Journal of Robotics, Networking and Artificial Life, Vol. 3, No. 3 (December 2016), 168-173

Analysis of fractional-order Chen System and Its application in Image Encryption
Hongyan Jia

Department of Automation, Tianjin University of Science and Technology,
1038 Dagunanlu Road, Hexi District, Tianjin300222, PR China
Qinghe Wang

Department of Automation, Tianjin University of Science and Technology,
1038 Dagunanlu Road, Hexi District, Tianjin 300222, PR China
E-mail: jiahy@tust.edu.cn
www.tust.edu

Abstract
In this paper, a fractional-order Chen system is discussed and analyzed by using the frequency-domain
approximation method. Its chaotic characteristic can be observed by numerical simulation when the order is as low
as 2.9. Then the existence of chaos is verified through the Lyapunov exponent diagrams, bifurcation diagrams, andphase orbits diagrams. At last, based on the fractional-order Chen system, a kind of double encryption algorithm isproposed and adopted to realize the image encryption. The experimental results and analysis show that the proposed
method not only has the ideal effect of image encryption, but also has a high security of resisting the statistical
attacks. That is, the proposed method is suitable and feasible for image encryption.
Keywords: fractional-order; Chen system; Lyapunov exponent diagrams; bifurcation diagrams; image encryption

1. Introduction
Although fractional calculus has a history of more than
300 years [1], the applications of fractional calculus tophysics and engineer have only become a hot topic in
recent years. Since the chaos phenomenon has beenfound in large number of fractional-order system, thedynamic characteristics of the fractional system hasbecome a research hotspot. Some fractional-orderchaotic systems have been proposed and researched,such as the fractional-order Lorenz system [2], thefractional-order Chen system [3-6], the fractional-order ofLü system [7].

With the rapid development of digital and network
technology, the significance of information security hasarisen more and more attention in recent years.
Especially for images, a kind of important information
carrier, it may contain much inferable information.
Therefore, its security has increasingly been importantand become one of research focus in the field of
information security. Image encryption technology is aneffective method to protect the transmission of digital
images [8]. So far, many kinds of image encryption technology have been reported, such as text encryption technology [9], quad-tree image encryption technology [10],chaotic encryption technology [11], image encryption

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Hongyan Jia, Qinghe Wang

technology based on DNA [12], and so on[13-15]. Intrinsic
properties of chaotic systems such as sensitivity toinitial conditions, unpredictability, pseudo-randomness,and ergodicity, are considered analogous to an ideal
cryptosystem. So the chaotic encryption technology has
increasingly highlighted its advantage and become
popular in the image encryption. Compared withinteger-order chaotic system, Fractional-order chaoticsystem has larger key space and more complex random
sequence [16, 17]. It is more suitable for image encryption,
but the research of image encryption based on
fractional-order chaotic system is very few and worthyof further study.
In this paper, a fractional-order Chen system isdiscussed and analyzed by using approximation of thefrequency domain method. Its chaotic characteristic canbe observed by numerical simulation when the order is
as low as 2.9. Then the existence of chaos is verified
through the Lyapunov exponents’ diagram, bifurcation
diagrams, and phase orbits diagrams. At last, based on
this fractional-order Chen system, a kind of double
encryption algorithm is proposed and adopted to realize
the image encryption. The experimental results andanalysis show that the proposed method is effective andpractical for image encryption.

may exist in the system when a ∈ [36,47] .When
selecting α = 1 , β = 1 , γ = 0.9 , a = 40 , c = 28 , andvarying b , the Lyapunov exponents’ diagram andbifurcation diagrams of the system are shown in Fig.2(a), and (b), respectively. From the Fig.2, we canobserve that the system may has chaotic behaviors
when b ∈ [0.8,5]. When fixingα = 1,β = 1,γ = 0.9,a =40 , b = 3 , and varying c , the Lyapunov exponents’
diagram and bifurcation diagram of the system areshown in Fig.3 (a), and (b), respectively. From the Fig.3,
chaotic behaviors may exist in the system when c ∈[23,32]. By the above analysis, we can conclude thatchaotic behaviors exist in the fractional-order Chensystem when the order is as low as 2.9. Furthermore,when selecting α = 1 , β = 1 , γ = 0.9 , a = 40 , b = 3 ,and c = 28 , a chaotic attractor can be obtained bynumerical simulation, which is shown in Fig.4. The
result is consistent with the theoretical analysis based on
the Lyapunov exponents’ diagram and bifurcation
diagrams. In this paper, we will mainly utilize thefractional-order Chen system with the initial conditionof α = 1 , β = 1 , γ = 0.9 , a = 40 , b = 3 , and c = 28 .
Under the above condition, the system has a typicalattractor and can be used in the proposed image
encryption algorithm.
52. Fractional-order Chen system0dα x⎧ dtα = a(y − x)⎪dβ y= (c − a)x − xz + cy
⎨ dtγβ⎪d z = xy − bz
⎩ γdt-5

LEThe fractional-order Chen system is described asfollows [18]-10-15-20(1)-25303234363840a42

44

4648

50

4850(a) Lyapunov exponents’ diagram

Where system parameters a,b,c ∈ R,α,β, andγ refer tothe fractional order. Generally speaking, the positive
Lyapunov exponent is an important characteristic toshow chaotic behavior of systems. Therefore, theLyapunov exponents’ diagrams and bifurcation
diagrams of the fractional-order Chen system are firstly
discussed in this paper. When fixingα = 1,β = 1,γ =0.9 , b = 3 , c = 28 , and varying a , the Lyapunov
exponents’ diagram and bifurcation diagram of thesystem are shown in Fig.1 (a), and (b), respectively.
From the Fig.1, we can obtain that chaotic behaviors

151050-5-10-15
30

32

34

36

3840
a424446(b) Bifurcation diagram
Fig. 1. Lyapunov exponents’ diagram and Bifurcation diagramwith b=3, c=28, and varying a.

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Analysis of fractional-order

20515
010-5yLE

5-10

0-15-5-20-10-25012435b

76

8910(a) Lyapunov exponents’ diagram-15-20-2010-15-10-50x5101520Fig. 4. Chaotic attractor with a=40, b=3, and c=28.

8642

3. Image Encryption Algorithm

0-2-4-6-8-10012345b678910(b) Bifurcation diagram
Fig. 2. Lyapunov exponents’ diagram and Bifurcation diagramwith a=40, c=28, and varying b.
50LE-5-10-15-20-25
2025c3035(a) Lyapunov exponentd’ diagram

4. Experimental Results

151050-5-10-152025c30

The steps of the proposed Algorithm are listed briefly as
follows:
Step 1: Generate three chaotic sequences x, y and z byusing fraction-order Chen system.
Step 2: Decompose the color image into R, G and Bchannels.
Step 3: The sequence x is sorted and used to scramble
the matrixes of R, G and B channels using confusion
operation to randomly shuffle the pixel position.
Step 4: The three chaotic sequences is combined withscrambled matrixes using diffusion operation to change
the pixel value.
Step 5: Obtain the encrypted image.
For decryption process, it is an inverse process ofencryption. That is to say, the operation of encryption
should be performed reversely.

35(b) Bifurcation diagram
Fig. 3. Lyapunov exponents’ diagram and Bifurcation diagramwith a=40, b=3, and varying c.

In this paper, a color Image of size 256 ∗ 256 isselected as the original image shown in Fig.5 (a). The
experimental simulation has been carried out whenfixing α = 1, β = 1, γ = 0.9 , a = 40, b = 3, c = 28 ,and setting the initial values of state variables
�x0 , y0, z0 � to(0.1,0.2,0.3). The encrypted image and thedecrypted images are shown in Fig.5 (b) and (c),respectively. From the Fig.5, we can observe theencrypted image is completely different from theoriginal image, and become an unrecognizable image.
While the decrypted image and the original image isidentical.

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(a)

(a)

(b)

(b)

(c)

(c)

Fig. 5. The experimental results (a) the original image, (b) theencrypted image (c) the decrypted image.

5. Performance Analysis
5.1. Key sensitivity analysis
An efficient or an ideal encryption scheme should havea high security; That is, it should be very sensitive to thesecret key. It means a tiny change in the key will lead toa significant change in the encrypted image, even thedecryption will fail when using the secret key with aslight change. In order to validate the key sensitivity, atest is performed using a key that is a bit different fromthe original key to decrypt the original image. The
decrypted results are shown in Fig.6 (a), (b), (c), and (d),respectively. Fig.6 (a), (b), and (c) show the decrypted
images with the deviation of any one of the statevariables values x0 ,y0 and z0 is up to the 10−9 , and keep
others unchanged. Fig.6 (d) shows the decrypted image
with the system parameter a=40.000000001, and keep
others unchanged. It is evident that the decryption isfailed as long as there is a tiny change of the original
key. So it can be concluded that the proposed algorithm
has high sensitivity to the secret key.

(d)Fig. 6. Decrypted image with (a) incorrect x0 , (b) incorrect y0 ,(c) incorrect z0 , (d) incorrect system parameter a.

5.2. Histogram analysis
Generally speaking, histogram can clearly reflect thedistribution of pixel values in the image, so it isconvenient for people to correctly evaluate the pixel
values distribution of an image. Here the histograms of
the original image are illustrated in Fig.7 (a), (b), and(c), respectively. The histograms of the encrypted image
are illustrated in Fig.8 (a), (b), and (c), respectively. It
can be seen that the histograms of the encrypted image
are flat enough, and uniformly distributed. It indicatesthat no useful information can be extracted about theoriginal image. Thus, we can conclude that the proposed

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Analysis of fractional-order

encryption method have good resistance against
statistical attack.
5.3 Correlation Coefficient analysis
Correlation coefficient is an important indicator tomeasure the relevancy between two adjacent pixels. The
correlation coefficient is generally high in the originalimage. That means it is vulnerable to statistical attack.
An image encryption should decrease the correlation
coefficient in the encrypted image as much as possible.
Here, we calculate correlation coefficient for an image
according to the following formula [19-21]:r_xy=(cov(x,y))/(√(D(x)) √(D(y))) ,(2)

where

1

cov(x, y)= ∑N
i=1(x i − E(x))(yi − E(y)) ,1

N

E(x) = ∑Ni=1 x i ,N
1

2D(x) = ∑N
i=1(x i − E(x)) ,N

(3)
(4)
(5)

The correlation coefficients of the original image andthe encrypted image are provided in Table 1. It can beseen from Tabel 1 that the correlation coefficient of theoriginal image is close to one, while that of theencrypted image is close to zero. It is obvious that theadjacent pixels in encrypted image are almost
uncorrelated. So, we can obtain that the proposed
encryption method has good resistance against statistical
attack.

(a)

(a)

(b)

(b)

(c)

(c)

Fig. 7. Histograms of the original image (a) R channel, (b) Gchannel, and (c) B channel.

Fig. 8. Histograms of the encrypted image (a) R channel, (b) Gchannel, and (c) B channel.

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Table 1 Correlation Coefficient
Original Image

Encrypted Image

R

0.95176

0.0025015

G

0.95910.0014247

B

0.971310.0012004

6. Conclusion
This paper firstly discusses and analyzes a fractionalorder Chen system. Its chaotic characteristic can beobserved by numerical simulation when the order is as
low as 2.9. Then the existence of chaos is verified
through the Lyapunov exponents’ diagram, bifurcation
diagrams, and phase orbits diagrams. At last, based on
the fractional-order Chen system, a kind of double
encryption algorithm is proposed and adopted to realize
the image encryption. The experimental results andanalysis show that the proposed method has goodencryption effect, and high sensitivity to the secret key.
It also has good resistance against statistical attack.
Besides, the encryption process is simple, and it isconvenient to encrypt image. So the proposed
algorithm is feasible and effective, and it is suitable forthe image encryption field.
AcknowledgementsThis work was supported in part by the Young
Scientists Fund of the National Natural Science
Foundation of China (Grant No.11202148, 61174094).
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