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978-1-4244-3986-7/09/$25.00 ©2009 IEEE 
Image Encryption Algorithm Based on Henon 
Chaotic System 
Chen Wei-bin#1, Zhang Xin*2 
#
College of Computer Science and Engineering, Wenzhou University  
Wenzhou, Zhejiang, China 
1sun@wzu.edu.cn 
*
School of Information and Engineering, Wenzhou Medical College 
Wenzhou, Zhejiang, China 
2zhangxin@wzmc.net 
 
Abstract— In this paper, a new image encryption algorithm is 
presented based on Henon chaotic system in order to meet the 
requirements of secure image transfer. Shuffling the positions 
and changing the grey values of image pixels are combined to 
shuffle the relationship between the cipher-image and the 
original-image. First, the Arnold cat map is used to shuffle the 
positions of the image pixels. Second, the shuffled-image is 
encrypted based on Henon’s chaotic system pixel by pixel. There 
are several parameters in this kind of chaos system and Arnold 
cat map. The results of several experimental, statistical analysis 
and key sensitivity tests show that the proposed image encryption 
scheme provides an efficient and secure way for image 
encryption. The distribution of grey values of the encrypted 
image exhibits a random-like behavior. 
 
Keywords— Image encryption; Arnold map; Henon’s chaotic 
system 
I. INTRODUCTION 
In order to transmit images data safely, numerous 
encryption algorithms have been developed. Due to desirable 
properties of non-linear dynamical systems such as ergodicity, 
sensitive dependence on 
initial conditions and good 
pseudo-random properties, the chaos-based encryption has 
suggested a new and efficient way to deal with the intractable 
problem of fast and highly secure image encryption. Therefore, 
chaotic dynamics are expected to provide a fast and easy way 
to build cryptosystems. In order to improve the security of the 
image encryption algorithm, many researchers prefer shuffling 
the positions and changing the grey values of image pixels 
simultaneously. In [1], a symmetric image encryption scheme 
based on 3D chaotic cat maps is presented. In [2], on the 
security of 3D Cat map based symmetric image is proposed. 
And combined image encryption algorithm based on diffusion 
mapped disorder and hyperchaotic systems encryption scheme 
are also presented [3-4]. However, the encryption arithmetic 
based on 3D chaotic cat maps is a computationally expensive 
process. And the key space is not independence. 
This paper is organized as follows. In Section 2, the design 
of the proposed image encryption scheme is discussed in 
details. In Section3, some simulation results are described. In 
Section 4, security analysis is given. In Section5, conclusion 
remarks are drawn. 
II. THE PROPOSED ENCRYPTION ALGORITHM 
The image encryption algorithm includes two steps. Firstly, 
the positions of the pixels of the original image are shuffled 
by Arnold cat map. Then the pixel values of the shuffled 
image are encrypted by Henon’s chaotic system. 
A. Encryption by Arnold cat map 
The Arnold cat map is a two-dimensional invertible chaotic 
map [5]. Without loss of generality, we assume the dimension 
of the original grayscale image I is 
M
M 
. Arnold cat map 
is described as following: 
)
mod(
1
1
1
1
M
y
x
cd
d
c
y
x
n
n
n
n












+
=






+
+
     (1) 
where c and d are positive integers. The 
)
,
(
1
1
+
+
n
n
y
x
is the 
new position of original image, and the 
)
,
(
n
n
y
x
 is the 
original position of the original image. where 

,
2
,1
,
0
=
n
. 
After iterating N times, there exist positive integersT ,such 
that
)
,
(
)
,
(
1
1
y
x
y
x
n
n
=
+
+
. The period T depends on the 
parameters c , d and the size M  of the original image. 
Thus the parameters c ,d and the number of iterations N all 
can be used as the secret keys. Since there only exists a linear 
transformation and mod function, it is very efficient to shuffle 
the pixel positions using the Arnold cat map. After several 
iterations, the correlation among the adjacent pixels can be 
disturbed completely. Some experiments are given in Section 
3 to demonstrate the efficiency of Arnold cat map. However, 
the periodicity of Arnold cat map should degrade the security 
the encryption, because the possible attacks may iterate the 
Arnold cat map continuously to reappear the original image. 
There are no differences between the original-image and 
cipher-image on the statistical properties [6-8]. At the same 
time, the key space of positive integers is limited. Therefore, 
we adopt Henon chaotic system to change the pixel values 
next to improve the security. 
B. Encryption by Henon chaotic system and Arnold cat map 
Henon chaotic map[9-10]is first discovered in 1978, which 
Authorized licensed use limited to: PSN College of Engineering and Technology. Downloaded on December 19, 2009 at 13:11 from IEEE Xplore.  Restrictions apply. 
is described as following: 

,
2
,1
,
0
,
1
1
2
1
=
=
+

=
+
+
i
bx
y
y
ax
x
i
i
i
i
i
      (2) 
The well-studied Henon map presents 
a 
simple 
two-dimensional map with quadratic non-linearity. This map 
gave a first example of the strange attractor with a fractal 
structure. Because of its simplicity, the Henon map easily 
lends itself to numerical studies. Thus a large amount of 
computer investigations followed. Nevertheless, the complete 
picture of all possible bifurcations under the change of the 
parameters a and b is far from completion. If one chooses 
4
.
1
,3
.
0
=
=
b
a
, the system is chaotic. 
In our scheme, two variables of the Henon chaotic map are 
adopted to encrypt the shuffled-image. The encryption process 
consists of there steps of operations. 
Step1: The Henon chaotic system is converted into 
one-dimensional chaotic map. The one-dimensional Henon 
chaotic map is defined as: 
i
I
i
bx
ax
x
+

=
+
+
2
1
2
1
      (3) 
Where 
3
.
0
=
a
, 
]
4
.
1
,
07
.
1
[
b
.The parameter a ,the 
parameter b , initial value
0
x and initial value
1
x may 
represent the key. 
Step2: After image shuffle, we adopt Henon chaotic map to 
change the pixel values of the shuffled-image. First, Henon 
chaotic map is obtained by formula(3). Then transform matrix 
of pixel values is created. 
Step3: The exclusive OR operation will be completed 
bit-by-bit between the transform matrix of pixel values and 
the values of the shuffled-image. We can obtain the 
cipher-image. 
Since the chaotic systems are deterministic, the receiver can 
reconstruct the same shuffled-image exactly using the same 
secret keys. Then the anti-process of image shuffle is 
described as: 
)
mod(
1
1
1
1
1
M
y
x
cd
d
c
y
x
n
n
n
n












+
=






+
+

   (4) 
The parameter is chosen the same as the process of image 
shuffle. After that, the original image can be obtained. The 
decrypted process is completed.  
III. EXPERIMENTAL ANALYSIS 
Some experimental results are given in this section to 
demonstrate the efficiency of our scheme. The original-image 
with the size 256256 is shown in Fig.1(a) and the histogram 
of the original-image is shown in Fig.1(b). Fig.2(a) is the 
shuffled-image and Fig.2(b) is 
the histogram of the 
shuffled-image. The Arnold cat map 
is chosen as 
2
=
= d
c
and 
10
=
M
.Image shuffle is first step of image 
encryption.  
Fig.2(b) illustrates the histogram of the original-image is 
the same as the histogram of the shuffled-image. As can be 
seen that, Arnold cat map only shuffle the pixel positions of 
the original- image. Fig.3(a) illustrates the cipher-image by 
Henon chaotic map and Fig.3(b) is the corresponding 
histogram. The parameters are selected as
3
.
0
=
a
, 
4
.
1
=
b
.The secret keys to change the pixel values of the 
original fusion-image are
01
.
0
0
=
x
,
02
.
0
1
=
x
. As we can 
see, the histogram of the ciphered image is fairly uniform and 
is significantly different from that of the original-image. The 
encryption procedure complicates. 
Fig.4(a) illustrates the decrypted shuffled-image and 
Fig.4(b) is the corresponding histogram. The parameters are 
selected as 
3
.
0
=
a
, 
4
.
1
=
b
. The secret keys to change the 
pixel values of the shuffled-image are 
01
.
0
0
=
x
,
02
.
0
1
=
x
. 
Since the chaotic systems are deterministic, the receiver can 
reconstruct the same shuffled-image exactly using the secret 
keys.Fig.4(c) illustrates the decrypted original-image and 
Fig.4(d) is the corresponding histogram. The anti-process of 
image shuffle is described as formula (4). 
 
  
(a) 
 
(b) 
Fig. 1. original-image and its histogram  (a) original-image;  (b) histogram 
of the original-image 
Authorized licensed use limited to: PSN College of Engineering and Technology. Downloaded on December 19, 2009 at 13:11 from IEEE Xplore.  Restrictions apply. 
  
(a) 
 
(b) 
Fig. 2. Encryption by using Arnold cat map: (a) shuffled image; (b) histogram 
of the shuffled-image. 
 
(a) 
 
(b) 
Fig. 3. Encryption by Henon’s chaotic system: (a) cipher-image; (b) 
histogram of the cipher-image. 
 
(a) 
 
(b) 
 
(c) 
 
(d) 
Fig.4. decrypted image and histogram of the decrypted image (a) decrypted 
shuffled-image (b) histogram of the decrypted shuffled-image (c) decrypted 
original-image (d) histogram of the decrypted original-image 
Authorized licensed use limited to: PSN College of Engineering and Technology. Downloaded on December 19, 2009 at 13:11 from IEEE Xplore.  Restrictions apply. 
IV. SECURITY ANALYSIS 
A good encryption scheme should be sensitive to the secret 
keys, and the key space should be large enough to make 
brute-force attacks infeasible. In our encryption algorithm, the 
key-image and the initial values of Henon chaotic map are 
used as secret keys. The key space is large enough to resist all 
kinds of brute-force attacks. The experimental results also 
demonstrate that our scheme is very sensitive to the secret key 
mismatch. Fig. 5 illustrates the sensitivity of our scheme with 
the secret key 
0
x and
1
x .The cipher-image is shown in Fig. 
3(a), which is decrypted using
01
.
0
0
=
x
,
02
.
0
1
=
x
,and 
2
=
= d
c
,
10
=
M
.As can be seen that, even the secret key 
0
x  is changed a little (
0100001
.
0
0
=
x
), the decrypted 
shuffled-image is absolutely different from the original 
shuffled-image. So it is difficult to obtain the original-image. 
Similar results for other secret keys
1
x  all can be obtained. As 
we can see, the decrypted image with wrong keys has a 
histogram with random behavior. The sensitivity to initial 
conditions which is the main characterization of chaos 
guarantees the security of our scheme. Furthermore, the step 
of image shuffle is double protection of secret key. 
Undoubtedly, the secret keys are secure enough. 
 
 
(a) 
 
 
(b) 
 
Fig.5. The sensitivity to the secret key
0
x
 (a) decrypted shuffled-image 
(
100001
.
0
0
=
x
) ; (b) histogram of the decrypted image. 
 
V. CONCLUSIONS 
In this paper, a new image encryption scheme is presented. 
Both theoretical analysis and experimental results show that 
the proposed cryptosystem has high security. It is found that 
such a design can enhance the randomness. The proposed 
algorithm has four merits: 1) the algorithm has a large enough 
key space to resist all kinds of brute force attacks. 2) The new 
encrypted arithmetic not only shuffle the pixel positions of the 
original-image, but also change the grey values of the 
original-image. 3) The encryption algorithm is very sensitive 
to the secret keys. 4) The operation time of the encryption 
algorithm is shorter than the 3D Arnold cat map. As 
demonstrated in our simulation, this approach is suitable in 
security.  
REFERENCES 
[1] G. Chen,Y. Mao, K. Charles, “A symmetric image encryption scheme 
based on 3D chaotic cat maps,” Chaos, Solutions & Fractals, pp. 
749-761, Dec. 2004. 
[2] K. Wang, W. Pei, “On the security of 3D Cat map based symmetric 
image encryption scheme,” Physics Letters A., pp. 432-439, May. 2005.  
[3] S.-M. Chang, M.-C. Li, W.-W. Lin, “Asymptotic synchronization of 
modified logistic hyper-chaotic systems and its applications,” 
Nonlinear Analysis, pp. 869–880, Jan. 2009.  
[4] H. Lian-xi, L. Chuan-mu, L. Ming-xi, “Combined image encryption 
algorithm based on diffusion mapped disorder and hyperchaotic 
systems,” Computer Applications, pp. 1892-1895, Aug. 2007. 
[5] Z.-H. Guan, F. Huang, W. Guan, “Chaos-based image encryption 
algorithm,” Physics Letters A., pp. 153–157, Aug. 2005. 
[6] Y. Heng-fu, W. Yan-peng, T. Zu-wei, “An image encryption algorithm 
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[7] Z. Qiong, S. Minfen, Z. Yikui, “3D Chaotic Cat Map Based Digital 
Image Encryption Method,” Journal of Data Acquisition &Processing, 
pp. 292-298, Sep. 2007. 
[8] S. Qiu-dong, M. Wen-xin, Y. Wen-ying, D. Hong. “A Random 
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the Technique Based on Arnold Transform,” Journal of shanghai 
second polytechnic university, pp. 159-163, Sep. 2008. 
[9] E. Petrisor, “Entry and exit sets in the dynamics of area preserving 
Henon map,” Chaos,Solitons and Fractals, pp. 651–658, Oct. 2003. 
[10] L. Guo-hui, Z. Shi-ping, X. De-ming, L. Jian-wen, “An Intermittent 
Linear Feedback Method for Controlling Henon-like Attractor,” 
Journal of Applied Sciences, pp. 288–290, Dec.2001. 
Authorized licensed use limited to: PSN College of Engineering and Technology. Downloaded on December 19, 2009 at 13:11 from IEEE Xplore.  Restrictions apply.