Research on compound chaos image encryption method
with clock- varying
MENG Xing1, WANG XiaoMan*, CHU Ying1
Department of electronics and communication
Chang Chun University of Science and Technology
Chang Chun,Jin Lin
Abstract—In this paper, we suggest one compound chaos image encryption method with time- varying multilevel initial
parameters. This algorithm combines Subsection-linearity mapping, Chebyshev mapping with Logistic mapping in order to
disturbing values of each image point and the array of original images based on chaotic second-scrambling way. The chaotic
system uses time-varying system clock to change the first chaotic mapping initial parameter, and takes those chaotic
sequences which generated by the first chaotic mapping as the initial parameters for the second chaotic mapping, then
generate two-dimensional image pixel position scrambling matrix, and uses the same system clock to lock Chebyshev
mapping initial parameter, then generate two-dimensional image pixel value transforming matrix. By this way for using the
time-varying system clock, The encryption system can randomly changes many initial parameters of chaotic image
encryption system to enhance image transmission security, has the capability of “one time one secret”. The simulation results
show that this algorithm is simple, safe, easy to achieve with software and has huge secret key space.
Keywords: Image encryption; Chebychev mapping; Logistic mapping; Secret key
1. Introduction
According with the rapid development of network
technology and defense technology, digital image
transmission and recognition in battlefield are more and
more widely. Safety and high-efficiency encryption
method has become hotspot topic. America meteorologist
called Lorenz has originally lodged chaos theory. Chaos is
the complexity movement patterns of a deterministic
nonlinear system, it has too much useful characteristic,
such as: good pseudo-random characteristics,
unpredictability track, inherent randomness, the overall
stability and local instability, extremely sensitive to initial
conditions. Those excellent characteristics make it very
easy to construct cipher system.
There are many different approaches for image
encryption. Traditional methods for image encryption are
based on cryptograph concept such as Data Encryption
Standard (DES) [1] and Advanced Encryption Standard
(AES) [2, 3].They consider image as a data sequence or
stream and encrypt them byte by byte or block by block.
However, their encryption/decryption processes have
huge computation complexity.
The principium of chaos encryption communication is
that the sender disturb image information by using chaos
signal in channel, this makes the image information like
stochastic yawp. The receiver wipe off chaos signal and
resume image information[4].This paper combines
Subsection-linearity mapping, Chebyshev mapping with
Logistic mapping and use system clock to control chaos
system parameters, suggest one compound chaos image
encryption method with time- varying, this method not
only solves small key space and low security of chaos
encryption system, but also makes the encryption system
very complexity and highly stands against cipher attack
capability, and attains “one time one secret”.
2. Chaos Image Encryption Theory
with Time- Varying Parameters
In this paper, use clock information to change initial
parameters of compound chaos encryption system. For the
sender, obtain the clock information, encrypt clock
information by RSA system and send this information in
common channel. At the same time, use time-varying
clock to change multilevel parameters of chaos system and
finish the image encryption. Cryptograph will be sent in
secret channel to receiver. The Fig.1 is the encryption and
decryption scheme.
Compound Chaos
Encryption SystemCrytographCrytograph Plaintext
System ClockRSA Encryption Common Key
Decryption
Compound Chaos
Encryption System
Change control
parameter
Image Decryption
Channel
Channel
Figure 1. Chaos image encryption/ decryption principle with
time-varying parameters
Open Journal of Applied Sciences
Supplement：2012 world Congress on Engineering and Technology
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A. Compound Chaos Encryption System
Firstly we introduce three chaos models in this system.
1) Subsection-linearity mapping [5]
pxx
nn
/
1 
),0( pxn 1
)1/()1(
1pxx nn  
)1,( pxn 2
When 0< p
<1, the Lyapunov exponent is over zero and
will happen chaos phenomena.
2) Logistic mapping [5]
)1(
1nnn xuxx 
3
In above equation, 0 dd u
1,
u
is called control
parameter, when 3.5699456< u
<4, the system will go into
chaos status.
3) Chebychev mapping [6]
)] arccos(cos[
1nn
xkx

4
The definition range of this equation is (-1㧘1), when
the parameter
k
is equal to six, the Lyapunov exponent of
Chebychev system is 1.7917333, so the mapping works in
chaos status.
So, base on the above three chaos mapping, suggest
chaos image encryption method with time-varying
parameters. Fig.2 shows the process of image encryption
and decryption of a simple chaotic system for this method.
Subsection-linearity
Mapping
Logistic
Mapping
Position
Scrambling
Matrix
PixelValue
Transforming
Matrix
Clock Numerical
Value
Cryptograph
Chebychev
Mapping
Original Image
Change control
parameter
0x
u
Chebychev Initial
Paramter
M
6HFUHW.H\ 6HFUHW.H\
Index
Number
M
Members
Initial
Parameter
For
Chaos
Mapping
Figure 2. Compound chaos image encryption systemwith
time-varying parameters
B. Compound Chaos Encryption Process
The main technology of this method describe in
following.
Firstly, we use system clock information to disturb
control parameter p of Subsection-linearity mapping, for
the same secret
0
x
, the firstly Subsection-linearity
mapping can be considered as many different models as to
enhance chaos system capability for attack
parameter p .The last, by using clock information to lock
initial parameter of second chaos Chebychev mapping, it
can enhance security exponent of encryption system, it
means that when both the secret
0
x and control
parameter p
not change, the system clock can make index
number difference ,then make two-dimensional image
pixel value transforming matrix difference. By the above
method, it makes chaos driver system more complexity,
and achieves both pixel position scrambling matrix and
pixel value transforming matrix changed in real time. Even
the same image, if we encrypt it in different time, it will
have different cryptograph. The following is the encrypt
method, the system clock contains year, month, date, hour,
minute, second.
The first step: change year, month, date into numerical
value for hundred bit, ten bit, entries bit, and add those
values, and map result for the value between zero and one.
By the initial secret key, the Subsection-linearity mapping
generates chaos sequence, the number of chaos sequences
are the same as the row numbers (such as M) of image,
and take those values as the initial parameters for Logistic
mapping. For the Logistic mapping will be iterative of N
detracting one times, and generate N sequences, and obtain
NMu
chaos sequences.
The second step: use the
NMu
chaos sequences
which generate in the first step to obtain
NMu
bits
matrix J according to the row first principle. Late, arrange
the elements of J from the large to small, and generate
matrix G by the row first principle. Then, we obtain
one-dimensional pixel position scrambling matrix C by
recording the position coordinate for the elements of
matrix G in matrix J.
The third step: as the first step, make the hour, minute,
second information to numerical values and add the
integral part of those values and map the result for the
value between zero and M. Then, take it as the one index
number of M chaos sequences in the step one and take the
value by this index number as initial parameter of
Chebychev mapping. For the Chebychev mapping will be
iterative of N times, and generate
NMu
chaos
sequences, and change those chaos sequences to pixel
value transforming matrix H by the row first principle. A
last, system uses matrix H and matrix C to finish image
two-dimensional encryption.
Decryption arithmetic is the symmetrical reverse
process. Firstly, operate XOR between the gray matrix H
and cryptograph image, obtain position transforming
image P and use one-dimensional position transforming
matrix C to transform image P. Then , we can decrypt
original image.
3. Simulation Results
In this paper, all the results are simulated on image
“cameraman.tif”(
256256u
) by using Matlab.For the
encryption process, the values of system secret keys are as
following:
0
x
is equal to 0.17,
u
is equal to 3.581, k is
equal to five, other is time-varying clock information. The
clock information is obtained by the clock function in
Matlab before the start of encryption arithmetic. For the
encryption process, the system time is equal to
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2011-5-15-10-05-45.234. For the wrong parameter
decryption process, not change the parameter
0
x
,
u
and
k
, only make the initial parameter difference of
Chebychev mapping by the clock information. The clock
value is 2011-5-15-10-10-13.187 at the starting of
decryption arithmetic. The Fig.3 shows the simulation
results. (a) and (b) are corresponding to original image and
cryptograph, (c) and (d) are corresponding to the
histogram of original image and the histogram of
cryptograph, (e) and (f) are corresponding to decryption
image by right parameters and decryption image by the
wrong parameters, (g) and (h) are corresponding to the
histogram of decryption image by right parameters and
histogram of decryption image by the wrong parameters .
 
(a):Original image (b): Encryption image
 
(c): Histogram of (a) (d): Histogram of (b)
 
(e):Right decryption (f)Wrong decryption

(g): Histogram of (e) (h): Histogram of (f) 
Figure 3. Image encryption / decryption simulation results
Form the above results, we can conclude that original
image has been transformed and replaced and can not be
seen clearly in frank. When use the right parameter
encryption, the decryption image is the same as the
original image, when the parameter has the little difference,
the decryption image is very different from the original
image.
4. Capability Analysis
A. Principle Analysis
In this system, use one-dimensional chaos mapping
drive two two-dimensional chaos mappings and achieve to
transform and replace. Comparing to literature[7,8], use
time-varying clock information to change control
parameter of one-dimensional chaos mapping, So, change
chaotic parameter one time, it means replacing chaos
equation one time. For one encryption process, not only
change control parameter of one-dimensional chaos
mapping, but also change the initial parameter of
two-dimensional chaos mapping. Then it can enhance
complexity of chaos sequence.
B. Secret Key Space and Secret Key Sensitivity Analysis
Secret key of this chaos system contains four members,
initial parameter
0
x of Subsection-linearity mapping,
control parameter u
of Logistic mapping, control
parameter k of Chebychev system and time-varying
secret key(in this system is clock information).By this ,it
can huge enhance secret key numbers and secret key space.
The secret key space is many times than single one-
dimension chaos mapping. It can fully resist infinitude
attack.
Sensitivity analysis: an ideal image encryption
algorithm should be sensitive to both the cryptic key and
plaintext image. Change on one pixel bit in cryptic key or
the plaintext will produce one different encryption image.
It also means that the small difference on decryption
cryptic key will not correctly decrypt the cryptograph
image. In this system, only change clock information and
obtain clock information before encryption algorithm start,
one is 2011-5-15-22-7-48.218, another is
2011-5-15-22-14-23.203. Then compare two cryptograph
images. Fig.4 shows the difference of two cryptograph
images by the different secret key encryption. From the
results, we can see that the cryptograph will be quite
different by the little change on the secret key for the same
original image. The more times results show that the
cryptograph will be quite different even little change on
the same secret key
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Figure 4. Pixel value difference distributing of two Ciphertexts
C. Histogram Analysis
The histogram is used for describing gray number of
one image, it can give the statistical characteristic of one
image. After comparing (c) and (d) of Fig.3, we can see
that the pixels of original image are very asymmetric in all
Due to big secret key space, sensitive to the secret key,
and strong statistical analysis of resisting the attack. Chaos
image encryption technology has been widely used, and
also reflected good encryption effect in practice. In this
paper, describe one chaos image encryption method with
the gray value, but the pixels of cryptograph image is very
symmetrical in the gray value. So, the encryption
algorithm destroys statistical characteristic of original
image and fully diffuse and confuse cryptograph and
plaintext.
D. Correlation of pixel
We also analyze the correlation of two adjacent
vertical direction pixels, two adjacent horizontal pixels and
two diagonal adjacent pixels of plaintext image and
cryptograph image. There, analyze the correlation of
original image 3(a) and cryptograph image 3(b). Analysis
process is as following: Firstly, randomly select 16384
pairs of adjacent pixels from image. Then use the
following two formulas calculating their correlation
coefficients:
))(())((),cov( yEyExExEyx 5
)() (),cov( yDxDyxrxy 6
Where,
x
and
y
are the adjacent pixels of the image.
Among the numerical value counting.
Fig.5 describes the correlation of two adjacent
horizontal direction pixels of original image and
cryptograph image. In the picture, the abscissa represents
the pixels value of original and cryptograph in position
(i,j), the y-axis represents the pixels value of original and
cryptograph in position (i,j+1). At the same time, Table 1
gives the correlation of three directions. From the table1,
we can see small correlativity of cryptograph, but high
correlativity of original image.
(a): original image (b): Encryption image
Figure 5. Correlation of Horizontal Direction Adjacent Pixel
TAB.1 THE CORRECTION COEFFICIENTS OF PLAINTEXT IMAGE AND
CRYPTOGRAPH IMAGE
Direction of
Adjacent Pixels
Original Image Cryptograph Image
Horizontal 0.9931 0.0307
Vertical 0.9806 0.0295
Diagonal 0.9685 0.0304
5. Conclusions
time-vary multilevel parameters. The outstanding
characteristic is that use time-vary clock information to
change many parameters of chaos system. It not only
enhances secret keys and secret space, but also achieves
the encryption requirement of “one time one secret”. The
simulation results show that this algorithm is simple, safe,
easy to achieve with software and has huge secret key
space.
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