# Multimedia Security Application of a Ten-Term Chaotic System without Equilibrium

Authors: Xiong Wang; Akif Akgul; Sezgin Kacar; Viet-Thanh Pham
Journal: Complexity
 (2017)

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## Abstract

A system without equilibrium has been proposed in this work. Although there is an absence of equilibrium points, the system displays chaos, which has been confirmed by phase portraits and Lyapunov exponents. The system is realized on an electronic card, which exhibits chaotic signals. Furthermore, chaotic property of the system is applied in multimedia security such as image encryption and sound steganography.


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## Body

## 1. Introduction

Recently, there is an increased interest in multimedia communication, which is vital in various areas ranging from entertainment industries, economics, and medical applications to militaries [1, 2]. Several attempts have been made to provide confidentiality, identity, and integrity. A considerable amount of studies about multimedia security has been introduced, for example, data encryption, steganographic messages, watermarking, or multimedia authentication [1–6].Extensive researches have shown that the use of chaos for multimedia communication is a potential application [7–11]. Because of the similar characteristics between chaotic systems and cryptographic primitives, chaotic cryptography is a major area of interest [12]. Liu et al. presented the audio encryption scheme with confusion and diffusion [13]. They based on a multiscroll chaotic system and one-time keys [13]. By using complex hyperchaotic systems, color image encryption was proposed [14]. Chaos maps with Markov properties were applied to construct a new encryption algorithm [15]. In addition, Ghebleh and Kanso introduced a robust chaos scheme to image steganography [16]. In order to generate message authentication codes, suitability of chaos functions was studied in [17].In this work, we study a ten-term chaotic system without equilibrium and its multimedia security application. The chaotic attractors in this system are “hidden attractors” because the basin of attraction for a hidden attractor is not connected with any unstable fixed point [18–21]. It is noted that the concept of hidden attractors has been suggested in connection with the occurrence of unpredictable attractors in multistable systems [21]. Researchers have shown that multistability is connected with the occurrence of unpredictable attractors [21–30]. Recently, hidden attractor has been investigated in numerous systems such as Chua system [19], drilling system [31], Lorenz-like system [32], Goodwin oscillator [33], electromechanical systems [34], two-dimensional maps [35], phase-locked loop circuits [36], and Rabinovich–Fabrikant system [37].


## 2. A System without Equilibrium and Its Implementation

### 2.1. A System without Equilibrium

Chaotic systems without equilibrium attract have been attracting a lot of interest [38–42]. When mentioning a chaotic system without equilibrium, it means that there is no real equilibrium in such a system. In this work, we study a general three-dimensional form described by(1)x˙=a1y,y˙=a2x+a3y+a4z+a5xz,z˙=a6x+a7z+a8xy+a9xz+a10,in which ai(i=1,…,10) are parameters and ai≠0.We can solve the following three equations to find the system’s equilibrium:(2)a1y=0,a2x+a3y+a4z+a5xz=0,a6x+a7z+a8xy+a9xz+a10=0.From (2), we have(3)Ax2+Bx+C=0,in which(4)A=a2a5a9-a52a6,B=a2a5a7-a4a5a6-a52a10,C=-a4a5a10,for(5)a5a7-a4a9≠0.As a result, there is no equilibrium in general form (1) for(6)Δ=B2-4AC<0.Based on conditions (5) and (6) and ai≠0, we have applied a search procedure [40] to find the set of parameters for which general form (1) does not have equilibrium. A set of parameters has been found as follows:(7)a1=a,a2=a7=a8=-1,a3=b,a4=c,a5=a6=a10=1,a9=d,where a,b,c,d>0. General form (1) is rewritten by(8)x˙=ay,y˙=-x+by+cz+xz,z˙=x-z-xy+dxz+1.It is simple to see that system (8) has no equilibrium for a=2, b=0.1, c=2.5, and d=0.3. However, it displays chaos as shown in Figure 1. In this case, calculated Lyapunov exponents of the system are illustrated in Figure 2. Lyapunov exponents are L1=0.2563, L2=0, and L3=-0.5762 for t=10,000. Positive Lyapunov exponent L1=0.2563 verifies the chaotic behavior of system (8). Attractors of system (8) are hidden attractors according the introduced definition in [21]. It is noted that the hidden attractors are observed in systems without equilibria, with no unstable fixed points or with one stable fixed point [21].Figure 1
Projections of attractors without equilibrium in (a)x-y plane, (b) x-z plane, and (c) y-z plane for a=2, b=0.1, c=2.5, and d=0.3 and initial conditions (x(0),y(0),z(0))=(1,1,1). The attractor of system (8) is “hidden” according to the definition introduced in [21].
(a)
(b)
(c)Figure 2
Lyapunov exponents of system without equilibrium (8) for a=2, b=0.1, c=2.5, and d=0.3 and initial conditions (x(0),y(0),z(0))=(1,1,1): L1 (red), L2 (green), and L3 (blue). Positive value of Lyapunov exponent L1 indicates chaos of system (8).


### 2.2. Electronic Circuit Implementation of the Chaotic System without Equilibrium

Electronic implementation of a chaotic model is useful for confirming the model’s feasibility as well as realizing applications [43–45]. The main aim of this section is to design a circuit to realize the system without equilibrium by using electronic elements [46–51]. The chaotic system must be rescaled for electronic circuit implementation. The amplitude values of system (x, y, and z) are in the interval of (-80,60). They are higher than the interval of (-15,15) which are electronic materials limitations. For scale process, let X/2=x, Y/6=y, and Z=z/4 and then adjust the original state variables x, y, z instead of the variables X, Y, Z. As a result, the scaled system finally becomes of the following form:(9)X˙=3aY,Y˙=-13X+bY+23cZ+43XZ,Z˙=12X-Z-3XY+2dXZ+14.An electronic circuit is designed for the scaled system as shown in Figure 3.Figure 3
The circuit schematic of the scaled system.As can be seen in Figure3, we selected C1 = C2 = C3 = 1 nF, R1 = 66 kΩ, R2 = R3 = R8 = R9 = R15 = R16 = 100 kΩ, R4 = 1200 kΩ, R5 = 4000 kΩ, R6 = 240 kΩ, R7 = 30 kΩ, R10 = 800 kΩ, R11 = 400 kΩ, R12 = 24000 kΩ, R13 = 67 kΩ, and R14 = 14 kΩ. The scaled chaotic system was done on the electronic card (see Figure 4). All phase portraits on the oscilloscope are shown in Figure 5, which illustrate the feasibility of the system.Figure 4
The experimental circuit of the scaled chaotic system.
(a)
(b)Figure 5
All phase portraits of the scaled chaotic system on the oscilloscope: (a)X-Y plane, (b) X-Z plane, and (c) Y-Z plane.
(a)
(b)
(c)


## 2.1. A System without Equilibrium

Chaotic systems without equilibrium attract have been attracting a lot of interest [38–42]. When mentioning a chaotic system without equilibrium, it means that there is no real equilibrium in such a system. In this work, we study a general three-dimensional form described by(1)x˙=a1y,y˙=a2x+a3y+a4z+a5xz,z˙=a6x+a7z+a8xy+a9xz+a10,in which ai(i=1,…,10) are parameters and ai≠0.We can solve the following three equations to find the system’s equilibrium:(2)a1y=0,a2x+a3y+a4z+a5xz=0,a6x+a7z+a8xy+a9xz+a10=0.From (2), we have(3)Ax2+Bx+C=0,in which(4)A=a2a5a9-a52a6,B=a2a5a7-a4a5a6-a52a10,C=-a4a5a10,for(5)a5a7-a4a9≠0.As a result, there is no equilibrium in general form (1) for(6)Δ=B2-4AC<0.Based on conditions (5) and (6) and ai≠0, we have applied a search procedure [40] to find the set of parameters for which general form (1) does not have equilibrium. A set of parameters has been found as follows:(7)a1=a,a2=a7=a8=-1,a3=b,a4=c,a5=a6=a10=1,a9=d,where a,b,c,d>0. General form (1) is rewritten by(8)x˙=ay,y˙=-x+by+cz+xz,z˙=x-z-xy+dxz+1.It is simple to see that system (8) has no equilibrium for a=2, b=0.1, c=2.5, and d=0.3. However, it displays chaos as shown in Figure 1. In this case, calculated Lyapunov exponents of the system are illustrated in Figure 2. Lyapunov exponents are L1=0.2563, L2=0, and L3=-0.5762 for t=10,000. Positive Lyapunov exponent L1=0.2563 verifies the chaotic behavior of system (8). Attractors of system (8) are hidden attractors according the introduced definition in [21]. It is noted that the hidden attractors are observed in systems without equilibria, with no unstable fixed points or with one stable fixed point [21].Figure 1
Projections of attractors without equilibrium in (a)x-y plane, (b) x-z plane, and (c) y-z plane for a=2, b=0.1, c=2.5, and d=0.3 and initial conditions (x(0),y(0),z(0))=(1,1,1). The attractor of system (8) is “hidden” according to the definition introduced in [21].
(a)
(b)
(c)Figure 2
Lyapunov exponents of system without equilibrium (8) for a=2, b=0.1, c=2.5, and d=0.3 and initial conditions (x(0),y(0),z(0))=(1,1,1): L1 (red), L2 (green), and L3 (blue). Positive value of Lyapunov exponent L1 indicates chaos of system (8).


## 2.2. Electronic Circuit Implementation of the Chaotic System without Equilibrium

Electronic implementation of a chaotic model is useful for confirming the model’s feasibility as well as realizing applications [43–45]. The main aim of this section is to design a circuit to realize the system without equilibrium by using electronic elements [46–51]. The chaotic system must be rescaled for electronic circuit implementation. The amplitude values of system (x, y, and z) are in the interval of (-80,60). They are higher than the interval of (-15,15) which are electronic materials limitations. For scale process, let X/2=x, Y/6=y, and Z=z/4 and then adjust the original state variables x, y, z instead of the variables X, Y, Z. As a result, the scaled system finally becomes of the following form:(9)X˙=3aY,Y˙=-13X+bY+23cZ+43XZ,Z˙=12X-Z-3XY+2dXZ+14.An electronic circuit is designed for the scaled system as shown in Figure 3.Figure 3
The circuit schematic of the scaled system.As can be seen in Figure3, we selected C1 = C2 = C3 = 1 nF, R1 = 66 kΩ, R2 = R3 = R8 = R9 = R15 = R16 = 100 kΩ, R4 = 1200 kΩ, R5 = 4000 kΩ, R6 = 240 kΩ, R7 = 30 kΩ, R10 = 800 kΩ, R11 = 400 kΩ, R12 = 24000 kΩ, R13 = 67 kΩ, and R14 = 14 kΩ. The scaled chaotic system was done on the electronic card (see Figure 4). All phase portraits on the oscilloscope are shown in Figure 5, which illustrate the feasibility of the system.Figure 4
The experimental circuit of the scaled chaotic system.
(a)
(b)Figure 5
All phase portraits of the scaled chaotic system on the oscilloscope: (a)X-Y plane, (b) X-Z plane, and (c) Y-Z plane.
(a)
(b)
(c)


## 3. Image Encryption and Sound Steganography Applications of the System without Equilibrium

After considering the circuit implementation of the system, in this section, image encryption and hiding of encrypted image data in a sound file have been implemented to show that the no-equilibrium chaotic system can be used in multimedia security applications. In order to realize these applications, firstly random number generator design has been done.


### 3.1. Random Number Generator (RNG) Design

One of the most basic structures used in chaos-based encryption and stenography applications is RNG. In this study, before the security applications, a RNG design has been implemented for use in these applications. In RNG design,mod(256) operation has been performed by taking 13, 14, and 15 digits after the point of state variables obtained by using Runge–Kutta 4 (RK4) algorithm. In this case, 8 bits of each state variable are obtained in each iteration of the RK4 algorithm. Finally, each 8-bit sequence obtained from 3 state variables (x, y, z) has been XORed at each iteration to obtain a random bit sequence of RNG. In Table 1, the statistical NIST-800-22 test results of the random number sequence of 1,000,000 bits obtained from the designed RNG are shown.Table 1
NIST-800-22 test results of the new chaotic system based RNG.
Statistical tests
P value
Result
Frequency (Monobit) Test
0,4556674150378
Successful
Block-Frequency Test
0,312738896039824
Successful
Cumulative-Sums Test
0,5441395972238
Successful
Runs Test
0,117478755093071
Successful
Longest-Run Test
0,88635602631487
Successful
Binary Matrix Rank Test
0,594206193094231
Successful
Discrete Fourier Transform Test
0,783086624373691
Successful
Nonoverlapping Templates Test
0,0586441453317821
Successful
Overlapping Templates Test
0,868314176679646
Successful
Maurer’s Universal Statistical Test
0,349319372138117
Successful
Approximate Entropy Test
0,0419351614775444
Successful
Random-Excursions Test(x=-4)
0,746783846542712
Successful
Random-Excursions Variant Test(x=-9)
0,810174242469133
Successful
Serial Test-1
0,366073450623053
Successful
Serial Test-2
0,333453364381209
Successful
Linear-Complexity Test
0,992656091838689
SuccessfulThe statistical NIST-800-22 test is known as the internationally accepted best random test. The NIST test is a comprehensive test consisting of 15 different tests. In order to be able to speak of a complete randomness, obtainedP values from the all of 15 tests must be provided with the condition of 1 > P value > 0.01. When the values in Table 1 are examined, it is seen that the RNG based on the new system is successful in all tests.


### 3.2. Image Encryption Application

In this application, image encryption is performed using RNG obtained in Section3.1. In the application 128×128 pixel size image with gray scale (“pepper”) is used. All pixels of the original image have converted to binary and all binary bits have been subjected to XOR processing with a random bit sequence obtained from the RNG to perform encryption. The decryption process is also performed by applying the reverse of the encryption process. Figure 6 shows the original, encrypted, and decrypted images. As seen in the figure, the encrypted image is not understood. So, the encryption is visually very successful. Correlation, histogram, entropy, and differential attack analyses were also performed to analyze the performance of the encryption. Figures 7 and 8 show correlation distributions and histogram graphs. In these graphs, it is seen that encryption process provides a very homogeneous distribution. The homogeneity of the encrypted image distributions shows the encryption is good. In addition, we have implemented some existing image encryption methods to compare with our encryption method (see Tables 2 and 3). The entropy, correlation, and differential attack (NPCR and UACI) results and encryption time show the good performance of the proposed image encryption method.Table 2
Security analysis result and comparisons (256 × 256 “pepper” image).
Entropy
Correlation
NPCR
UACI
This work
7.9972
0.0042
99.9802
30.0634
Ref. [52]
7.9560
0.5210
99.6289
31.8345
Ref. [53]
7.9972
0.0520
99.6109
12.7548
Ref. [54]
7.9820
0.0052
99.5773
34.1402
Ref. [55]
7.9958
0.0068
99.6170
25.125
Ref. [56]
7.9991
0.0082
99.028
33.10
Ref. [57]
7.998
0.0071
99.50
33.39Table 3
Encryption time and comparisons (256×256 “pepper” image).
Encryption time (s)
This work
0.4865
Ref. [52]
1.6734
Ref. [58]
3.704
Ref. [59]
0.712
Ref. [60]
5.6544
Ref. [61]
0.5630Figure 6
Original, encrypted, and decrypted images.
(a)
(b)
(c)Figure 7
Correlation distributions of original and encrypted images.
(a)
(b)Figure 8
Histograms of original and encrypted images.
(a)
(b)


### 3.3. Sound Steganography Application

In this section, the128×128 encrypted picture obtained in Section 3.2 has been hidden in the original sound as in Figure 9. In this hiding process, firstly, the pixel values of the encrypted picture are converted into binary numbers. Secondly, the float values obtained from the sound data are converted to 32-bit binary numbers. Finally, all the bits belonging to the encrypted image are hidden by placing them in the LSB bits of the 32-bit binary numbers of the selected sound data using the RNG obtained in Section 3.1. As a result, the sound data in which the encrypted image is hidden is shown in Figure 9. There is no visual difference between the original and embedded sound data seen in Figure 9. This shows that the steganography application has a good performance. Since it is not enough that there is no difference visually between the original and embedded sounds, analyses of correlation, histogram, entropy, mean square error (MSE), maximum absolute error (MAXERR), and the energy ratio (L2RAT) were also performed. As seen in Figures 10 and 11, correlation distributions and histograms of the original and embedded sounds are the same. In addition, the results of the analyses in Table 4 indicate that there is no difference between the original and embedded sounds. This means that the sound steganography application based on the chaotic system without equilibrium has a good performance.Table 4
Security analysis of the steganography process.
Analysis
Original sound
Embedded sound
Correlation
0.9994
0.9994
Entropy
13.4926
13.4926
MSE
0
MAXERR
0
L2RAT
1Figure 9
Original and embedded sound.
(a)
(b)Figure 10
Correlation distributions of original and embedded sounds.
(a)
(b)Figure 11
Histograms of original and embedded sounds.
(a)
(b)


## 3.1. Random Number Generator (RNG) Design

One of the most basic structures used in chaos-based encryption and stenography applications is RNG. In this study, before the security applications, a RNG design has been implemented for use in these applications. In RNG design,mod(256) operation has been performed by taking 13, 14, and 15 digits after the point of state variables obtained by using Runge–Kutta 4 (RK4) algorithm. In this case, 8 bits of each state variable are obtained in each iteration of the RK4 algorithm. Finally, each 8-bit sequence obtained from 3 state variables (x, y, z) has been XORed at each iteration to obtain a random bit sequence of RNG. In Table 1, the statistical NIST-800-22 test results of the random number sequence of 1,000,000 bits obtained from the designed RNG are shown.Table 1
NIST-800-22 test results of the new chaotic system based RNG.
Statistical tests
P value
Result
Frequency (Monobit) Test
0,4556674150378
Successful
Block-Frequency Test
0,312738896039824
Successful
Cumulative-Sums Test
0,5441395972238
Successful
Runs Test
0,117478755093071
Successful
Longest-Run Test
0,88635602631487
Successful
Binary Matrix Rank Test
0,594206193094231
Successful
Discrete Fourier Transform Test
0,783086624373691
Successful
Nonoverlapping Templates Test
0,0586441453317821
Successful
Overlapping Templates Test
0,868314176679646
Successful
Maurer’s Universal Statistical Test
0,349319372138117
Successful
Approximate Entropy Test
0,0419351614775444
Successful
Random-Excursions Test(x=-4)
0,746783846542712
Successful
Random-Excursions Variant Test(x=-9)
0,810174242469133
Successful
Serial Test-1
0,366073450623053
Successful
Serial Test-2
0,333453364381209
Successful
Linear-Complexity Test
0,992656091838689
SuccessfulThe statistical NIST-800-22 test is known as the internationally accepted best random test. The NIST test is a comprehensive test consisting of 15 different tests. In order to be able to speak of a complete randomness, obtainedP values from the all of 15 tests must be provided with the condition of 1 > P value > 0.01. When the values in Table 1 are examined, it is seen that the RNG based on the new system is successful in all tests.


## 3.2. Image Encryption Application

In this application, image encryption is performed using RNG obtained in Section3.1. In the application 128×128 pixel size image with gray scale (“pepper”) is used. All pixels of the original image have converted to binary and all binary bits have been subjected to XOR processing with a random bit sequence obtained from the RNG to perform encryption. The decryption process is also performed by applying the reverse of the encryption process. Figure 6 shows the original, encrypted, and decrypted images. As seen in the figure, the encrypted image is not understood. So, the encryption is visually very successful. Correlation, histogram, entropy, and differential attack analyses were also performed to analyze the performance of the encryption. Figures 7 and 8 show correlation distributions and histogram graphs. In these graphs, it is seen that encryption process provides a very homogeneous distribution. The homogeneity of the encrypted image distributions shows the encryption is good. In addition, we have implemented some existing image encryption methods to compare with our encryption method (see Tables 2 and 3). The entropy, correlation, and differential attack (NPCR and UACI) results and encryption time show the good performance of the proposed image encryption method.Table 2
Security analysis result and comparisons (256 × 256 “pepper” image).
Entropy
Correlation
NPCR
UACI
This work
7.9972
0.0042
99.9802
30.0634
Ref. [52]
7.9560
0.5210
99.6289
31.8345
Ref. [53]
7.9972
0.0520
99.6109
12.7548
Ref. [54]
7.9820
0.0052
99.5773
34.1402
Ref. [55]
7.9958
0.0068
99.6170
25.125
Ref. [56]
7.9991
0.0082
99.028
33.10
Ref. [57]
7.998
0.0071
99.50
33.39Table 3
Encryption time and comparisons (256×256 “pepper” image).
Encryption time (s)
This work
0.4865
Ref. [52]
1.6734
Ref. [58]
3.704
Ref. [59]
0.712
Ref. [60]
5.6544
Ref. [61]
0.5630Figure 6
Original, encrypted, and decrypted images.
(a)
(b)
(c)Figure 7
Correlation distributions of original and encrypted images.
(a)
(b)Figure 8
Histograms of original and encrypted images.
(a)
(b)


## 3.3. Sound Steganography Application

In this section, the128×128 encrypted picture obtained in Section 3.2 has been hidden in the original sound as in Figure 9. In this hiding process, firstly, the pixel values of the encrypted picture are converted into binary numbers. Secondly, the float values obtained from the sound data are converted to 32-bit binary numbers. Finally, all the bits belonging to the encrypted image are hidden by placing them in the LSB bits of the 32-bit binary numbers of the selected sound data using the RNG obtained in Section 3.1. As a result, the sound data in which the encrypted image is hidden is shown in Figure 9. There is no visual difference between the original and embedded sound data seen in Figure 9. This shows that the steganography application has a good performance. Since it is not enough that there is no difference visually between the original and embedded sounds, analyses of correlation, histogram, entropy, mean square error (MSE), maximum absolute error (MAXERR), and the energy ratio (L2RAT) were also performed. As seen in Figures 10 and 11, correlation distributions and histograms of the original and embedded sounds are the same. In addition, the results of the analyses in Table 4 indicate that there is no difference between the original and embedded sounds. This means that the sound steganography application based on the chaotic system without equilibrium has a good performance.Table 4
Security analysis of the steganography process.
Analysis
Original sound
Embedded sound
Correlation
0.9994
0.9994
Entropy
13.4926
13.4926
MSE
0
MAXERR
0
L2RAT
1Figure 9
Original and embedded sound.
(a)
(b)Figure 10
Correlation distributions of original and embedded sounds.
(a)
(b)Figure 11
Histograms of original and embedded sounds.
(a)
(b)


## 4. Conclusions

This paper introduces a 3D system without equilibrium, which exhibits chaotic behavior. The system includes ten terms and has been implemented easily by an electronic circuit. The findings of this work add to a growing list of systems with hidden attractors. This work enhances our understanding of practical applications using systems with hidden attractors. We have used the system without equilibrium for image encryption and sound steganography. According to our knowledge, this is the first time that the 128 kbit data can be encrypted and hidden in sound files. Therefore the findings of this work have important implications for future practice. Other chaotic systems without equilibrium will be discovered in our future researches.



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