension, is time delay.
2. Reconstruct the time series as , , …, .
3. Increase and rearrange . When , if the two values are equal, rearrange by subscript.
4. is redefined to . Therefore, there will be permutations.
5. Define the probability distribution of all symbols as , , …, , .
6. The permutation entropy of the time series can be calculated by the following formula:
, that is to say that when the probability of each symbol is equal, then the stream has the maximum permutation entropy. To facilitate data analysis,
will be normalized.
The results of the permutation entropy test are shown in
Table 2.
4.5. Approximate Entropy
Approximate entropy (ApEn) is used to measure the law of motion and unpredictability of a quantized time series, which is often used in nonlinear dynamics. It is characterized by the use of a non-negative number to represent the complexity of a time stream, which can reflect the possibility of new information in the stream. Therefore, the higher the approximate entropy is, the higher the complexity of the time series is. The algorithm description is shown as follows:
1. Define a time series , , …, .
2. is the length of the comparison vector.
3. is the measure of similarity.
4. Reconstruct the dimension vector , , …,, and .
5. When
, calculate the number of vectors satisfying the following conditions:
6. The function is defined as:
Here,. represents the element of the vector ; represents the distance between and , whose value is determined by the maximum difference value of the corresponding element; , and is allowed to exist in the case of equality.
From the above, the definition of Approximate Entropy (ApEn) can be obtained. In general, the value of the parameter
or
and
are determined by the actual application. Here
, and
represents the standard deviation of the original time series. Normally,
. The more complex the time series is, the greater the corresponding approximate entropy is. The ApEn here is shown in
Table 3, which means that the time series are of good unpredictability and can be used in nonlinear dynamics.