An Image Encryption & Decryption Approach Based on Pixel Shuffling Using 3D Logistic Map & 3D Henon Map

An Image Encryption & DecryptionApproach Based on Pixel ShufflingUsing 3D Logistic Map & 3D Henon

Map

Project ID: 3005

B.Tech. Project ReportSubmitted in partial fulfillment of the requirements for the

Degree of Bachelor of TechnologyUnder Biju Pattnaik University of Technology

Submitted by

AMIT KUMAR PRADHAN Roll No. # 201014583

ANSHUMAN PANDA Roll No. # 201018214

2013 - 2014

Under the guidance of

Mr. Agyan Kumar Prusty

NATIONAL INSTITUTE OF SCIENCE & TECHNOLOGYPalur Hills, Berhampur- 761008, Orissa, India

2

ABSTRACT

Encryption is mainly used to secure the data that will be

transmitted over networks. There are so many techniques

introduced which are used to protect the confidential image

data from any unauthorized access. Multimedia data contains

different types of data that includes text, audio, video,

graphic, images with the increasing use of multimedia data

over internet, here comes a demand of secure multimedia data.

Most of the encryption algorithm available is generally used

for text data and not suitable for multimedia data. In our

project we have implemented approach for an image encryption

and decryption by 3D logistic and 3D Henon's maps approach

based on Pixel shuffling using chaos theory. We create

combinations between two adjacent pixels; to create linear

independence relationships in the same row. Where, the keys

will be stored in the first column in an encrypted manner by

using a Henon's map with initial condition known as Key1.

After that, we used another key called Key2 for the logistic

map; to change the position of the pixels by shuffling the

pixels position and it’s used for image scrambling.

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ACKNOWLEDGEMENT

It is our proud privilege to epitomize our deepest sense of

gratitude and indebtedness to our guide, Mr. Agyan Kumar

Prusty for his valuable guidance, keen and sustained interest,

intuitive ideas and persistent endeavor. His inspiring

assistance, laconic reciprocation and affectionate care

enabled me to complete my work smoothly and successfully.

I shall be failing in my duties if, I do not express my

deep sense of gratitude towards Dr. Diptendu Sinha Roy, B-

Tech Project Coordinator, who has been a constant source of

inspiration for me throughout this report.

I acknowledge with immense pleasure the sustained interest,

encouraging attitude and constant inspiration rendered by

Prof. Sangram Mudali, Director, and Prof. Geetika Mudali,

Placement Director of N.I.S.T. His continued drive for better

quality in everything that happens at N.I.S.T. and selfless

inspiration has always helped us to move ahead.

Last but not the least I cannot forget the co-operation myfriends in this endeavor.

Amit kumar pradhan

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Anshuman panda

TABLE OF CONTENTS

ABSTRACT.....................................................i

ACKNOWLEDGEMENT.............................................ii

TABLE OF CONTENTS..........................................iii

LIST OF FIGURES..............................................v

LIST OF TABLES..............................................vi

LITERATURE REVIEW..........................................vii

CHAPTER 1....................................................1

INTRODUCTION.................................................1

CHAPTER 2....................................................3

THE LOGISTIC MAP.............................................3

2.1 Logistic map as a random number, generator –.................3

CHAPTER 3....................................................4

HÉNON MAP....................................................4

iv

CHAPTER 4....................................................5

INDEPENDENCE BETWEEN PIXELS..................................5

4.1 Creating Linear Independence Pixels..........................5

PRESENT WORK.................................................9

CHAPTER 5....................................................9

EXPERIMENTAL RESULTS.........................................9

5.1 Keys Space analysis.........................................10

5.2 Information Entropy Analysis................................10

5.3 Key Sensitivity Test........................................10

5.4 Histogram Analysis..........................................11

CHAPTER 6...................................................12

CHAOS THEORY................................................12

CHAPTER 7...................................................13

PSEUDORANDOM NUMBER GENERATOR...............................13

6.1 Why need Pseudorandom Number Generator......................13

CHAPTER 8...................................................14

FUTURE WORK.................................................14

CONCLUSIONS.................................................15

REFERENCE...................................................16

APPENDIX....................................................17

Henon_Map (Rendom Number).......................................17

Logistic_map....................................................19

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LIST OF FIGURES

FIGURE 2.1 BIFURCATION DIAGRAM OF THE LOGISTIC MAP........................2FIGURE 3.1 THE STRANGE ATTRACTOR OF HENON MAP.............................3FIGURE 4.1: ENCRYPTION WITH BYTE SEQUENCE.................................6FIGURE 4.2 PROPOSED ENCRYPTION ALGORITHMS.................................7FIGURE 5.1 ENCRYPTION BY HENON CHAOTIC SYSTEM AND SHUFFLING PIXEL POSITION8FIGURE 5.2 DECRYPTION BY HENON CHAOTIC SYSTEM AND LOGISTIC FUNCTION.......9FIGURE 5.4: HISTOGRAM ANALYSIS: (A) HISTOGRAM OF ORIGINAL IMAGE; (B)

HISTOGRAM OF SHUFFLED IMAGE; (C) HISTOGRAM OF CIPHER IMAGE; (D) HISTOGRAM OF DECRYPTED IMAGE.........................................10

FIGURE 5.1 A PLOT OF THE LORENZ ATTRACTOR FOR VALUES R = 28, Σ = 10, B = 8/3..................................................................11

FIGURE 8.1 THE BIFURCATION DIAGRAM OF THE SECOND PROPOSED LOGISTIC MAP (A) VERSUS A FOR DIFFERENT VALUES OF.....................................13

vi

α = (0.1, 0.4, 0.7, 1.0, 100) and (b) versus a for different values λ = (1.5, 2.5, 4.0)......................................................13

LIST OF TABLES

TABLE.1: ORIGINAL IMAGE 5 X 5.............................................4TABLE.2: X-OR THE ADJACENT PIXELS........................................5TABLE.3: ENCRYPTING THE FIRST COLUMN......................................5Table.4: Example of the shuffling approach................................8

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LITERATURE REVIEWCryptography is standard field of encryption that could be

help for secure data form adversary. It encompasses many

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problems like encryption, authentication, and key distribution

to name a few. The field of modern cryptography provides a

theoretical foundation based on which one can understand what

exactly these problems are, how to evaluate protocols that

purport to solve them and how to build protocols in whose

security one can have confidence SSL (Secure Sockets Layer) is

a cryptographic protocol serving to encrypt the connection

between the hosting server that stores information and the web

client (the browser of the visitor of the website). This

encrypted connection provides secure access to personal and

business information which should not be available to third

parties. SSL is a technology already used by millions of

websites through which they provide protection for online

transactions with their customers.

Digital images are widely used in various applications, that

include military, legal and medical systems and these

applications need to control access to images and provide the

means to verify integrity of images. So, depend upon critical

situation we can secure the particular image before send to

destination and an image could be encrypt by different type of

method.

The logistic map is one of the simplest and thus more widely

used chaotic maps. Introduced first in 1845 by Verhulst as a

model for the population growth of a species it is expressed

as a recurrence equation.

We will study the dynamics of the 3D Hénon map and prove

Theorem 2. The dynamics can be complex, especially when |B| is

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close to one, essentially because when |B| = 1 the map has,

for certain values of the parameters, fixed points with three

multipliers are on the unit circle.

In recent years there is a tremendous boom in chaos-based

cryptography research, there are still some limitations that

prevent its wider application. Emphasizing, a big drawback is

its relatively slow speed.

x

CHAPTER 1

INTRODUCTION

Data Encryption is one of the important fields of the

cryptograph, so before sending the information to another side

the user needs to make his data unreadable by using any

encrypting methods e.g: encrypting the E-mail password. On

another hand, image is one of the most important information

that used in new applications like video streaming websites,

phones and satellites. Moreover, protecting the image is not

like protecting the data since the image needs special rules

to encrypt it. Until now, various data encryption algorithms

have been used over the internet; to protect the data

transmissions between the users such as: RSA, IDEA or DES. If

we apply those methods on the image, we need a long of time to

encrypt and decrypt the image and some of them may not be

useful to encrypt the data in a real time communication.

Furthermore, the chaos theory was firstly used in the computer

system by Edward Lorenz 1963.

The deterministic behavior of the chaos system,

initial sensitivity, parameter sensitivity and

unpredictability are the main reasons that bring chaos on the

image encryption. Many researchers try to solve this problem

by using different chaos systems, like they used a hybrid

method for image encryption by applying multi-chaotic systems;

to increase the key space and make system’s breaking very

difficult. Where, the calculations and the executions time to

1

encrypt the image are very large. Therefore, in they used

Rossler chaotic system to encrypt the image by applying

changes in the pixels value and their position to increase the

uncertainty in the cipher image. The one time pads with the

logistic map (as a chaotic function) are used in to encrypt

the image and increase the size of the encrypted keys. Where,

in an improved DES and the logistic map are used to encrypt

the image. Others, like proposed new modifications to the

Advanced Encryption Standard (MAES) to increase the security

level by using a chaotic system.

Edward Lopez derived a Chaos theory which is a

part of mathematical physics. A chaotic system based on

confusion and diffusion was developed in 1989 Chaotic systems

are sensitive, non-liner, deterministic and easy to

reconstruct after filling in the image. Henon map is one of

the chaotic maps used for generating Pseudo-random sequence

required for encryption. Henon chaotic map discovered in 1978

is used as a symmetric key stream cipher cryptographic system.

It is mathematical in nature.

Two dimensional discrete-time nonlinear dynamical Henon

chaotic map generates pseudo-random binary sequence.

Chaos-based data encryption algorithm for images

and videos are proposed by using three chaotic systems to

encrypt the image and to enhance the encryption properties. On

another hand, Electrocardiogram signals of the persons are

used to generate the encryption keys; to encrypt the data by

extracting their features by using Honen map as a chaotic

2

system. There are many problems in applying chaos on the image

encryption, such as: the existing number of invalid and weak

keys and the keys are not sensitive to the initial conditions.

A linear independence between adjacent pixels and

to use a logistic map in the image shuffling and henon map in

the image encryption, which makes the cryptosystem very

efficient and robust.

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CHAPTER 2

THE LOGISTIC MAP

2.1 Logistic map as a random number, generator –

For the largest value of the control parameter, the

logistic map is able to generate an infinite chaotic sequence

of numbers. Here we describe a simple method for obtaining a

random number generator based on this property of the logistic

map. Comparing to usual congruential random generators, which

are periodic, the logistic random number generator is

infinite, a periodic and not correlated. A periodic random

number generator is a valuable tool for computer simulation

methods.

The logistic map depends on a parameter λ ∈ R. It is

defined on the interval [0, 1] as Lλ(x) = λx(1 − x). The graph

for λ = 4 is shown on the left. Increasing λ from λ = 0 λ = 4

this map undergoes a series of period-doubling bifurcations in

much the same way as what we have seen for the Rössler system.

4

Figure 2.1 Bifurcation diagram of the logistic map

CHAPTER 3

HÉNON MAP

The Hénon map is a discrete-time dynamical system. It is one

of the most studied examples of dynamical systems that exhibit

chaotic behavior. The Hénon map takes a point (xn, yn) in the

plane and maps it to a new point

Xn+1 = Yn+1-aX2n (1)

Yn+1=bX n(2)

5

The map depends on two parameters, a and b, which for the

canonical Hénon map have values of a = 1.4 and b = 0.3. For

the canonical values the Hénon map is chaotic. For other

values of a and b the map may be chaotic, intermittent, or

converge to a periodic orbit. An overview of the type of

behavior of the map at different parameter values may be

obtained from its orbit diagram.

The map was introduced by Michel Hénon as a simplified

model of the Poincaré section of the Lorenz model. For the

canonical map, an initial point of the plane will either

approach a set of points known as the Hénon strange attractor,

or diverge to infinity.

The Hénon attractor is a fractal, smooth in one direction and

a Cantor set in another. Numerical estimates yield a

correlation dimension of 1.25 ± 0.02 and a Hausdorff dimension

of 1.261 ± 0.003 for the attractor of the canonical map.

It is known that for parameter ranges α∈[1.16,1.41] and

β∈[0.2,0.3] map generates chaotic behaviour i.e. sensitive

dependence on initial

conditions.

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Figure 3.2 The strange attractor of Henon Map

CHAPTER 4

INDEPENDENCE BETWEEN PIXELS

4.1 Creating Linear Independence Pixels

The linear independence between two variables a and b

can be written as a relationship that combined a and b without

describing a and b. To do that, many methods can be used such

as a mathematical combination that describes the system or by

using XORing operation between two variables. In this paper,

we apply the XORing between neighbors’ pixels to create a

linear independence relationship. Our approach is described as

follow:

1- XOR the adjacent pixels in the same row.

2- Store the result of the XORing in the second

adjacent pixel to the current one. The second XOR will

use the previous one to apply the XOR to its value.

3- After repeating 1-2 for all rows in the image the

first column consists of the keys; to solve the created

relations between the adjacent pixels.

4- To encrypt the keys in the first column, we apply the

logistic map and create N-keys (N number of rows); to

hide the keys in the image.

For Example: assume the image 5 x 5, as shown in Table.1 our system is applied in the following steps:

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Table.1: Original Image 5 x 5

1- XORing the adjacent pixels: XOR 11, 12 and store the

result in pixel 2, then XOR the result of the previous

XORing with 13 and store the result in pixel 3

….etc. Until the XORing for all of the adjacent pixels

are completed in the same row. Then, repeat the process

to each row, the results will be as follows:

Table.2: X-OR the adjacent Pixels

11

7 10

4 11

9 7 12

1 10

17

1 19

5 16

25

3 24

4 17

36

6 37

1 36

11

12

13

14

15

9 14

11

13

11

17

16

18

22

21

25

26

27

28

21

36

34

35

36

37

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2- The first column will contain the key linear independencerelations. we a map in the first column and XOR generatedkeys (described in point B), keys in the image.

Table.3: Encrypting the first column

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7 10

4 11

32

7 12

1 10

10

1 19

4 17

43

3 24

4 17

50

6 37

1 36

1- Encrypting the keys:

To encrypt the first column that contains the keys that

solved the linear independence relationships between the

adjacent pixels. We scale the output to be in the range [0

255] which we called it Pixel Key(PK) as in the following

steps:

Step 1: choose the initial value of (X1,Y1) for Henon map.This value works as an initial secret symmetric key for Henonmap.

Step 2: Henon map work as a key stream generator forcryptosystem. The size of sequence depends upon the size ofimage. If the image size is m×n then the number of henonsequence will be 8×m×n obtained by equation (1).

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Step 3: Experimental analysis conclude that cut-off point,0.3992, has been determined so that the sequence is balanced.The decimal values are then converted into binary valuesdepending upon this threshold value, as given in equation (3)where Z is a binary sequence.

Zi = (3)

Step 4: Henon sequence is then reduced by combining each

consecutive 8 bits into one decimal value.

Figure 4.1: Encryption with Byte Sequence

2-Shuffling the pixels position:

To increase the randomization in the encrypted image, we

shuffled the pixels position to new position by using the

0 if Xi <= 0.3992

1 if Xi => 0.3992

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logistic map in equation 1. In the following steps, the

shuffled approach will be described:

1- Choice the initial conditions of the logistic map.

2- Transfer the M x N image to (MN) x 1 vector image.

3- Generate the chaotic sequence MN x 1 called C, shuffle the

encrypted image by sorting the C vector so we get S shuffled

vector.

4- Transfer the Vector S to M x N matrix image. The shuffling

approach is shown in the Table 4, where, E (Encrypted image

from the step 1, C (chaotic sequences), S (shuffled) Matrices.

Table.4: Example of the shuffling approach

E 1 2 3 4 5 6 7 8 9 10

C .64 .96 .06 .87 .18 .38 .60 .44 .04 .91

S 9 3 5 6 8 7 1 4 10 2

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Henon Map

Figure 4.2 Proposed Encryption Algorithms

3 -Decryption of Encrypted Image

Since, the chaotic system behavior is deterministic so

reconstruction of image using the same key (X1, Y1) at

decryption end gives the shuffled image. This shuffled image

is further arranged in an order exactly opposite of the way

done for encryption. Finally, the original image is obtained

at receiver’s end. Where, the decryption process is done in

the reverse order of the encryption process.

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PRESENT WORK

CHAPTER 5

EXPERIMENTAL RESULTS

In this section, experimental results of the proposed image

encryption algorithm are illustrated to appreciate the

efficiency of existing algorithm. The MATLAB 7.6 software was

used for implementing this code. The initial parameters for

Henon map are chosen as a = 1.4 and b = 0.3 to make the system

chaotic. Secret symmetric key for encryption is a combination

of X1=0.01 and Y1=0.02. Lena image with a size 256 x 256 are

used in our experimental. In Fig.2 (a –b) shows the Lena image

and cipher image respectively. With input keys Key1= 2 x10-14

and Key2=1 x10-14 for an images. It is clear that the cipher

images are totally different from the plain-text images.

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(a)

(b)

(c)

(a)

(b)

(c)Figure 5.1 Encryption by Henon Chaotic System and Shuffling pixel position

(A) Original Image; (B) Shuffled Image; (C) Cipher Image

Figure 5.2 Decryption by Henon Chaotic System and Logistic function

(A) Cipher Image; (B) Shuffled Image after Decryption; (C)

Original Image

5.1 Keys Space analysis

For Key1 and Key2 serve as encryption keys, the space of the

keys is at least 1030, so it’s large enough to resist the brute

force attacks.

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5.2 Information Entropy Analysis

Information entropy is defined by the degree of uncertainties

in the encryption system. It is used to calculate the

Effectiveness of image encryption algorithm. Statistical

measure of randomness to characterize the texture of the input

image is termed as entropy. It is calculated as given in

equation (5).

H= -sum (p.*log2 (p)) (5)

Ideal entropy of an encrypted image should be equal to 8,

which corresponds to a random source. Practically, ideal

information entropy cannot be achieved. It is always less than

the ideal value. The values calculated in Table 1 are very

close to the ideal value.

5.3 Key Sensitivity Test

For secure encryption, the key should be sensitive with large

space key size to resist all kind of brute force attack.

Randomness is the key point of Henon map. To test the

sensitivity of the key involved, a minute variation was done

in original secret key by changing it from x(1)=0.01 and

y(1)=0.02 to x’(1)=0.010001 and y’(1)=0.020001. As a result,

it was not possible to obtain the original image at receiver’s

end.

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5.4 Histogram Analysis

The histogram of an image is graphical representation of pixel

intensity values. There are 256 different possible intensities

for a gray image, so in graphical representation of the

histogram will display 256 intensities and the distribution of

pixels amongst those intensity values.

Figure 5.4: Histogram Analysis: (A) Histogram of Original Image; (B)Histogram of Shuffled Image; (C) Histogram of Cipher Image; (D) Histogram

of Decrypted Image

It is analyzed from Figure 5.4, that the distribution of gray

scale values is uniform in cipher image, and significantly

different from histograms of original image. In the original

image some gray scale values do not exist in the range of 0 to

A

C

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255 but in encrypted image gray-scale values exist uniformly

in the range 0 to 255. Therefore, it is proved that the

encrypted image does not help intruders to employ statistical

attack on encryption procedure.

CHAPTER 6

CHAOS THEORY

Chaos theory is a field of study in mathematics, with

applications in several disciplines

including meteorology, physics,

engineering, economics and biology. Chaos theory studies the

behavior of dynamical systems that are highly sensitive to

initial conditions, an effect which is popularly referred to

as the butterfly effect. Small differences in initial

conditions (such as those due to rounding errors in numerical

computation) yield widely diverging outcomes for such

dynamical systems, rendering long-term prediction impossible

in general. This happens even though these systems

are deterministic, meaning that their future behavior is fully

determined by their initial conditions, with

no random elements involved. In other words, the deterministic

nature of these systems does not make them predictable. This

17

behavior is known as deterministic chaos, or simply chaos. This

was summarized by Edward Lorenz as follows:

Figure 5.1 A plot of the Lorenz attractor for values r = 28, σ = 10, b =8/3

CHAPTER 7

PSEUDORANDOM NUMBER GENERATOR

A pseudorandom number generator (PRNG), also known as a

deterministic random bit generator (DRBG), is an algorithm for

generating a sequence of numbers that approximates the

properties of random numbers. The sequence is not truly random

in that it is completely determined by a relatively small set

18

of initial values, called the PRNG's state, which includes a

truly random seed. Although sequences that are closer to truly

random can be generated using hardware random number

generators, pseudorandom numbers are important in practice for

their speed in number generation and their reproducibility,

and they are thus central in applications such as simulations

(e.g., of physical systems with the Monte Carlo method), in

cryptography, and in procedural generation. Good statistical

properties are a central requirement for the output of a PRNG,

and common classes of suitable algorithms include linear

congruential generators, lagged Fibonacci generators, and

linear feedback shift registers. Cryptographic applications

require the output to also be unpredictable, and more

elaborate designs, which do not inherit the linearity of

simpler solutions, are needed. More recent instances of PRNGs

with strong randomness guarantees are based on computational

hardness assumptions, and include the Blum Blum Shub, Fortuna,

and Mersenne Twister algorithms.

6.1 Why need Pseudorandom Number Generator

This stretching-and-folding does not just produce a

gradual divergence of the sequences of iterates, but an

exponential divergence (see Lyapunov exponents), evidenced

also by the complexity and unpredictability of the chaotic

logistic map. In fact, exponential divergence of sequences of

iterates explains the connection between chaos and

unpredictability: a small error in the supposed initial state

of the system will tend to correspond to a large error later

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in its evolution. Hence, predictions about future states

become progressively (indeed, exponentially) worse when there

are even very small errors in our knowledge of the initial

state. This quality of unpredictability and apparent

randomness led the logistic map equation to be used as a

Pseudo-random number generator in early computers.

CHAPTER 8

FUTURE WORK

Due to time consistency we couldn’t able to do that. So, ourfuture job is using 3D Logistic and 3D Henon map image encryptand decrypt by chaotic function.

Those types of technique are use in later and implementthrough 3D logistic and 3D henon map we secure our image of

different kind of attacks. Let’ s look e.g.

Figure 8.1 the bifurcation diagram of the second proposed logistic map (a)versus A for different values of

α = (0.1, 0.4, 0.7, 1.0, 100) and (b) versus a for different values λ =(1.5, 2.5, 4.0)

20

Three-dimensional Henon maps have not been as widely studied

as one– and two-dimensional maps, though in recent years there

has been a considerable increase in their study. One of

important reasons for this fact is that multi-dimensional

dynamical systems (the dimension of the phase space is at

least four for flows and three for maps) can exhibit

complicated dynamics that is decidedly distinct from lower-

dimensional cases. In particular they can possess a new

variety of strange attractor called a wild-hyperbolic

attractor

CONCLUSIONS

We applied to a plain image and results thus obtained

competitive to other level of security of images. Eavesdrop

cannot cryptanalysis the cipher image. Here, the security

relies on secret keys along with the image encryption

technique. Chaos is known for randomness, so it could be

highly secured. Confusion has been done by pixel of the image

by XORing the adjacent pixels form actual position to a new

position and create linear independence between them.

Diffusion has been done through Logistic sequence generated

and key encrypted through Henon map. So both the processes of

21

increasing confusion and diffusion resulted in increasing the

security of cryptosystem. Logistic map is a very simple but

widely used classic chaotic map, so it’s used for encrypt

whole matrix with increase confusion process of an image

pixel. The Hénon map of the second kind large scale, wild-

hyperbolic attractor, so it’s use encrypt for first column of

an image pixel. Those kinds of technique are securing our

image from adversary and protect different kind of attack.

22

REFERENCE

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services to the network’s edge,” IEEE Trans. Industrial

Electronics, vol. 50, pp. 404-411, 2003.

[2] T. H. Chen, G. Horng, and W. B. Lee, “A publicly

verifiable copyright-proving scheme resistant malicious

attacks,” IEEE Trans. Industrial Electronics, vol. 52, pp.

327-334, 2005.

[3] R. Brown and LO. Chua, “Clarifying chaos: examples and

counterexamples,” Int. J. Bifurcat Chaos, vol. 6, pp.219-242,

1996.

[4] J. Fridrich, “Symmetric ciphers based on two-dimensional

chaotic map,” Int. J. Bifurcat Chaos, vol. 8, pp.1259-1284,

1996.

[5] Hazem Mohammad Al-Najjar, “Digital Image Encryption

Algorithm Based on a Linear Independence Scheme and the

Logistic Map” Technical College of Arar, Department of

Computer Technical and Vocational Training Corporation Arar,

KSA.

[6] Image encryption using henon chaotic map with byte

sequencen. s. raghava & ashish kumar Department of Information

Technology, Delhi Technological University, New Delhi, India.

[7] S.V.Gonchenko, J.D.Meiss and I.I.Ovsyannikov, “Chaotic

dynamics of three-dimensional

hénon maps that originate from a homoclinic bifurcation,”

DOI: 10.1070/RD2006v011n02ABEH000345, Received October 3,

2005; accepted November 12, 2005.

23

[8] Yun Cao，Runhe Qiu and Yuzhe Fu, “Color image encryption

based on hyper-chaos,” Information and Technology Department,

Donghua University Shanghai, China, pp. IEEE, 2009.

[9] Rakesh S, Ajitkumar A Kaller, Shadakshari B C and Annappa

B, “Multilevel image encryption,” Department of Computer

Science and Engineering, National Institute of Technology

Karnataka, Surathkal, {rakeshsmysore, ajitkaller,

shadsbellekere}@gmail.com,[email protected].

[10] Pawan N. Khade and Prof. Manish Narnaware, “3D chaotic

functions for image encryption,” IJCSI International Journal

of Computer Science Issues, Vol. 9, Issue 3, No 1, May 2012

APPENDIX

Henon_Map (Rendom Number)

ans =

Columns 1 through 21

0 1 -1 1 -1 0 0 0 0 1 -2 -1 0 0 1 -1 0 -1 0 -1 0


-1 1 -1 0 0 0 0 0 1 -1 0 0 0 0 1 -2 -1 0 0 0 -1

24


1 -2 -1 -1 0 0 0 1 -1 1 -1 0 0 0 0 1 -1 1 -1 0 -1


1 -2 -1 0 0 0 0 -1 1 -2 -1 -1 0 0 0 0 1 -1 1 -1 0


-1 1 -2 -1 0 0 0 0 0 1 -2 -1 0 0 0 0 1 -2 -1 0 0


0 -1 1 -2 -1 -1 0 0 1 -1 0 0 1 -1 0 0 0 0 0 1 -1


0 0 0 0 1 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0


1 -1 0 0 0 -1 1 -2 -1 0 0 0 1 -1 0 -1 1 -1 0 -1 1


-2 -1 0 0 0 -1 1 -2 -1 -1 0 0 1 -1 0 0 1 -1 1 -1 0

25


0 1 -1 0 -1 0 -1 1 -1 0 -1 1 -2 -1 0 0 0 0 0 0 -1


1 -2 -1 -1 0 0 0 0 0 1 -1 0 0 0 1 -1 1 -1 0 -1 1


-2 -1 0 0 0 0 -1 1 -2 -1 -1 0 0 0 0 0 0 0 1 -1 1


-2 -1 0 0

26

Logistic_map

Random Number


0.4612 0.9940 0.0240 0.0936 0.3393 0.8967

0.3706 0.9330 0.2500 0.7501 0.7499 0.7503


0.7495 0.7510 0.7480 0.7541 0.7418 0.7661

0.7167 0.8122 0.6102 0.9514 0.1850 0.6030


0.9576 0.1625 0.5444 0.9921 0.0313 0.1213

0.4263 0.9783 0.0850 0.3109 0.8570 0.4901


0.9996 0.0016 0.0062 0.0246 0.0962 0.3476

0.9071 0.3369 0.8936 0.3802 0.9426 0.2164


0.6783 0.8729 0.4438 0.9874 0.0498 0.1894

0.6140 0.9480 0.1972 0.6332 0.9290 0.2639


27

0.7769 0.6932 0.8507 0.5080 0.9997 0.0010 0.0041

0.0165 0.0647 0.2422 0.7342 0.7806


0.6850 0.8631 0.4726 0.9970 0.0119 0.0472

0.1800 0.5903 0.9674 0.1262 0.4412 0.9862


0.0546 0.2063 0.6550 0.9039 0.3474 0.9068

0.3380 0.8950 0.3759 0.9384 0.2311 0.7109


0.8221 0.5849 0.9711 0.1121 0.3981 0.9585

0.1591 0.5351 0.9951 0.0197 0.0771 0.2846


0.8144 0.6045 0.9563 0.1672 0.5568 0.9871

0.0510 0.1937 0.6247 0.9378 0.2335 0.7159


0.8136 0.6066 0.9545 0.1737 0.5741 0.9780

0.0859 0.3141 0.8618 0.4764 0.9978 0.0089


0.0354 0.1364 0.4712 0.9967 0.0132 0.0520

0.1973 0.6335 0.9287 0.2649 0.7789 0.6889

28


0.8572 0.4896 0.9996 0.0017 0.0069 0.0274

0.1066 0.3810 0.9433 0.2138 0.6723 0.8812


0.4186 0.9735 0.1031 0.3700 0.9324 0.2522

0.7544 0.7412 0.7673 0.7141 0.8166 0.5990


0.9608 0.1508 0.5123 0.9994 0.0024 0.0096

0.0381 0.1464 0.5000 1.0000 0.0000 0.0000


0.0000 0.0000 0.0000 0.0000 0.0001 0.0006

0.0023 0.0094 0.0371 0.1429 0.4900 0.9996


0.0016 0.0064 0.0255 0.0993 0.3578 0.9191

0.2974 0.8359 0.5488 0.9905 0.0377 0.1451


0.4963 0.9999 0.0002 0.0009 0.0035 0.0138

0.0544 0.2059 0.6540 0.9052 0.3434 0.9019

29


0.3540 0.9147 0.3121 0.8588 0.4851 0.9991

0.0036 0.0142 0.0559 0.2110 0.6659 0.8899


0.3919 0.9532 0.1783 0.5860 0.9704 0.1149

0.4069 0.9653 0.1339 0.4639 0.9948 0.0208


0.0813 0.2989 0.8382 0.5426 0.9928 0.0288

0.1118 0.3971 0.9576 0.1624 0.5440 0.9922


0.0308 0.1194 0.4206 0.9748 0.0983 0.3546

0.9155 0.3095 0.8548 0.4964 0.9999 0.0002


0.0008 0.0033 0.0132 0.0522 0.1980 0.6352

0.9269 0.2711 0.7904 0.6627 0.8941 0.3787


0.9921 0.0315 0.1221 0.4287 0.9797 0.0797

0.2933 0.8291 0.5668 0.9822 0.0701 0.2608

30

Up to

0.4295 0.9801 0.0779 0.2872 0.8189 0.5932

0.9653 0.1341 0.4645 0.9950 0.0200 0.0784


0.2892 0.8222 0.5848 0.9713 0.1117 0.3969

0.9575 0.1630 0.5456 0.9917 0.0330 0.1276


0.4454 0.9881 0.0472 0.1798

31

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An Image Encryption & Decryption Approach Based on Pixel Shuffling Using 3D Logistic Map & 3D Henon Map