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Chaotic encryption system based on pixel value and
position transformation for color images
Atheer J. Mansoora, Hikmat N. Abdullahb , M. F. Al-Gailanib, And Hadi T. Ziboonc
a Al-Turath University College, Baghdad, Iraq,
atheer.jabar@turath.edu.iq
b Al-Nahrain University, College of Information Engineering, Baghdad, Iraq
dr.h.abdullah@ieee.org
m.falih@coie-nahrain.edu.iq
c University of Technology, Baghdad, Iraq
Dr.hadi.zeboon@gmail.com
Abstract—In this paper, a full image encryption
system is proposed. The proposed system depends on
changing the value and coordinates of the pixels at
the same time. The system has the ability to be
applied on all types of images with any dimensions.
The results of the system were tasted and compared
with the traditional algorithms. The results that have
been obtained from the simulation show that the
proposed system has higher degree of security and
faster encryption time.
Keywords—security, image encryption, image
scrambling, chaotic sequences, pixels coordinate.
I INTRODUCTION
Image encryption is a method for protecting the
information in a digital image. There are two ways for
image encryption :- the first one is changing image
pixels' value and the other is changing image pixels'
position (scrambling). The Highest degree of image
encryption is by changing both the pixels' values and
coordinates. The resulting image is an encrypted image
that has the same size of the original image with different
histogram and redistributed of pixels in the original
image[1]. Chaotic systems are used to produce the
random numbers which are used for the encryption
process because of its characteristics of random behavior
and aperiodicity[2-3]. Many algorithms are proposed for
even changing pixels' value [3], or changing pixels
position. Many algorithms depend on using two chaotic
systems. The first one for changing the positions such as
ACM. While the second one for pixel’s value alternating.
[2][4].
In this paper, a new encryption algorithm for image
encryption based on changing image pixels' value and
position using chaotic flow sequences is proposed. At
first, the proposed system is used to change the M*N
image to a vector of 1*(M*N). then the user has to select
one of five chaos flow systems which are: Lorenz,
Rössler, Chua, Nien and CL for the encryption process.
The encryption process depends on the 3-D chaotic
system. Where the algorithm depends on the first
dimension for pixels position changing. While the other 2
dimension used to change the value of the pixels. The
main contributions of the proposed work are as follows:
the use of the chaotic flows increases the immunity
against the intruders as compared with the chaotic Maps
because it is consists of three bands or higher. Also, the
proposed system is applicable to equal and non-equal
dimensional images. Changing pixels' value and position
is done in one iteration and using one chaotic system for
the whole process.
The rest of the paper is organized as follows: in
section 2, the chaotic systems that are used in the
proposed system are described in details. In section 3, the
quality measurements of encryption that are used to
evaluate the performance of the proposed system are
reviewed. section 4 the proposed algorithm and the
simulation results. The last section is the conclusions
drown throughout the work.
II DESCRIPTION OF CHAOTIC SYSTEMS
In this section, the mathematical models of the
chaotic systems used are reviewed as shown in table1.
First, the classical Arnold map used for image square
images .Then, the chaotic flow sequences used in the
proposed scheme are described.
TABLE I. CHAOTIC SYSTEMS
Reference equations
Arnold
cat map
𝑋
𝑌1𝑎
𝑏𝑎𝑏1
𝑋
𝑌 𝑚𝑜𝑑𝑁 (1)
1- scrambling by changing the pixels positions
2- applied on images with size of N*N
3- a and b are positive integers.
4- For better scrambling, repeating process is necessary
to get better result.[5]
Lorenz
system x σ
y
x
𝑦 𝑟∗𝑥𝑦𝑥∗z (2)
𝑧 𝑥∗𝑦𝛽∗𝑧 ,
1- σ, r and β are the parameters of the system. The
values of σ, r and β must be greater than 0, σ > β+1,
and𝑟𝑟𝑐
[3]
Rössler
system 𝑥 𝑧𝑦
𝑦 𝑥𝑎𝑦 (3)
𝑧 𝑏𝑧𝑥𝑐
1- a, b and c are the parameters of the system.[6]
Chua
system 𝑥 𝛿𝑦𝑥ℎ
𝑥
𝑦 𝑥𝑦𝑧 (4)
𝑧𝑏𝑦𝛾𝑧 ,
1- 𝛿, 𝑏 𝑎𝑛𝑑 𝛾 are the system parameters, and
2 ℎ𝑥 𝑚1 0.5𝑚0 𝑚1|𝑥 1| |𝑥 1|
developed from the Chua electrical circuit that had non-
linear behavior.[7]
Nien 𝑥 𝛿𝑥𝑦𝑓
𝑥
2021 18th International Multi-Conference on Systems, Signals & Devices (SSD'21)
978-0-7381-4392-7/21/$31.00 ©2021 IEEE 433
2021 18th International Multi-Conference on Systems, Signals & Devices (SSD) | 978-1-6654-1493-7/21/$31.00 ©2021 IEEE | DOI: 10.1109/SSD52085.2021.9429368
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system 𝑦 𝛽𝑥𝑦𝛾𝑧 (5)
𝑧
y
1- 𝛿,𝑏 𝑎𝑛𝑑 𝛾 are the system parameters
2- 𝑓𝑥𝑏𝑥 0.5
𝑎𝑏|𝑥 𝐼𝑜||𝑥 𝐼𝑜|
3- δ= 6.3; β=0.7; γ= 7; a=-1.143; b=-0.714; Io=3 [8]
III SCRAMBLING QUALITY MEASUREMENTS
Many measurements are used to evaluate the
strength of the scrambling algorithm quality. This
section briefly discusses these measurements.
A.
Distance scrambling factor (DSF)
This is the first theory for scrambling measurement.
Here, the distance factor is calculated by comparing the
pixels positions between the original image and the
scrambled one. The calculation of distance scrambling
factor degree could be described as follows:
For original image of pixels positions (i, j) and
scrambled image of pixels positions (i',j'), the moving
distance could be calculated as:
D(i,j)=𝑖𝑖𝑗𝑗 (6) ,
The average (mean) moving distance will be:
𝐴𝐷
∗ (7)
Where M*N is the dimension of the image. The
maximum distance could be calculated from:
Amax (D) =𝑀1𝑁1 (8)
So, the Distance scrambling factories
DSF=A(D)/Amax(D ) (9)
The greater DSF means higher degree of scrambling [9-
10].
B. Average distance change (ADC)
This method is used to calculate the average distance
change in image pixels. The ADC for non-equal
dimensions image is :
𝐴𝐷𝐶𝑖,𝑗
𝐷𝑖1,𝑗,𝑖1,𝑗 𝐷𝑖
1,𝑗,𝑖1,𝑗 𝐷𝑖,𝑗1,𝑖,𝑗1
𝐷𝑖,𝑗1,𝑖,𝑗1 , (10 )
where
𝐷𝑖,𝑗,𝑖,𝑗𝑖𝑖𝑗𝑗 (11)
where (i', j') is the position of scrambled image pixel and
(i,j) is the position of original image pixel. The average
distance of the whole image could be calculated from the
following equation:
𝐴𝐷𝐶
∗∑∑𝐴
𝐷𝐶𝑖,𝑗 ,(12)
If the result from equation (12) equal zero, that’s
mean the cipher image is the plaintext image. Bigger
ADC means higher scrambling [11]
C. 2-D correlation coefficient
This method is used to obtain the degree of
similarity between two images (matrices) by calculating
the correlation coefficient between the two images with
the same size M*N by the following equation:
𝑟 ∑∑
∑∑
∑∑
(13)
Where X and Y are two images (matrices), X
and Y
are the mean value of the elements of X and Y
respectively. The range of the r values is between +1 and
-1, where if r=+1 that’s mean X and Y are identical,
while if r=0 the two image are totally different, and if
r=-1 that's mean that X and Y are identical but with
phase shift of 180ᵒ (mirrored) [12-14].
D. Peak Signal-to-Noise Ratio (PSNR)
Peak Signal-to-Noise Ratio (PSNR) can be used to
evaluate the encryption scheme strength by indicating
the pixel's value between the original image and the
encrypted image. It can be calculated by using the
following equation
PSNR10∗log ∗∗
∑∑
,, (14)
Where O and E are the original and the encrypted
image respectively, (i,j) is the coordinate of the pixel and
M,N are the image size.
Lower PSNR means higher encryption effectively[11-
13]
E. Measurement Based on the Value Changing
This test depends on the comparison between the
pixel's value before encryption with the pixel's value
after encryption to obtain the changes whether regular or
irregular.
The encryption quality could be obtained by
measuring the deviation between the plain image and the
cipher image as shown in the equation below:-
EQ ∑||
... (15)
Where, HC and HP are the number of pixels at
gray level i of the cipher and plain images respectively
There is another encryption quality measurement
proposed by Luo et al., which could be evaluated by
calculating the relative error of each pixel in the cipher
image with its counterpart in the plain image, which is
done by using the following equation
ARE
∗∑∑|,,|
|,|
...(16)
Where P and C are the plain and cipher images of
size M*N respectively [15-16]
F. Maximum Deviation Analysis
Maximum Deviation analysis is a statistical analysis
which is used to calculate the deviation between the
original and the encrypted images. This analysis could
be calculated according to the following equation:-
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MD
∑h
(17)
h=|H-H'| (18)
Where H and H' are the histogram distribution of the
original and the encrypted image. Higher MD means
higher encryption degree and the encrypted image
faraway (deviated) from the original one[17]
G. Irregular Deviation Analysis
The irregular deviation is used to measure how the
encryption process changes the values of the original
image irregularly and randomizes it in uniform manner
to make the statistical distribution of changing the pixels'
values uniform
The irregular deviation analysis can be obtained
from the following steps:-
Calculate the absolute difference between the
original and the encrypted images:-
D=|O-E| (19)
Where O and E represent the original and encrypted
images respectively.
Obtain the histogram of the absolute difference
H=histogram(D) (20)
Calculate the average value of the histogram
deviation Av
∑
(21)
Evaluate the absolute differences between the
histogram deviation and the average deviation value
H|hAv| (22)
Compute the irregular deviation factor value
I∑H
(23)
Smaller ID means higher encryption strength, where
it is a pointer for the uniform distribution between the
original and encrypted pixels' values[18]
H. Measurement Based on the Value and Position Changing
This test depends on the comparison between the
pixel's value before encryption with the pixel's value
after encryption to obtain the changes whether regular or
irregular.
The encryption quality could be obtained by
measuring the deviation between the plain image and the
cipher image as shown in the equation below:-
EQ ∑||
(24)
where, HC and HP are the number of pixels at
gray level i of the cipher and plain images respectively
This measurement could be found by the flowing
steps:-
Obtain the relationship between each pixel and its
four neighbors as shown below
D1i,j|Ei1,jEi,jOi1,jOi,j|
D2i,j|Ei1,jEi,jOi1,jOi,j|
D3i,j|Ei,j 1Ei,jOi,jOi,j1|
D4i,j|Ei,j 1Ei,jOi,j1Oi,j|
Ri,j D1i, j D2i,j D3i,j D4i, j (25)
Calculate the scrambling degree S as below
S∑∑,
∗∗ (26)
The value between 0 and 1, and greater S means
better encryption results.[19]
I. Entropy
The encryption quality could be measured by
calculating the entropy of the plain image and the
entropy of the cipher image, then comparing between
them. Image entropy could be found from:-
E∑ Pi∗ log
(27)
The maximum entropy equals 8, and it is referred to
as an ideal case of randomness. Practical calculation of
the entropy must be less than the maximum value.[20]
IV THE PROPOSED ALGORITHM
The proposed scheme performs the image
encryption by using chaotic systems. The algorithm of
the proposed scheme could be explained throughout the
following flow chart that is shown in figure 1.
The decryption process is done by following the
same steps of the encryption process with using the same
chaotic system, same initial values and same parameters.
Any difference in any parameter or initial value makes
the decryption process incorrect.
It is worth noted that the decryption processes for
vectors R, G and B, shown in figure1 as three successive
blocks, are identical. The encryption process for each
vector could be explained as following:-
1- Label each pixel with 0
2- Check the label of the pixel if
a- Equal 1, move to the next pixel (it has been
encrypted) and return to step 2.
b- Equal 0, continue the process and change the
pixel’s label to 1
3- Select a pixel randomly using the chaotic system
4- Check the label of the pixel if
c- Equal 1, (it has been encrypted) return to step
3.
d- Equal 0, continue the process and change the
pixel’s label to 1
5- Exchange the two pixels that have been selected
6- Xoring the two pixels with the other two number
that have been obtained from the chaotic system.
7- Repeat steps 2 to 6 until all pixels are encrypted
Figure 2 shows the process of encryption
implementation on each vector.
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Figure1. The Flowchart of the proposed algorithm
Figure.2. The flow chart of encryption process for vector R block of
figure 1
V Simulation Results
For equal dimensions, the image sizes that are
taken for the system implementation is (256*256). Table
2 clarify the parameters for each system used in the
proposed system
TABLE II. C
HAOTIC SYSTEMS PARAMETERS
For a fair comparison, the initial value for
systems used is the same (X0=0.1, Y0=0.1 and Z0
=0.1).Figure 3 shows the simulation results of the
proposed system.
a) Lane image b) Histogram of original image
c) Lorenz system result d) Histogram of encrypted image
e) Rossler system result f)Histogram of encrypted image
g) Chua system result h) Histogram of encrypted image
i) Nien system result j) Histogram of encrypted image
k) CL system result l) Histogram of encrypted image
Figure 3 . Results of the proposed algorithm for image of size 256*256 using
chaotic flow sequences.
Table 3 shows the statistical measurements of
the encryption system results comparing with the best
results of the traditional scheme (the best value of each
case is taken for each iteration). The yellow cells are the
best result of the tests, while green cells show the
number of iterations in the traditional schemes to reach
the best result.
As shown from the table, it is very clear to
discover that :-
1- the proposed system reaches the same degree of
immunity in one iteration with one chaotic system
2- the proposed system could be implementing by
converting the 2-D image into 1-D by two ways:
a- row by row
b- column by column
Chaotic Flow System Parameters
Lorenz a=10, b=24, c=8/3
Rössler a=0.2, b=0.2, c=5.7
Chua δ =10, b=14.78, γ=0.0385, m
1
=-1.27, m
0
=-0.68
Nien δ =6.3, β=0.7, γ=7, b=-0.714, a=-1.143, I
o
=-3
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3- The traditional method needs 2 chaotic systems
(Arnold for scrambling and another system for
changing pixel’s value)
4- The traditional method needs more than 1 iteration to
get the highest degree of security.
5- Better result in the traditional scheme could be
obtained after testing each iteration (more wasted
time and more complexity in the process not in the
result)
Figure 4 shows the required time for encryption
and decryption process of the proposed algorithm and
the traditional scheme of each image size. It is very clear
that all the proposed algorithm systems are faster than
the traditional one about 4 times.
Figure 4 The required time for (encryption and decryption processes) of
different implemented systems
TABLE III. THE STATISTICAL MEASUREMENTS OF THE ENCRYPTION SYSTEM
RESULT COMPARING WITH TRADITIONAL METHOD
system type type /
iteration ARE EQ CORR. ID MD PSNR S ENTROPY
Rossler
p
ro
p
ose column 131.241 11.310 -0.00523 2214.208 2877.333 8.8227 0.6618 7.98203
Arnold
Rossler 19 131.244 11.492 -0.00612 1867.417 2922.333 8.7929 0.6818 7.98261
Chua
p
ro
p
ose column 131.235 11.409 -0.00442 2764.625 2901.000 8.8404 0.6554 7.98413
Arnold
Lorenz 3131.230 11.589 -0.00979 2616.875 2945.333 8.7525 0.6685 7.98690
Best S
Best ENTRO PY
system type type /
iteration ARE EQ CORR. ID MD PSNR S ENTROPY
Nien propose column 131.250 11.357 0.00078 2649.417 2884.000 8.8582 0.6512 7.98327
Arnold Chua 48 131.281 11.552 0.01148 2570.042 2932.833 8.8699 0.6520 7.98354
system type type /
iteration ARE EQ CORR. ID MD PSNR S ENTROPY
Rossler
p
ro
p
ose column 131.242 11.586 -0.00441 2278.708 2942.167 8.7960 0.6524 7.98381
Arnold
Lorenz 23 131.252 11.948 -0.00532 3284.458 3036.333 8.7281 0.6544 7.98258
system type type /
iteration ARE EQ CORR. ID MD PSNR S ENTROPY
Nien propose column 131.250 11.357 0.00078 2649.417 2884.000 8.8582 0.6512 7.98327
Arnold Chua 60 131.270 11.354 0.00010 2888.625 2888.000 8.8450 0.6534 7.98145
system type type /
iteration ARE EQ CORR. ID MD PSNR S ENTROPY
CL proposed column 131.228 11.523 -0.01279 2639.042 2926.833 8.7817 0.6611 7.98333
Arnold CL 25 131.222 11.706 -0.03308 3007.917 2976.833 8.605 0.66920 7.98163
Best ARE and ID
Best EQ and MD
Best CORR.
Best PSNR
An advantage of the proposed system is the ability of the
implementation on non-equal dimensions image. The
traditional system cannot be implemented only on equal
dimension images. For non-equal dimensions, the image
size that is taken for the system implementation is
473*846. Figure 5 shows the original image of size
473*846 with the resulting encrypted images after
implementing the proposed algorithm. After one
iteration only, the encrypted image is decrypted using
any type of the chaotic flow sequences considered as
shown.
a) original image b) Histogram of the original
c) encrypted image (initial x0=y0=z0=0.1) d) Histogram of encrypted image
Figure 5 Lena original and the encrypted image using 4th propose algorithm
(473*846)
Table 4 shows the statistical measurements of
the encryption system results for non-equal dimension
images. CL column has the best results according to the
statistical measurements shown in table 4 (the yellow
cells). According to the statistical analysis of the
algorithm, it has an excellent performance with high
encryption degree and a short time required for the
implementation.
TABLE IV. THE STATISTICAL MEASUREMENTS OF THE ENCRYPTION SYSTEM
RESULT FOR NON-EQUAL DIMENSION IMAGE
system type ARE EQ CORR ID MD PSNR S
column 1385.708 93.302 0.00130 300905.255 353553.000 8.562 0.66939
raw 1388.555 93.305 0.00084 297734.667 354278.833 8.557 0.66937
column 1387.849 93.303 0.000 305964.667 354106.167 8.557 0.670
raw 1387.617 93.302 ‐0.001 299280.432 354063.000 8.552 0.669
column 1385.073 93.307 0.00076 299108.432 353388.333 8.567 0.66809
raw 1385.073 93.305 ‐0.00048 292406.313 353374.333 8.557 0.66745
column 1387.891 93.303 0.000 292806.589 354114.000 8.553 0.67030
raw 1386.398 93.305 0.00184 298029.844 353704.000 8.565 0.66801
column 1390.299 93.300 0.000 298424.901 354729.500 8.558 0.669
raw 1386.753 93.304 ‐0.00007 301607.922 353819.833 8.559 0.669
Lorenz
Rossler
Chua
Nien
CL
Figure 6 shows the required time for (encryption
and decryption processes) of the proposed algorithm on
Lena non-equal dimensions image. For the results that
are shown in figure 7, it could be noticed that the
algorithm has relatively very high speed even with
images with large size. Lorenz column is the fattest
system in the proposed system. While
1,18 1,26 1,07 1,03 0,98 0,95
1,33 1,30
0,92 0,98
4,14
0,00
1,00
2,00
3,00
4,00
5,00
timeinsecond
437
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Figure 6 The required time for encryption and decryption processes for non-
equal image dimensions of different implemented systems
VI. CONCLUSIONS
A pixel value and position transformation for color
images is proposed. The first step is converting the 2-D
image into 1-D vector. Then the positions and values of
the pixels are changed according to the generated
number from the chaotic system. The image then is
reconstructed after changing all pixels' locations and
values. The scheme performance is effectively evaluated
by applying three types of tests which are Distance
Scrambling Factor (DSF), Average Distance Change
(ADC), 2-D correlation coefficient, Entropy required
Time, Maximum Deviation , Peak Signal To Noise
Ratio, Irregular Deviation and Measurement Based On
The Value And Position Changing (S) The experimental
result shows that the proposed system has a higher
degree of encryption from the first iteration comparing
with the traditional encryption system such as Arnold
Cat Map. Another advantage of the proposed system is
the implementation on equal and non-equal dimension
images. Future improvement of the proposed system can
include using hybrid chaotic flows and maps, changing
the value of the pixels in the same time when changing
their position to obtain higher degree of security.
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