Three Dimensional Chaotic System for Color Image Scrambling Algorithm

Three Dimensional Chaotic System for Color
Image Scrambling Algorithm
S. N. Lagmiri1
, N. Elalami2
, J. Elalami3
1
Information and Production System, Mohammadia School Engineering, Mohamed V University in
Rabat, Morocco
2
LAII, Mohammadia School Engineering, Mohamed V University in Rabat, Morocco
3
LASTIMI, Higher School of Technology of Sale, Mohamed V University in Rabat, Morocco
Abstract ̶ With the development of information security, the traditional image encryption methods have become
outdated. Because of amply using images in the transmission process, it is important to protect the confidential image
data from unauthorized access. This paper presents a new chaos based image encryption algorithm, which can improve
the security during transmission more effectively utilizes the chaotic systems properties, such as pseudo-random
appearance and sensitivity to initial conditions. Based on chaotic theory and decomposition and recombination of pixel
values, this new image scrambling algorithm is able to change the position of pixel, simultaneously scrambling both
position and pixel values. Experimental results show that the new algorithm improves the image security effectively to
avoid unscramble, and it also can restore the image as same as the original one, which reaches to the purposes of image
safe and reliable transmission.
Keywords: Color image, chaotic system, decomposition, image scrambling, recombination
I. INTRODUCTION
Recently, security of multimedia data is receiving more and more attention due to the transmission over
various communication networks. In order to protect personal information, many image encryption algorithms
are designed and proposed such as two-dimensional cellular automata based method [2], Henon chaotic map
[10, 13], Chen's hyperchaotic system [12], Arnold transform [3, 4]. Chaotic functions are blessed with properties
like sensitivity to the initial conditions, and ergodicity which make them very desirable for encryption [1].
Image scrambling is one of the methods for securing the image by scrambling it into a disordered one beyond
recognition, making it hard for those who get the image in unauthorized manner to extract information of the
original image from the scrambled images. Further, image scrambling technology depends on data hiding
technology which provides non-password security algorithm for information hiding. Now, the mainly used three
kind of image scrambling types are scrambling in the space domain, scrambling in the frequency domain, and
scrambling in the color or grey domain. In a great quantity of all kind of image scrambling algorithms, the
image scrambling algorithms based on chaos have attracted more and more attention since they can provide a
high level of security [5, 6, 7, 8].
International Journal of Computer Science and Information Security (IJCSIS),
Vol. 16, No. 1, January 2018
8 https://sites.google.com/site/ijcsis/
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This paper focuses on a new image scrambling algorithm which introduces a new chaotic system. Image
scrambling using chaotic properties is an application for providing security to the images from getting into the
hands of unauthorized user. The proposed image scrambling scheme generates the permuting address codes by
sorting the chaotic sequence directly. This paper analyzed that the scrambling performance of the new algorithm
is statistic. The conclusion of this paper indicates that the new algorithm can provide a high level security. The
paper results in good performance of the proposed algorithm that can also be applied in the real-time
applications and digital communications as it is a straightforward mechanism and easy to implement.
The rest of the paper is organized as follows: proposed chaotic system in section 2, image scrambling
algorithm based on chaos theory in section 3, experimental details and results are analyzed in section 4. The
paper is observed by a conclusion in section 5.
II. PROPOSED CHAOTIC SYSTEM
In this section, we describe the new chaotic system used in this work.
2.1. New Chaotic System
Recently, Chen and Lee [9] introduced a new chaotic system, which is described by the following nonlinear
differential equation:
{
̇ = −
̇ = − +
̇ = − +
(1)
� =
Where:
- , and are the state variables and , and are positive constants.
- = .
- � is the system measured output.
When = . , = = , the system (1) is chaotic.
2.2. Lyapunov exponent
By linearizing the Jacobian matrix � round the equilibrium point � and solving the following equation:
|λ� − �| = (2)
Therefore, the new chaotic system (1) has three eigenvalues shown in figure 1.
� = . � = − .99 � = − .999
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Figure 1: Lyapunov exponent of new chaotic system
2.3. Sensitivity to initial conditions
Sensitivity to initial conditions means that each point in a chaotic system is arbitrarily closely approximated
by other points with significantly different future paths, or trajectories. Thus, an arbitrarily small change, or
perturbation, of the current trajectory may lead to significantly different future behavior. The next figure
compares the time series for two litely different initial conditions. The two time series stay close together for
about 2 iterations. But after that, they are pretty much on their own.
(a)
(b)
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(c)
Figure 2: Sensitivity to two initial conditions [-2, 2, 1] and [-1.9, 1.9, 0.9]
(a): (b): (c):
III. IMAGE SCRAMBLING ALGORITHM BASED ON CHAOS THEORY
3.1. Proposed algorithm
The proposed chaotic system is now used in the design of color image encryption algorithm. The proposed
images encryption algorithm input is an original image whilst the output is a scrambled one. Figure 3 illustrate
the proposed algorithm scheme.
Figure 3: Principle of chaotic scrambling algorithm for color image
Original
Image
Scrambled color
Image
Red
Component
Green
Component
Blue
Component
R- Key
G- Key
B-Key
lx
ly
lz
Scrambled Red
component
Scrambled Green
component
Scrambled Blue
component
RGB three-color
separation block
RGB three-color
combination block
Chaotic
Generator Key
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IV. EXPERIMENTAL DETAILS AND RESULTS
A good quality scrambled algorithm should be strong against all types of attack. Some experiments are given
in this section to demonstrate the efficiency of the proposed technique. In this section, the proposed technique is
applied on two color images "Gallery" and "Alice", of resolution of " 256*256". We analyze the results by
calculating histogram and correlation coefficient, to test the performance of the proposed technique. The next
figures show the results of scrambled algorithm.
Figure 4: "Alice" image corresponding for different step of the scrambling process
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Figure 5: "Gallery" image corresponding for different step of the scrambling process
4.1. Statistical analysis
In order to resist attacks, the scrambled images should possess certain random properties. To prove the
robustness of the proposed algorithm, a statistical analysis has been performed by calculating the histograms and
the correlation coefficients for the original image and the scrambled image. For the two images that have been
tested, it has been determined that their quality is good.
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4.1.1. Histogram Analysis
An image histogram is a commonly used method of analysis in image processing. The advantage of a
histogram is that it shows the shape of the distribution for a large set of data. Thus, an image histogram
illustrates how pixels in an image are distributed by plotting the number of pixels at each color intensity level. It
is important to ensure that the encrypted and original images do not have any statistical similarities.
The experimental results of the original image and its corresponding scrambled image and their histograms
are shown in Fig. 6. The histogram of each original image illustrates how the pixels are distributed by graphing
the number of pixels at every color of RGB [14]. It is clear that the histogram of the scrambled image is
different from the respective histograms of the original image.
(a) (b)
Figure 6: "Alice" image histogram in three channels RGB
(a): Original (b): Scrambled
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(a) (b)
Figure 7: "Gallery" image histogram in three channels RGB
(a): Original (b): Scrambled
4.2. Correlation of two adjacent pixels
In addition to the histogram analysis, we have also analyzed the correlation between two vertically adjacent
pixels, two horizontally adjacent pixels and two diagonally adjacent pixels in plain image and cipher image
respectively.
A correlation is a statistical measure of security that expresses a degree of relationship between two adjacent
pixels in an image or a degree of association between two adjacent pixels in an image. The aim of correlation
measures is to keep the amount of redundant information available in the scrambled image as low as possible
[11, 15].
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Equation (3) is used to study the correlation between two adjacent pixels in the horizontal, vertical, diagonal
and anti-diagonal orientations:
� =
∑ ( �× �)−∑ �×∑ �
�
�=
�
�=
�
�=
√ ∑ �
�
�= − ∑ �
�
�= × ∑ �
�
�= − ∑ �
�
�=
(3)
where and are the intensity values of two adjacent pixels in the image and is the number of adjacent
pixels selected from the image to calculate the correlation. Results for the correlation coefficients of two
adjacent pixels are shown in tables 1and 2.
In the experiments results, 3000 pairs of two adjacent pixels are randomly selected. Fig. 8 shows the
distribution of two adjacent pixels in the original image and the encrypted-image. There is very good correlation
between adjacent pixels in the image data [16, 17], while there is only a small correlation between adjacent
pixels in the scrambled image.
(a) (b)
Figure 8: Horizontal, vertical and diagonal correlation of original and scrambled "Alice" image
(a): Original image (b): Scrambled image
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(b) (b)
Figure 9: Horizontal, vertical and diagonal correlation of original and scrambled "Gallery" image
(a): Original image (b): Scrambled image
Table I
Correlation coefficient corresponding to original and scrambled images
Direction Horizontal Vertical Diagonal
Alice image
Original 0.9759 0.9855 0.9677
Scrambled 0.5632 0.5656 0.5612
Gallery image
Original 0.9878 0.9777 0.9679
Scrambled 0.3977 0.3921 0.3832
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Table II
Correlation coefficient corresponding to original and recovered images
Direction Horizontal Vertical Diagonal
Alice image
Original 0.9759 0.9855 0.9677
Renew 0.9759 0.9855 0.9677
Gallery image
Original 0.9878 0.9777 0.9679
Renew 0.9674 0.9665 0.9290
4.3. PSNR
Peak Signal to Noise Ratio (PSNR) criterion is used to test the unobservable factor. This measure indicates
the degree of similarity between the watermark images and a watermark images. PSNR is expressed
mathematically in the following form:
[ ] = log
55
� ��,��
(4)
where EQM is the mean square error between the two images ( original , recovered .
� , = ∑ ∑ , − ,
−
=
−
=
To recover the two images, we apply the inverse of the proposed algorithm in figure 2. The result is shown in
figure 10.
Figure 10: "Alice" and "Gallery" recovered image
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PSNR high means: Mean square error between the original image and reconstructed image is very low. It
implies that the image been properly restored. In the other way, the restored image quality is better; in our case,
the values of PSNR are as follow:
PSNR (Alice) = Inf
PSNR (Gallery) = 51.75
The result is much closed with the correlation coefficient.
- For "Alice", the correlation coefficient for the original and renew image are identical. The PSNR equal to
INF, that means the renew image is identical to original image.
- For "Gallery", the correlation coefficient for the renew image is at 40% of the original image, that justify
the corresponding PSNR value.
V. CONCLUSION
In this paper, a new image scrambling algorithm, by using image scrambling to encrypt the image to improve
the security of image. The new algorithm based on chaotic system and decomposition and recombination of
pixel values is able to scramble pixel positions and pixel values of images. Analysis of the statistical information
of scrambled images in the experimental tests shows that the present algorithm provides reasonable security.
Owing to the strong irregularity of the sorting transformation that improves the effect of the scrambling. The
experimental results show that the algorithm is effective to scramble the image and can provide high security. It
simulates scrambling under Matlab 7 to confirm it.
REFERENCES
[1] Pawan N. Khade and Prof. Manish Narnaware, “3D Chaotic Functions for Image Encryption,” IJCSI International Journal of
Computer Science Issues, Vol. 9, Issue 3, No 1, May 2012, pp: 323- 328.
[2] P. Ping, Z. J. Wang, and F. Xu, “A two-dimensional cellular automata based method for multiple image”, International Conference on
Computer Science & Service System, vol. 112, pp. 101-104, 2014.
[3] F. Chen, K. W. Wong, X. Liao, and T. Xiang, “Period distribution of generalized discrete Arnold cat map”, Theoretical Computer
Science, vol. 552, no. 4, pp. 13-25, 2014.
[4] L. Chen, D. Zhao, and F. Ge, “Mage encryption based on singular value decomposition and Arnold transform in fractional domain”,
Optics Communications, vol. 219, no. 6, pp. 98-103, 2013.
[5] Zhu Liehuang, Li Wenzhuo, Liao Lejian and Li Hong. “A Novel Image Scrambling Algorithm for Digital Watermarking Based on
Chaotic Sequences” [J]. International Journal of Computer Science and Network Security, 6(8B): 125-130, 2006.
[6] Matthews R. “On the derivation of a ‘chaotic’ encryption algorithm [J]”. Cryptogia, 1989, 13:29-42.
[7] Scharinger J. Fast encryption of image data using chaotic Kolmogorov flow [J]. J. Electronic Imaging, 1998, 7(2): 318-325.
[8] Yu X Y, Zhang J, Ren H E, Xu G S and Luo X Y. “Chaotic Image Scrambling Algorithm Based ob S-DES [J]”. Journal of Physics:
Conference Series, 2006, 48: 349-353.
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[9] S.N. Lagmiri, M. Amghar and N. Sbiti, “Synchronization between a new chaotic system and Rössler system by a high gain observer”,
IEEEXplore, May ,2015.
[10] J. Khan, J. Ahmad, and S. O. Hwang, “An efficient image encryption scheme based on: Henon map, skew tent map and s-box,”
International Conference on Modeling, Simulation and Applied Optimization, vol. 10, pp. 1- 6, 2015.
[11] Burger, W. and M. Burge, Digital image processing: “an algorithmic introduction using Java”. 2008: Springer-Verlag New York Inc.
[12] D. L, “Color image encryption algorithm based on chua's circuit and chen's hyper-chaotic system,” Journal of Information &
Computational Science, vol. 12, pp. 1021-1028, 2015.
[13] S. Liu, J. Sun, and Z. Xu, “An improved image encryption algorithm based on chaotic system”, Journal of Computers, vol. 4, no. 11,
pp. 1091- 1100, 2009.
[14] Abderrahim, N.W., F.Z. Benmansour, and O. Seddiki, “Integration of chaotic sequences uniformly distributed in a new image
encryption algorithm.” 2012.
[15] Jolfaei, A. and A. Mirghadri, “Survey: image encryption using Salsa20”. International Journal of Computer Science Issues, 2010. 7(5):
pp. 213-220.
[16] 17. H. El-din. H. Ahmed, H.M.K., O. S. Farag Allah, “Encryption quality analysis of the RC5 block cipher algorithm for digital
images”. Optical Engineering, 2006. 45( 10).
[17] 18. Ibrahim S I Abuhaiba and M.A.S. Hassan, Image Encryption Using Differential Evolution Approach in Frequency Domain. Signal
& Image Processing : An International Journal(SIPIJ), 2011. 2, No.1: p. 51-69.
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