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.
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 [
Extensive researches have shown that the use of chaos for multimedia communication is a potential application [
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 [
Chaotic systems without equilibrium attract have been attracting a lot of interest [
We can solve the following three equations to find the system’s equilibrium:
From (
Based on conditions (
It is simple to see that system (
Projections of attractors without equilibrium in (a)
Lyapunov exponents of system without equilibrium (
Electronic implementation of a chaotic model is useful for confirming the model’s feasibility as well as realizing applications [
The circuit schematic of the scaled system.
As can be seen in Figure
The experimental circuit of the scaled chaotic system.
All phase portraits of the scaled chaotic system on the oscilloscope: (a)
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.
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,
NIST-800-22 test results of the new chaotic system based RNG.
Statistical tests |
|
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 |
0,746783846542712 | Successful |
Random-Excursions Variant Test |
0,810174242469133 | Successful |
Serial Test-1 | 0,366073450623053 | Successful |
Serial Test-2 | 0,333453364381209 | Successful |
Linear-Complexity Test | 0,992656091838689 | Successful |
The 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, obtained
In this application, image encryption is performed using RNG obtained in Section
Security analysis result and comparisons (256 × 256 “pepper” image).
Entropy | Correlation | NPCR | UACI | |
---|---|---|---|---|
This work | 7.9972 | 0.0042 | 99.9802 | 30.0634 |
Ref. [ |
7.9560 | 0.5210 | 99.6289 | 31.8345 |
Ref. [ |
7.9972 | 0.0520 | 99.6109 | 12.7548 |
Ref. [ |
7.9820 | 0.0052 | 99.5773 | 34.1402 |
Ref. [ |
7.9958 | 0.0068 | 99.6170 | 25.125 |
Ref. [ |
7.9991 | 0.0082 | 99.028 | 33.10 |
Ref. [ |
7.998 | 0.0071 | 99.50 | 33.39 |
Encryption time and comparisons (
Encryption time (s) | |
---|---|
This work | 0.4865 |
Ref. [ |
1.6734 |
Ref. [ |
3.704 |
Ref. [ |
0.712 |
Ref. [ |
5.6544 |
Ref. [ |
0.5630 |
Original, encrypted, and decrypted images.
Correlation distributions of original and encrypted images.
Histograms of original and encrypted images.
In this section, the
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 | 1 |
Original and embedded sound.
Correlation distributions of original and embedded sounds.
Histograms of original and embedded sounds.
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.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors acknowledge Professor GuanRong Chen, Department of Electronic Engineering, City University of Hong Kong, for suggesting many helpful references. The author Xiong Wang was supported by the National Natural Science Foundation of China (no. 61601306) and Shenzhen Overseas High Level Talent Peacock Project Fund (no. 20150215145C).