Color image encryption scheme using coupled hyper chaotic system with multiple impulse injections

doi:10.1016/j.ijleo.2016.10.036

Optik

Volume 129, January 2017, Pages 231-238

https://doi.org/10.1016/j.ijleo.2016.10.036 Get rights and content

Abstract

The paper proposes a color image encryption scheme based on coupled hyper chaotic Lorenz systems. The novelty is to randomly inject impulse signals into coupled Lorenz system during iterations to enhance the complexity of trajectory. Six state variables sequences are generated to encrypt the red, green and blue components by bitwise operations of exclusive or (XOR) and left or right cyclic shift. Six initial values and indeterminate multiple impulse signals can make the cryptosystem own larger variable key space to resist against exhaustive attack, even the attack from quantum computer. Simulation results show that the mean encryption speed is stable, which means the speed depends solely upon hardware equipment and algorithm. Statistical analysis demonstrates the high effectiveness of the proposed image encryption algorithm.

Introduction

In design of symmetric cryptographic algorithms, some important aspects, such as easy to implement, high security, resistant to crypt analysis, real-time encryption and decryption speed [1], must be considered.

Generally, hyper chaotic systems have more complex dynamical characteristics than their chaotic systems[2], and have been applied in designing image encryption algorithms. Niu et al. [3] presented an image encryption scheme on the basics of formal model of DNA computing-splicing system and hyper-chaotic system, while programming the method, the instinct properties of hyper-chaotic system and splicing model are utilized.Yuan et al. [4] designed an efficient image encryption scheme based on 2D hyper-chaotic system, the confusion and the diffusion procedures of the proposed scheme are interacted on each other, the cryptosystem with the interacted structure is steadier and harder to decipher. Tan et al. [5] designed a quantum color image encryption algorithm based on a hyper-chaotic system, the quantum Fourier transform is exploited to fulfill the encryption.

Short-period behavior of chaotic system after too many times iteration can lead to degeneration of dynamics [6], these problems can be solved by random noise impulse injection during the iteration process [7], [8], because chaotic system is extremely sensitive to the initial conditions, a tiny deviation in iterating can result in significantly different trajectory. The injection times can be determined by the size of plain-text, and the injection moments can be random numbers, and the value of state variable after injection should not exceed the original interval of the state variable.

Some image encryption schemes based on chaotic systems or hyper chaotic systems have designed the mechanism of keys changing and distribution [9]. According to Shannon’s theory, only those schemes based on one-time keys can be regarded as theoretically absolute security and can make differential attack ineffective. The best quantum attack against generic symmetric-key systems is an application of Grover's algorithm [10], which requires work proportional to the square root of the size of the key space. It is clear that symmetric-key systems offer the smallest key sizes for post-quantum cryptography [11]. One-time keys can be sampled from natural noise [12], such as the call of cicadas, the sounds of wind, thunder and rain.

The paper proposes a color image encryption scheme using coupled hyper chaotic Lorenz system, to enhance the complexity of trajectory, the impulse noise signals of true random numbers are sampled from noise and being injected into one of the state variables, the injection moment and times are random. Fast and bitwise operations of XOR and left or right cyclic shift are applied to encrypt three color components. Larger and variable key space can make the algorithm more secure. Security and statistical analysis demonstrate the effectiveness of the image encryption scheme.

Section snippets

The coupled hyper chaotic Lorenz system

The coupled hyper chaotic Lorenz system was obtained by coupling two identical Lorenz systems, which is proposed by Grassi et al., [13], and the corresponding differential equation can be described in Eq. (1). ${\begin{cases} {\dot{x}}_{1} = a (y_{1} - x_{1}) \\ {\dot{y}}_{1} = b x_{1} - y_{1} - x_{1} z_{1} + k_{1} (x_{2} - y_{2}) \\ {\dot{z}}_{1} = x_{1} y_{1} - c z_{1} \\ {\dot{x}}_{2} = a (y_{2} - x_{2}) \\ {\dot{y}}_{2} = b x_{2} - y_{2} - x_{2} z_{2} + k_{2} (x_{1} - y_{1}) \\ {\dot{z}}_{2} = x_{2} y_{2} - c z_{2} \end{cases} .$ Here $Y = {[x_{1}, y_{1}, z_{1}, x_{2}, y_{2}, z_{2}]}^{T}$ is the state variable vector, and $a$ , $b$ and $c$ are control parameters, $k_{1}, k_{2} > 0$ are coupling parameters. When $a = 10$ , $b = 28$ , $c = 8 / 3$ , and $k_{1} = k_{2} = 0.05$ , the system exhibits

Design of encryption algorithm

Input: Color plain image $P$ with the size of $W_{p} \times H_{p}$ ; randomly sampled noise signals $N S$ .

Output: The ciphered color image $C$ with the same size.

Step 1. Randomly select six signals of $x_{1} (0)$ , $x_{2} (0)$ , …, $x_{6} (0)$ from $N S$ to serve as six initial values of system (3), and two other values to serve as impulse signals of $Δ x_{1} (1)$ and $Δ x_{1} (2)$ .

Step 2. Iterating system (3) for 1200 times to remove the transient process, and then continue to iterate it for $N = W_{p} \times H_{p}$ times. When the iterating times $t_{1} = ⌊ N / 3 ⌋$ and $t_{2} = ⌊ 2 N / 3 ⌋$

Experimental results

The keys are noise signal intercepted in randomly sampled environmental noise through digital voice recorder, and the corresponding time domain and frequency domain characteristic charts of sampled noise signal are shown in Fig. 6.

The plain images have identical size of $512 \times 512$ , as shown in Fig. 7, and their corresponding ciphered images are shown in Fig. 8.

System evaluation

In this section, security and statistical analyses are applied to test the cryptographic algorithm, including the variable key space, key sensitivity, correlation coefficients of adjacent pixels, differential attack and information entropy evaluation.

Conclusion

The paper proposes a color image encryption scheme based on coupled hyper chaotic Lorenz system, and the novelty is multiple impulse injections of true random signals sampled from noise during the iteration process, to make the coupled hyper chaotic Lorenz system produce more complex trajectory. Time-saving operations of bitwise XOR and cyclic shift are applied to encrypt the red, green and blue components. The injection times can be designed to depend upon the image size. The simulation

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Nos: 61363082, 61662073) and Minority Nationality Technology Talent Cultivation Plan of Xinjiang (No.201123116).

References (19)

L. Zhou et al.
On the analysis of local bifurcation and topological horseshoe of a new 4D hyper-chaotic system
Chaos Solitons Fractals
(2016)
G. Grassi et al.
Multi-wing hyperchaotic attractors from coupled Lorenz systems
Chaos Solitons Fractals
(2009)
L.Y. Wang et al.
A novel hybrid color image encryption algorithm using two complex chaotic systems
Opt. Laser. Eng.
(2016)
X. Wu et al.
A new color image encryption scheme based on DNA sequences and multiple improved 1D chaotic maps
Appl. Soft Comput.
(2015)
J. Chen et al.
A fast chaos-based image encryption scheme with a dynamic state variables selection mechanism
Commun. Nonlinear Sci. Numer. Simulat.
(2015)
A. Pande et al.
A chaotic encryption scheme for real-time embedded systems: design and implementation
Telecommun. Syst.
(2013)
H. Niu et al.
Splicing model and hyper?chaotic system for image encryption
J. Electr. Eng.
(2016)
M. Yuan H et al.
A new image cryptosystem based on 2D hyper-chaotic system
Multimedia Tools Appl.
(2016)
C. Tan R et al.
Quantum color image encryption algorithm based on a hyper-Chaotic system and quantum fourier transform
Int. J. Theor. Phys.
(2016)

There are more references available in the full text version of this article.

Cited by (58)

Construction of algebraic complex 9-bit lookup tables using non-chain-ring and its applications in data security
2024, Integration
A substitution box (S-box) is the only nonlinear component of a symmetric key encryption cipher that directly affects an encryption scheme’s performance and security level. So it is appealing to produce S-boxes with good performance and efficiency. Customarily, in algebra, Galois fields are used to generate such structures. It requires the storage memory of size ${8 \times 2}^{8}$ bits. Similarly, in $G F (2^{9})$ this figure reaches to ${9 \times 2}^{9}$ bits. In this study, a novel construction of lookup tables over the algebraic structure of non-chain ring $R = \frac{Z_{8} [u]}{< u^{3} >} = Z_{8} + u Z_{8} + u^{2} Z_{8}, (u^{3} = 0)$ of cardinality 512 is presented that holds ${9 \times 2}^{8}$ computer memory calls. Moreover, the use of non-chain ring operations in the building of lookup tables results in a more complex S-box. Afterward, we use this nonlinear layer to produce confusion in a novel image-enciphering scheme. A nonlinear encryption scheme such as a non-chain-ring S-box construction is susceptible to adversaries because of its complex connections between key ciphertexts and cryptographic primitives. With non-chain ring encryption, you can benefit from enhanced security, complexity of algorithm, resistance to modern attacks, as well as improved computing performance. Texture, algebraic, and statistical analyses were performed on the suggested S-box encryption technique. A comparison of the encryption data using the proposed S-box with the literature available S-boxes demonstrates that the suggested S-box is superior and can be employed in various ciphers. The results of the statistical, differential, and security analyses also reveal that our approach is more effective for image encryption.
IES: Hyper-chaotic plain image encryption scheme using improved shuffled confusion-diffusion
2022, Ain Shams Engineering Journal
Achieving a secured encrypted image is challenging in sensitive data communications. An accurate and secure encryption scheme for colour images requires the generation of intricate secret keys. To attain this goal, we proposed an improved shuffled confusion-diffusion based colour Image Encryption Scheme (IES) using hyper-chaotic plain images. First, five different sequences of random numbers were generated. Two of these sequences were used to mix image pixels and bits, while the remaining three were used to XOR the values of image pixels. The performance of the developed IES was evaluated in terms of various measures such as key space size, correlation coefficient, entropy, mean squared deviation (MSD), Peak Signal-to-Noise Ratio (PSNR) and differential analysis. The values of the correlation coefficient (0.000732), entropy (7.9997), PSNR (7.61), and MSE (11,258) were determined to be better compared to with the art of technologies that exist. The proposed IES outperformed the other methods in terms of computational efficiency and other performance metrics when compared. The proposed may be beneficial for encrypting big data sets on parallel machines.
A chaos based image encryption algorithm using Rubik's cube and prime factorization process (CIERPF)
2022, Journal of King Saud University - Computer and Information Sciences
Citation Excerpt :
2D-Logistic-Adjusted sine map is implemented for confusion and diffusion of image pixels in Hua and Zhou (2016), and also the random values are padded with plain image to increase the security. For color images, a combined hyper chaotic system with various impulse-based image encryption is suggested in Kadir et al. (2017). A novel dynamic index related substitution with block image permutation is implemented for encryption of images in Xu et al. (2017).
A novel Rubik’s cube based pixel level scrambling and simple XOR based diffusion is proposed in this paper to safely transmit multimedia information (images) through an untrusted channel, such as adaptive image content (i.e., plain image related) based initial random value generation is introduced to achieve high plain image sensitivity in order to overcome plain image related attacks. From this random value, the initial vectors of Henon map is obtained and this is iterated to obtain the key sequences to be applied over the Rubik’s cube row and column confusion processes. Also, the same random seed is involved in the key generation process based on prime factorization to be applied in diffusion. For each and every encryption of different plain image, the random list is changed dynamically and it is shown that the CIERPF method is secure against differential attacks. The CIERPF method is checked against various simulations to illustrate the security level of the technique. From the simulations, it is shown that the proposed methodology has good key space, high key sensitivity, and uniform distribution of cipher image pixels. Also differential cryptanalysis is performed for the proposed cryptosystem to prove its effectiveness towards differential attacks.
An image encryption algorithm based on dynamic row scrambling and Zigzag transformation
2021, Chaos, Solitons and Fractals
This paper introduces two new one-dimensional chaotic systems. They can be considered as an improvement of Logistic map, Sine map and Tent map. One is the chaotic system composed of Sine and Tent maps (STCS), the other is Logistic-Logistic chaotic system (L-LCS). Based on the two, an image encryption algorithm using dynamic row scrambling and Zigzag transformation is proposed. Firstly, the image is traversed in different directions of odd and even rows, and the results are scrambled with the chaotic sequence generated by STCS. Secondly, based on standard Zigzag scrambling idea, a special traversal method is used, that is, for the different permutation of the lower triangle and the upper triangle of the matrix (each triangular matrix does not contain the values on the main and sub diagonals) in order and direction. Then, combined with the chaotic sequence generated by L-LCS, two different diffusion formulas are used to diffuse the segmented image. Finally, this encryption scheme can also be extended to color images. Experimental simulation and performance analysis show that the encryption algorithm has better security.
A novel hybrid permutation substitution base colored image encryption scheme for multimedia data
2021, Journal of Information Security and Applications
A novel hybrid permutation substitution-based color image encryption scheme is presented. The permutation is desirable property to have which make any security system diffusing. In present article, permutation is applied through p-box. The substitution process is also vital because it create confusion. In presented work substitution is applied through S-p-box. 3D mixed chaotic map is utilized to make our algorithm more diffusing and confusing. The permutation box, permutation substitution box and initial conditions of 3D mixed chaotic map are also utilized as a key. The experimental outcomes of proposed technique have been critically analyzed and a comprehensive comparison with existing schemes is presented.
Simple colour image cryptosystem with very high level of security
2020, Chaos, Solitons and Fractals
To obtain a more secure colour image cryptosystem without complex construction, this paper presents a general simple colour image encryption model with a very high level of security, and that is based on two nearby orbits of chaotic systems. First, the initial value of a one-dimensional (1D) chaotic map is obtained using plaintext. Then, we obtain two nearby orbits of 1D chaotic maps to generate three new chaotic signals. Next, the generated systems independently encrypt the red, green and blue components (RGB) of the colour image. Finally, the three encrypted images are combined to obtain the final encrypted image. Simulation results show that our method is simple, effective and passed all NIST tests and part of TestU01 test. Since the proposed method is related to plaintext, it has much higher security level compared with the most recently reported chaos-based image algorithms. More importantly, the 1D logistic cryptosystem based on our method has a large key space with higher security.

View all citing articles on Scopus

View full text

Original research articleColor image encryption scheme using coupled hyper chaotic system with multiple impulse injections

Abstract

Introduction

Section snippets

The coupled hyper chaotic Lorenz system

Design of encryption algorithm

Experimental results

System evaluation

Conclusion

Acknowledgments

Chaos Solitons Fractals

Chaos Solitons Fractals

Opt. Laser. Eng.

Appl. Soft Comput.

Commun. Nonlinear Sci. Numer. Simulat.

A chaotic encryption scheme for real-time embedded systems: design and implementation

Telecommun. Syst.

Splicing model and hyper?chaotic system for image encryption

J. Electr. Eng.

A new image cryptosystem based on 2D hyper-chaotic system

Multimedia Tools Appl.

Quantum color image encryption algorithm based on a hyper-Chaotic system and quantum fourier transform

Int. J. Theor. Phys.

Original research article
Color image encryption scheme using coupled hyper chaotic system with multiple impulse injections