COLOR IMAGE ENCRYPTION SOLUTION BASED ON THE
CHAOTIC SYSTEM OF LOGISTIC AND HENON
Zhang Yunpeng
1*
, Sun Peng
2
, Xie Jing
1
and Huang Yunting
1
1
College of Software and Microelectronics, Northwestern Polytechnical University
P.O.Box 732, 127#, West YouYi Road, Xi’an, 710072, China
2
College of Computer, Northwestern Polytechnical University, 127#, West YouYi Road, Xi’an, 710072, China
Keywords: Cryptography, Chaotic system, Color image, Encryption.
Abstract: The security of color image has become an important network information security research field. To meet
the security requirements of color image and according to the characteristics of the image coding and
chaotic system, the paper presented a color image encryption solution based on the chaotic systems. With
the help of Logistic system, the solution generates the chaotic sequence, which is used to the parameters and
the number of iterations of Henon system. And then, encrypt the color image through multiple iterating the
Henon system. At last, we analyse and prove the solution in theory and experiment. The results show that
the encrypted image has a uniform distribution of the pixel value, has a good solution diffusion, can
effectively resist the phase space reconstruction attacks, and has a good security and reliability.
1 INTRODUCTION
At present, the conventional image encryption
solutions based on chaotic revealed many security
risks, mainly showed in the weakness of lacking of
ability to resist against the common attack methods.
In literature (Li, SJ, Zheng, X 2002, et al), the
security of image encryption solutions were
explored and analyzed. Meanwhile, showed the
weakness that some solutions are difficult to resist
the common attacks. In literature (Li, Q, Zhao, K, et
al 2007, p.45), Li Quan etc successfully deciphered a
chaotic image encryption solution by using the phase
space reconstruction method and the method of
exhaustion. To solve these problems, some scholars
made some methods resist conventional attacks
during designing image encryption solutions. For
example, Wu Xu etc proposed a method through
introducing an auxiliary key to resist the known
plaintext in the process of image encryption (Wu, X,
Chen, ED & Hu, JS* 2004, p.754), but it could do
nothing to the emerging phase space reconstruction
attacks. The solution in this paper is mainly used in
the two-dimensional chaotic system, and it applied T
generated in the encryption process as one of the
decryption keys in the decryption process. Expect
the final solution can be effective against the phase
space reconstruction attacks.
2 PROGRAMME DESCRIPTION
The chaotic systems (Logistic(Pareek, NK, Vinod, P ,
Sud, KK 2006, p.926
), Henon (Zhang, H, Wang, XF, et al
2005, p.2137
) and Arnold (Zou, JC, Tie XY 2000,
p.182
) ) this paper have used are suitable to
encryption solution, which has been recognized by
scholars in the field of Chaos.
Input: original image I (size r×r
),
encryption key (ab
μ
0
x
).
Where a, b for the Arnold transformation parameters;
μ
0
x
for the Logistic system parameters.
1. In the original image I (r×r), the gray value of
each pixel is A(x,y). Make the image into a
linear character array, for
A(x,y)= A
(r*x+y),
that
is, the gray value of each point in turn recorded
as A
1
,A
2
,…,A
n
,n=r×r.
2. By the
μ
0
x
of secret key, using logistic
system formula generates real-valued chaotic
sequence P = {p
ij
| ij= 012,… ,N - 1 }, and
generates the integers sequence D = {d
i
| i = 0
275
Yunpeng Z., Peng S., Jing X. and Yunting H. (2010).
COLOR IMAGE ENCRYPTION SOLUTION BASED ON THE CHAOTIC SYSTEM OF LOGISTIC AND HENON.
In Proceedings of the 5th International Conference on Software and Data Technologies, pages 275-278
DOI: 10.5220/0002923602750278
Copyright
c
SciTePress
12,…,N-1d
i
[2,3,4, …,62]}; here, if
the scope of d takes 1, it Will be broken by
exhaustive key flow; if the scope of d is too
large, iterative process will lead to be
time-consuming. If the scope of d takes 0, the
number of iterations takes 0, so pixel value does
not get encrypted. Through several experiments,
ultimately determine d is able to combine the
efficiency and safety in the above scope. The
specific description of generating method is in
the back for the two sequences.
3. The gray-scale transformation matrix generated
by the P: G = {gij | gij
[0,1,2, …,255] ij =
012N-1}
4. Using Arnold formula, key a, b for the
parameters, we can get the image I' after the
Arnold scrambling transformation of the
original image I. Set
1
x
1
y
for the
horizontal and vertical coordinates before the
transformation of the first pixel. Set
'
1
x
'
1
y
for the horizontal and vertical coordinates after
the transformation of the first pixel. Similarly,
k
k
y
'
k
x
'
k
y for the horizontal
and vertical coordinates before and after the
transformation of the K pixel. shown in Figure
1:
Figure 1: Transformation process.
5. Encrypt a image through iteration encryption by
using Henon system. The number of iterations d
determined by the generated chaotic sequence in
step 2, the parameters of Henon system which a
= 2, b = 1. Concrete steps are as follows:
1) Order
X
0
1
=
'
1
A
, Y
0
1
=
'
2
A
.let X
0
1
,Y
0
1
have
d1 iterative times by using Henon system, get
X
d
1
,Y
d
1
,
and then make X
d
1
assign
''
1
A
2) Order X
0
2
=Y
d
1
Y
0
2
=
'
3
A
, let X
0
2
Y
0
2
have d
2
iterative times by using Henon system, get X
d
2
Y
d
2
, and then make X
d
2
assign
''
2
A .
3) This chain iteration, order X
0
k+1
=Y
d
k
Y
0
k+1
=
'
2k
A
+
,
let X
0
k+1
,Y
0
k+1
have d
k+1
iterative times by using
Henon system, get X
d
k+1
,Y
d
k+1
, and then make
X
d
k+1
assign
''
1k
A
+
.
4) Until the order X
0
n-1
=Y
d
n-2
, Y
0
n-1
=
'
n
A , let
X
0
n-1
,Y
0
n-1
have d
n-1
iterative times by using
Henon system, get
X
d
n-1
,Y
d
n-1
, and then make
X
d
n-1
assign
''
1n
A
.
5) Order X
0
n
= Y
d
n-1
,Y
0
n
=
1
''
A
, let X
0
n
Y
0
n
have
d
n
iterative times by using Henon system, get
X
d
n
,Y
d
n
, and then make X
d
n
assign
''
n
A , make
Y
d
n
assign T.
So we can get
''
1
A
''
2
A
''
3
A 3
,…,
''
n
A
that are encrypted plaintext by Henon
system. T is used as one of the decryption keys
in the decryption process..
5. Gray-scale transformation matrix generated
according to the general linear pixel was
recorded as G1, G2, ..., Gn.
Make
''
i
A
and
the corresponding Gi do the bitwise XOR
operation, get
^
A
. As shown in Figure 2:
Figure 2: Gray matrix XOR process.
^
1
A
^
2
A
^
3
A 3
,…,
^
n
A
are
encrypted ciphertext which are got by XOR
operation. It is also the final encrypted plaintext.
Then make
^
i
A revert to grayscale value matrix.
So we got the encrypted image
I
*
.
Output: encrypted image I*T.
Decryption process is the inverse of the encryption
process, here no longer explain.
ICSOFT 2010 - 5th International Conference on Software and Data Technologies
276
3 EXPERIMENTAL RESULTS
AND PERFORMANCE
ANALYSIS
3.1 Analyse the Phase Space
Reconstruction Attacks
Phase space reconstruction break method means: If
the chaos encrypted image is a ciphertext which is
computed directly with the plaintext image pixels
and the corresponding chaotic sequence value, the
pixel value of the ciphertext after computing must
retain the characteristics of the chaotic equation in
the data features, the reason of which is that the
pixel value of the plaintext image pixels changes
more relaxed, but the changes of chaotic sequence
are larger. We can determine which kind of kinetic
equation the encryptor has used, as long as we can
re-drawed Phase space diagram of the ciphertext,
and find out the most similar one in the dynamic
characteristics through the comparison with the
original phase space diagram of all kinds of chaos
dynamics equations. Based on the idea, some
scholars (
Li, Q, Zhao, K, et al 2007, p.45) proposed
some successful examples that using phase space
reconstruction method deciphers stream cipher
encryption method based on chaotic sequence. This
section conducted phase space reconstruction to a
ciphertext generated in the solution by the use of
delay time and embedding dimension, based on the
method of attack proposed in the literature. The
results shown in Figure 3
Just B Component, G and R
Component have been omitted.
Figure 3: B Component of the Phase Space
Reconstruction.
From the above diagram, it is not difficult to see
that there's no meaning of the reconstructed phase
space, and we can not derive any useful information
of the chaotic system. So, the encryption solution
based on chaotic system proposed in this paper has a
very strong immune to the attack methods based on
phase space reconstruction.
3.2 Statistical Characteristics of
Histogram Analysis
Figure 4 shows the histogram statistics of the B
component of the original color map. It can be easily
seen that, the distribution of Blue primary colors is
extremely uneven. While Figure 5 shows the
histogram statistics of the encrypted color image.
Obviously, the histogram of the encrypted image and
the original are very different. The histogram of the
encrypted image is very uniform. It is far better than
the original and covers the distribution law very well.
Therefore, this algorithm can effectively resist
statistical attacks.
Figure 4: Lena image statistic histogram (B).
Figure 5: Encrypted Lena image statistic histogram (B).
3.3 Comparison with other Similar
Solutions
The proposed solution in this paper, although its
scrambling encryption based on Arnold system can
be similarly cracked, as generating decryption
parameter T which is depends on the plaintext in the
COLOR IMAGE ENCRYPTION SOLUTION BASED ON THE CHAOTIC SYSTEM OF LOGISTIC AND HENON
277
second part of the encryption process, so it is
difficult to obtain the parameter of the second part.
Even if we obtained the parameter in the second part,
but we do not know the iterative times and
encryption procedure, we also can not calculate the
parameter T. Therefore, compare with the analyzed
solutions in literature (
Cahit, C, Ercan, S 2008), the
solution in this paper is more safety.
4 CONCLUSIONS
This paper proposed a color image encryption
solution based on Logistic and Henon chaotic
system. Theoretical analysis shows the new solution
has a good diffusion and it can effectively resist
phase space reconstruction; experiment verification
and analysis explain solution method have good
statistical characteristics of histogram and the
distribution of encrypted pixel value is uniformity;
compare with some solutions which are proposed in
literature, we can know that the solution has good
safety and reliability.
ACKNOWLEDGEMENTS
This work is supported by Aero-Science Fund of
China (2009ZD53045), and Innovation Project of
Northwestern Polytechnical University (W016141).
REFERENCES
Li, S J, Zheng, X 2002, ‘On the security of an image
encryption method [A]’, Proceedings of IEEE 2002
International Conference on Image Processing[C].
Omni Press, pp.453-487.
Li, S J, Zheng, X 2002, ‘Cryptanalysis of a chaotic image
encryption method [A]’, Proceedings of IEEE 2002
International Symposium on Circuits and Systems [C].
Phoenix-Scottsdale: Omni Press, pp. 231-262.
Wu, X, Chen, E D & Hu, J S* 2004, ‘An improved chaotic
image encryption method’, Journal of Dalian
University of Technology, vol.44 (5), pp. 754-757.
He, X P, Zhu, Q S & Gu, P 2006, ‘A New Chaos-Based
Encryption Method for Color Image’, RSKT, vol.4062,
pp. 671–678.
Li, Q, Zhao, K, et al 2007, ‘Research on Deciphering
Method of a Kind of Chaotic Encrypting Picture’,
Journal of National University of Defense Technology,
vol. 29(3), pp. 45-49.
Pareek, N K, Vinod, P, Sud*, KK 2006, ‘Image encryption
using chaotic logistic map’, Image and Vision
Computing, vol.24, pp. 926–934.
Zhang, H, Wang, X F, et al 2005, ‘A fast image encryption
algorithm based on chaos system and henon map’,
Journal of Computer Research and Development,
vol.42(12), pp.2137-2142.
Zou, J C, Tie X Y 2000, ‘Arnold transformation of digital
image with two dimensions and its periodicity’, North
China University of Technology, vol.25 (5),
pp.182-189.
Cahit, C, Ercan, S* 2008, ‘Cryptanalysis of a chaos-based
image encryption algorithm’, Physics Letters A, vol.11
(25).
ICSOFT 2010 - 5th International Conference on Software and Data Technologies
278