Content uploaded by Hikmat N. Abdullah

Author content

All content in this area was uploaded by Hikmat N. Abdullah on Jan 12, 2022

Content may be subject to copyright.

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

Authorized licensed use limited to: Carleton University. Downloaded on June 15,2021 at 07:58:42 UTC from IEEE Xplore. Restrictions apply.

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:-

434

Authorized licensed use limited to: Carleton University. Downloaded on June 15,2021 at 07:58:42 UTC from IEEE Xplore. Restrictions apply.

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.

435

Authorized licensed use limited to: Carleton University. Downloaded on June 15,2021 at 07:58:42 UTC from IEEE Xplore. Restrictions apply.

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

436

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

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.

REFERENCES

[1] PENG Jing-yu, "Efficient Color Image Encryption and

Decryption Algorithm,"

I

nternational Journal of Digital

Content Technology and its Applications(JDCTA), vol. 7,

no. 6, pp. 129-136, 2013.

[2] Yaobin Mao, Liu Cao, and Wenbo Liu, "Design and

FPGA Implementation of a Pseudo-Random Bit

Sequence Generator Using Spatiotemporal Chaos," in

I

nternational Conference on Communications, Circuits

and Systems, vol. 3, Guilin, China, June 2006, pp. 2114–

2118.

[3] Tiegang Gao and Zengqiang Chen, "A new image

encryption algorithm based on hyper-chaos," Physics

Letters, Elsevier, pp. 394–400, 2008).

[4] Li Wenzhuo , Liao Lejian, Li Hong Zhu Liehuang, "A

N

ovel Image Scrambling Algorithm for Digital

Watermarking Based on Chaotic Sequences,"

I

nternational Journal of Computer Science and Network

Security (IJCSNS), vol. 6, no. 8B, pp. 125-130, 2006.

[5] Hikmat N Abdulah , Zeboon Hadi T Mansoor Atheer J,

"High Speed Non-equal Dimension Color Image

Scrambling using Chaotic Flow Sequences," in

Proceedings of International Conference on Change,

I

nnovation, Informatics and Disruptive Technology

(ICCIDT'16), London , 2016, pp. 419-433.

[6] Pierre Gaspard, "R¨ossler systems," Encyclopedia o

f

Nonlinear Science, pp. 808-811, 2005.

[7] Atheer J Mansor, Hikmat N Abdalla, and Hadi T Ziboon,

"Digital Image Scrambling Using Chaotic Systems Based

on FPGA," in Third Scientific Conference of Electrical

Engineering (SCEE), baghdad, 2018, pp. 19-24.

[8] C.K. Huang , S.K. Changchien, H.W. Shieh, C.T. Chen,

Y.Y. Tuan H.H. Nien, "Digital color image encoding and

decoding using a novel chaotic random generator,"

Chaos, Solitons and Fractals, vol. 32, no. 3, pp. 1070–

1080, 2007.

[9] Min Li, Ting Liang, and Yu-jie He, "Arnold Transform

Based Image Scrambling Method," in 3rd International

Conference on Multimedia Technology, China, 2013, pp.

1309-1316.

[10] Anna Brzozowska Ibrahiem MM El Emary, Shaping the

F

uture of ICT: Trends in Information Technology,

Communications Engineering, and Management.: Crc

Press, 2017/9/19.

[11] Alireza Jolfaei and Abdolrasoul Mirghadri, "A New

Approach to Measure Quality of Image Encryption,"

(IJCNS) International Journal of Computer and Network

Security, pp. 38-43, 2010.

[12] Zhenjun Tang and Xianquan Zhang, "Secure Image

Encryption without Size Limitation Using Arnold

Transform and Random Strategies,"

J

OURNAL OF

MULTIMEDIA , pp. 202-206, 2011.

[13] Priyanka Gupta, Sonia Singh, and Isha Mangal, "Image

Encryption Based On Arnold Cat Map and S-Box,"

I

nternational Journal of Advanced Research in Computer

Science and Software Engineering, pp. 807-812, 2014.

[14] Lahcene Merah, Adda Ali-Pacha,

N

aima Hadj Said, and

Mustafa Mamat, "Design and FPGA Implementation o

f

Lorenz Chaotic System for Information Security Issues,"

Applied Mathematical Sciences, pp. 237 - 246, 2013.

5,45 5,54 5,68 6,22

7,81 7,98

6,31 6,87 6,71 6,51

0,00

1,00

2,00

3,00

4,00

5,00

6,00

7,00

8,00

9,00

time in seconds

438

[15] Matt S. Willsey, Kevin M. Cuomoy, and Alan V. Opp

Oppenheim, "Selecting the Lorenz Parameters for

Wideband Radar Waveform Generation,"

I

NTERNATIONAL JOURNAL OF BIFURCATION AND

CHAOS, pp. 1-12, 2010.

[16] Atheer J. Mansor, Hikmat N. Abdullah, and Hadi T.

Zeboon, "Robust Encryption System Based on Novel

Chaotic Sequence," Research Journal of Applied

Sciences, Engineering and Technology, vol. 14, no. 1, pp.

48-55, 2017.

[17] Hadi T Zeboon, Atheer J Mansor Hikmat N Abdullah,

"Digital Image Encryption by Random Pixel Selecting

Using Chaotic Sequences," Al-Ma'mon College Journal,

vol. 26, pp. 228-236, 2015.

[18] Sudhir Keshari and S. G. Modani, "Image Encryption

Algorithm based on Chaotic Map Lattice and Arnold cat

map for Secure Transmission," International Journal o

f

Computer Sci ence and Technology (IJCST), vol. 2, no. 1,

pp. 132-135, march 2011.

[19] Manjunath Prasad and K.L. Sudha, "Chaos Image

Encryption using Pixel shuffling," Computer Science &

Information Technology (CS & IT), vol. 2, pp. 169–179,

2011.

[20] Jianghong Bao and Qigui Yang, "Period of the discrete

Arnold cat map and general cat map," Nonlinear

Dynamic, vol. 70, pp. 1365–1375, august 2012.

439