63

CHAPTER 4

EFFECT OF PIXEL SCRAMBLING METHOD

ON IMAGE ENCRYPTION

4.1 TWO CHAOTIC MAPS CSDP BASED IMAGE ENCRYPTION

USING PIXEL SCRAMBLING ALGORITHM

4.1.1 Introduction

In the present chapter a novel image encryption method using

simple logical and scrambling operations are introduced. In the first section of

the chapter, a simple transformation, permutation and two chaotic map CSDP

based scrambling (Ville et al 2003, Xiangdong et al 2008, Jiankun and

Fengling 2009) of image encryption is discussed. In the second section, a

pixel scrambling and simple logical operation based chaotic map image

encryption is shown. In the third section, a combination of above sections as

pixel scrambling, simple logical operation and transformation based two

chaotic map image encryption is exhibited. Finally, the present novel method

has been used to analyze and evaluate its performances through the statistical

measures Viz., histogram, cross correlation co - efficient, entropy, PSNR,

scrambling distance and key sensitivity.

64

4.1.2 The Types of Maps used in the CSDP based Image Encryption

using Pixel Scrambling

Chaotic Map

The chaos theory indicates the behaviour of certain nonlinear

dynamic system that under specific conditions exhibit dynamics that are

sensitive to initial conditions. The two basic properties of chaotic systems are:

The sensitivity to initial conditions and Mixing Property (Wu and

Rulkov 1993) proposed 1 D chaotic map to produce the chaotic sequence and

used to control the encryption processes. In the present work the chaotic maps

as Logistic map and Bernoulli map are used and described as below.

Logistic Map

A simple and well-studied example (Parker and Chua 1995, Wu

and Rulkov 1993, Kuo and chen 1991) of a 1D map that exhibits complicated

behavior is the logistic map from the interval [0,1] in to [0,1], parameterised

by and as mentioned in equations (3.11) and (3.12).

Bernoulli Map

The Bernoulli map which is used in the present novel method

(Parker and Chua 1995, Wu and Rulkov 1993) as mentioned in

equation (3.15).

4.1.3 The Present Novel Method for Image Security System to the

CSDP based Image Encryption using Pixel Scrambling

The present encryption algorithm belongs to the category of

combined value transformation and position permutation. Therefore, one

65

should initially define two bit-circulation functions with two parameters in

each function. One first bit circulation function is used to control the shift

direction and second bit circulation function is used to control the shifted bit-

number on the data transformation. In the present work, first bit circulation

function and second bit circulation function based scanning is used for their

performances and analyzed for the evaluation. The images in the present case

are treated as a 1D array by performing Raster scanning and zigzag scanning

as studied by Bourbakis and Alexopoulos (1991).

Figure 4.1 Chaos based Image Cryptosystem

Figure 4.1 shows the typical schematic of the present method,

where the scanned arrays are divided into various sub blocks. In each sub

block, position permutation and value transformation are performed and

finally scramble to produce the cipher image.

In continuation, a sub key is generated by applying the suitable

chaotic maps. Based on the initial conditions of chaotic map, the chaotic maps

are generated and allowed to iterate through various orbits. Hence, for each

sub block various chaotic sequence patterns are applied which are further

Plainimage

Secret key

Cipherimage

Seed

Key Generator - chaotic map-Logistic, Bernoulli map

DiffusionDiagonal shifting

Value mod Row

shifting

Position permutation

columnshifting

Scrambling

66

used to increase the efficiency of the key to be determined by the brute force

attack.

Further, the chaotic system based binary sequence is generated to

control the bit-circulation functions to perform successive data transformation

on the input data as presented in the section 3.1.4. In order to demonstrate the

correct functionality of the present signal security system, the simulation on

the present scheme has to be made. The following are the steps used for the

implementation of present chaos based mapping method.

4.1.4 The Algorithm used for Pixel Scrambling in the Present Work

Step 1: Covert 2 image into 1 array and then performs the Raster

scanning and the zigzag scanning.

Step 2: Consider a block size of 8 × 8 and convert them in to binary values.

Step 3: Sub key size is 20 bits, hence it is extracted from the chaos maps as

Bernoulli map. The Secret key is SEED, which are the initial conditions of the

each map. From the initial conditions the chaotic maps are allowed to iterate

through various orbits. Then, based on the chaotic system, binary sequence is

generated to control the bit-circulation functions to perform the successive

data transformation on the input data. A pair of and , the combination of

, , , , and resulting in the transformation pair may be non-unique used

as secret key.

Step 4: Convert the chaotic sub key in to binary values of 20 bits.

Step 5: Each 8 × 8 sub block of image pixel values circularly shifted by

chaos sequence generated from maps.

Step 6: The Circular shifting of Diagonal pixels are used as discussed in

section 3.2.5

67

Step 7: Perform the encryption by the chaotic sequence key values, which is

obtained from the orbits of chaos maps iteration.

Step 8: Chaos theory Based Image Scrambling (Xiangdong et al 2008)

transformation to a Gray scale image of size × pixels, can have an

arbitrary chaotic iteration = ( ) to generate a

chaotic sequence of real numbers.

The initial value is the secret key. The following scheme has

been applied to scramble and unscramble cipher image .

Step 8.1: Let an initial value be that is associated to the secret key.

Let = 1.

Step 8.2: Iterate from 0 1 times with the chaotic iteration 8.1, and get

the sequence of real numbers , … … , .

Step 8.3: Arrange the chaotic sequence , … … , in descending order, to

get the sorted sequence{ … . . , }.

Step 8.4: Determine the set of scrambling address codes , … … , ,

where {1,2 … . . }. is the new subscript of in the sorted sequence

{ … . . , }.

Step 8.5: Permute the column of the cipher image with permuting

address code , … … , , namely, replace the row pixel with the row

pixel for from1 .

Step 8.6: If = , end of iteration. Otherwise, let = , and = + 1.

Repeat from 8.2 to 8.5, to produce double encrypted cipher image data value

in 1 form.

Step 9: Transform the cipher image from1 Dimension to 2 Dimension.

68

Step 10: Transmit the chaotic sub key via secure channel using public key

algorithms.

Step 11: Decrypt the cipher image using the same chaotic sub key and SEED.

Step 12: Finally, performance analysis is carried out by doing correlation,

histogram and PSNR of the original, encrypted and decrypted image.

4.1.5 The Analysis and Evaluation of Present Method for Security

System to CSDP based Image Encryption using Pixel

Scrambling

The analysis of security system for two chaotic image encryption

are performed through the following types: Histogram analysis, correlation

analysis, PSNR and speed analysis and sensitivity analysis.

An image size of 256×265is considered as plain image for example x-ray of chest and should be performed with chaotic map along with orbit key. The most direct method to decide the disorderly degree of the encrypted image is through the sense of sight.

On the other hand, the arising correlation coefficient could provide quantitative measure on the randomness of the encrypted images. The logistic

map (1D) uses four parameters , , and (0) in generating the chaotic bit-stream. The four parameters could be viewed as the keys to the present signal security system. Among them, the parameters and can be fixed in both the transmitter and receiver according to the constraint shown in Step 1.

In order to apply the parameters and must be determined according to Step 1. The selection of and should follow the empirical law.

Based on the experimental experience, general combinations of and canalways result in very disorderly results. In the simulation, = 2 and = 2 are adopted in Step 1.

69

The initial conditions of chaotic maps used are, f (x) = 0.5 for Bernoulli map. The offset values for producing various orbits are chosen to be very less than the initial conditions. The visual inspection of Figure 4.2 shows the possibility of applying the algorithm successfully in both encryption and decryption. In addition, it reveals its effectiveness in hiding the information contained in them.

Figure 4.2 (a) Original (b) Cipher Image (c) Cipher Image (d) Decrypted Image with Maps and Scrambling

Histogram Analysis : An image histogram (Shubo et al 2009, Patidar et al 2009, Jiankun and Fengling 2009) illustrates how the pixels in an image are distributed by graphing the number of pixels at each level of intensity. One typical example among them is shown in Figure 4.3(a). The histogram of a plain image contains large spikes.

Figure 4.3 (a) Histogram of Original Image

Figure 4.3 (b) Histogram of Cipher Image

0 50 100 150 200 250

0

0.5

1

1.5

2

2.5

x 104

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100020003000400050006000700080009000

10000

70

The histogram of the cipher image as shown in Figure 4.3(b) is uniform, significantly different from that of the original image, and bears no statistical resemblance to the plain image. It is clear that the histogram of the encrypted image is uniform and significantly different from the respective histograms of the original image and hence does not provide any clue to employ any statistical attack on the present image encryption procedure.

Correlation Co-efficient Analysis : In addition to the histogram analysis

(Krishnamoorthi and Sheba Kezia Malarchelvi 2008, Shubo et al 2009,

Patidar et al 2009, Zhang et al 2007), the correlation between two vertically

adjacent pixels, two horizontally adjacent pixels and two diagonally adjacent

pixels in plain image and cipher image are considered in the present

investigation.

The correlation co-efficient analysis is computed as mentioned in

equations (3.4)-(3.6). Figure 4.4 shows the correlation distribution of two

horizontally adjacent pixels in plain image and cipher image for the all image.

The correlation co-efficients are found to be 0.9905 and 0.0308 for both plain

image and cipher image respectively.

Figure 4.4 a) Horizontal, Vertical and Diagonal Correlation of Plain Image

71

Figure 4.4 b) Horizontal, vertical and Diagonal Correlation of Cipher Image

The correlation coefficients (Shubo et al 2009, Patidar et al 2009,

Jiankun and Fengling 2009, Fishawy and Zaid 2007 of various maps are also

calculated and compared with each other. The results of correlation co

efficient are shown in the Table 4.1 for various plain, cipher images and maps

based correlation co efficient.

Table 4.1 Horizontal, Vertical and Diagonal Correlation coefficients of

Cipher Image

Original Image

Horizontal Correlation

Verticalcorrelation

DiagonalCorrelation

Cipher image with Maps and

scrambling Knee 0.2254 0.4400 0.0012 -0.00070115

Chest -0.0515 -0.0241 -0.0084 0.00010436

Human Head 0.5930 0.5759 -0.0646 -0.00094391

Lena 0.5254 -0.0241 0.0028 0.00095436

Correlation analysis of Scanning Pattern: The correlation coefficient is

found for the various directions of scanning patterns employed as presented in

section 3.2.6. The results as correlation co efficient for raster and zigzag

scanning are shown in Table 4.2.

72

Table 4.2 Horizontal Correlation Co-efficient for Raster Scanning and Zigzag Scanning

IMAGE Raster Scanning zigzag Scanning

Knee 0.0539 -0.00139

Chest -0.0535 -0.00590

Human Head 0.0174 -0.00230

Lena -0.0635 -0.00980

The observation shows from the task that the zigzag scanning is

more efficient than the raster scanning. In addition, the cipher image with

multiple maps is more resistant to crypt by the analyst attacks. The correlation

for plain and cipher image is shown in Table 4.3.

Table 4.3 Correlation Coefficient in Plain Image and Cipher Image

Direction of Adjacent Pixels Plain Image

Cipher image using Bernoulli

Map

Cipher Image with Maps and

Scrambling Horizontal 0.9670 0.0781 0.00887

Vertical 0.9870 0.0785 0.00923

Diagonal 0.9692 0.0683 0.00893

PSNR and Speed Analysis: The peak signal to noise (Krishnamoorthi and

Sheba Kezia Malarchelvi 2008, Jiri and Karel 2009) ratio of encrypted image

and original image is computed as mentioned in section 3.1.4. In the present

scheme of encryption, higher the visual quality of the cipher image, lesser are

the number of changed pixels and larger the value of PSNR it is around

9.3158 for the chest image, 9.0061for the knee image and 9.2709 for the head

image mentioned in the tables and shown in the Table 4.4.

73

Table 4.4 Encryption Speed

Image Size Speed (seconds)

PSNR

(dB)

Knee 256x256 0.1143 9.11

64x64 7.3168 9.32

8x8 117.06 9.26

Lena 256x256 0.1293 9.09

64x64 8.2722 9.31

8x8 132.35 9.20

Sensitivity Analysis : In differential attacks (Ahmed et al 2007) , to test the

influence of one-pixel change on the whole image encrypted by the present

algorithm, two common measures are NPCR and UACI as mentioned in the

section 3.1.4. The NPCR and UACI are calculated for knee, chest and Human

head and shown in the Table 4.5.

Table 4.5 NCPR AND UACI for Cipher Image

Image NPCR (%) UACI(%)

Knee 99.993 -0.00077

Chest 99.843 -0.00546

Human Head 98.430 -0.00839

Lena 98.80 -0.00330

74

4.2 A TWO CHAOTIC IMAGE ENCRYPTION USING PIXEL

SCRAMBLING

4.2.1 Introduction

In the previous section chaotic image encryption based on simple

scrambling and logical operation is discussed. However, always there is a

scope for improvement in the speed and security measures. Hence a new

chaotic image encryption based on simple scrambling and logical operation is

introduced. The present techniques is analyzed and its performances are evaluated

by statistical measures such as histogram, cross correlation, entropy, PSNR

,scrambling distance ,speed and key sensitivity analysis are discussed

4.2.2 The Present Novel Method for Image Security System Use of

Two Chaotic Image Encryption using Pixel Scrambling

4.2.2.1 The Present Image Encryption Method

The present encryption method is shown as a flow diagram in

Figure 4.5. Basically it uses pixel scrambling and XOR operation performed

in the present novel method. In the present work the key stream is generated

by PMMLCG and logistic chaotic maps.

In the first part of the algorithm which performs row and column

scrambling based on prime modulus linear congruential generator

(PMMLCG) chaotic map and then XOR operation is performed for scrambled

image with PMMLCG map. The scrambled image is again row and column

scrambling based on logistic map and then XOR operation performed for

scrambled image with logistic map.

75

Figure 4.5 Flow Diagram of the Present Novel Method

PMMLCG

Column scrambling

{ , , … , … }

{ , , … , … }

XOR pixels in adjacent rows

XOR pixels in adjacent columns

Scramble each pixel

Plain image{M ×N}

Chaos map

Column scrambling

Row scrambling

{ , , … , … }

{ , , … , … }

XOR pixels in adjacent rows

XOR pixels in adjacent columns XOR pixels in each row with chaos map

Cipher Image

Row scrambling

76

4.2.2.2 Key Stream Generator

The key stream is generated by two chaotic maps and the present

novel work uses PMMLCG and logistic map.The random sequence is

generated using PMMLCG. Linear Congruential Generators (LCG) are one of

the oldest and most studied random number generator (RNGs). A LCG is

parameterized by three integers , and .

Its basic form is

= ( + ) (4.1)

A special kind of LCG is called PMMLCG. Its parameters are

= 0 and being a prime. The advantage of PMMLCG is that eliminates an

addition which has an almost full period (of length ( 1)) and can be

subjected to the Spectral test. PMMLCG generate a random sequence by

using the above equation. PMMLCG is a secret key for the first part of the

algorithm and it is used for row, column scrambling and key XOR operations.

In the second part the algorithm uses logistic map as a secret key

and it is generated as follows. A random sequence from the logistic map has

been generated with secret key as mentioned in equation (3.3).

For (0,1) and (3.9876543210001,4) and are the

system control parameter and initial condition respectively. A secret key

value is , its typical value is 0.9876543219991. Depending on the value

of , the dynamics of the system could be changed dramatically.

The choice of in the equation above guarantees the system is in a

chaotic state and the output chaotic sequences have perfect randomness.

The Logistic map (Dachselt and Schwarz 2001, Parker and Chua 1995) has a

77

secret key for the second part of the algorithm and it is used for row, column

scrambling and key XOR operations.

4.2.2.3 The Algorithm for Present Novel Method

Step 1.1: Read an input plain image.

For the Gray scale image of size × pixels, can have an

arbitrary chaotic iteration

= ( ) , (4.2)

to generate two chaotic sequences of real numbers of lengths and

respectively. The initial seed value is derived from the secret key for one of

the sequence, and from the previous sequence for the other set.

Step 1.2: Similarly, generate two PMMLCG sequences of real numbers of

lengths M and N respectively as given by,

= (4.3)

The initial seed value is derived from the secret key for one of

the sequence and from the previous sequence for the other set.

Step 2: Permute the row of the image with the row obtained from

one of the PMMLCG sequences generated in step 1.2, and the column

with column obtained from the other sequence for all values of from

1 and from1 .

Step 3: If = and = , XOR adjacent pixels and end the iteration.

Otherwise, increment i and k and Repeat the previous step.

78

Step 4: Scramble each pixel ( , ) in the image to a position ( , )

determined from the random sequence generated in the earlier steps.

Step 5: Permute the row of the image with the row obtained from

one of the Chaotic sequences generated in step 1.1, and the column with

column obtained from the other sequence for from 1 and

from1 .

Step 6: If = and = , XOR adjacent pixels and end the iteration.

Otherwise, increment i and k and Repeat the previous step to produce double

encrypted cipher image.

Pseudo code

READ plain image and key.

GENERATE two PMMLCG sequences defined by

x = ax mod q, using an initial seed derived from the key.

PERMUTE ROWS of the plain image with

WHILE i!=M

I(i,j) I(xi,j)

I(i,j)= I(i,j) I(i+1,j).

PERMUTE COLUMNS of the resulting image

WHILE k!=N

I(i,j) I(i,xk)

I(i,j)= I(i,j) I(i,j+1).

SCRAMBLE Each Pixel in the Image

WHILE (i!=M && k!=N)

I(i,j) I(xi,xk).

GENERATE two Chaos sequences defined by

x = f( x ), using an initial seed derived from the key.

PERMUTE ROWS of the plain image with

79

WHILE i!=M

I(i,j) I(xi,j)

I(i,j)= I(i,j) I(i+1,j)

PERMUTE COLUMNS of the resulting image

WHILE k!=N

I(i,j) I(i,xk)

I(i,j)= I(i,j) I(i,j+1).

XOR pixels in each row with the chaos map.

WRITE the cipher image.

4.2.2.4 The Various Transformation of Pixel in the Plain Image during

Scrambling and flow Diagram of Present Image Encryption

Method

The image has been represented as 2D box and pixels of the plain

image are demonstrated by blue color shading. The scrambled image pixels

are represented by red color. After scrambling, the image is XORéd with

Logistic map, it is represented by grey color.

Stage I: Applying PMMLCG’s to transpose rows, columns and every

individual pixel in the image:

80

PMMLCG generator equation: z = ( z )mod q

where, a - multiplier q - Large Prime number

( , )

( , )

Stage II: Applying Logistic map to transpose rows and columns in the image:

Logistic map equation: x = ( x ) ( x )where – multiplier

81

4.2.3 The Analysis and Evaluation of Present Method for Security

System to Two Chaotic Map based Image Encryption using

Pixel Scrambling

In the present novel method of encryption the analysis of security

was also performed through the following types. Histogram analysis,

correlation co efficient analysis, key space analysis, PSNR and speed

analysis, average moving distance of scrambling, entropy and key sensitivity

analysis.

Histogram Analysis: The histograms of the several encrypted images as

well as its original images that have widely different content are analyzed as

mentioned in section 3.1.4. One typical example among them is shown in Figure

4.6 (b) as below.

CIPHER IMAGEEX-OR

Logistic Map

82

(a) (b) (c)

Figure 4.6 (a) Original Image (b) Histogram of Original Image (c) Histogram of Cipher Image

The histogram of a plain image contains large spikes, such spikes

correspond to gray values that appear more often in the plain image. The

histogram of the cipher image is shown in Figure 4.6 (c), is uniform,

significantly different from that of the original image and bears no statistical

resemblance to the plain image.

It is clear from the above that the histogram of the encrypted image

is fairly uniform and significantly different from the respective histograms of

the original image and hence does not provide any clue to employ any

statistical attack on the present novel method of image encryption procedure.

Correlation Coefficient Analysis : The cross–correlation coefficient

(Shubo et al 2009, Patidar et al 2009, Jiankun and Fengling 2009, Fishawy and

Zaid 2007, Zhang et al 2007) between the plain image A and the cipher image

B are calculated as presented in section 3.1.4.

The algorithm for present novel method produces highly

uncorrelated cipher text images with cross correlation values (horizontal,

vertical and diagonal correlations) that are lower than earlier chaos-based

image encryption schemes. The difference in correlation of plain and cipher

images obtained for other methods and present method is shown in the

Table 4.6

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 50 100 150 200 250

0

2000

4000

6000

8000

10000

12000

0 50 100 150 200 250

83

Table 4.6 The cross correlation analysis of the present algorithm

Image

Correlation between plain and cipher image Ahmed et al (2007), Xiangdong et al

(2008), Shubo et al (2009), Jiankun and Fengling (2009), Ismail et al (2007)

chaos based method

Present novel method

Lena.tif -0.00012303 -0.000061326Baboon.tif 0.00029386 0.000852120 Einstein.tif 0.00179810 -0.000423850Airplane.tif 0.00047238 -0.000291900Peppers.tif 0.00276760 -0.000651910

The statistical parameter of Vertical correlation, Horizontal

correlation and Diagonal correlation co efficient are obtained for other chaos

based method and present novel method for various pictures (Images).The

difference in values are shown in the Table 4.7.

Table 4.7 The Statistical Parameter Obtained for other Chaos based Method and Present Method

Statistical parameters

Images

Lena.tif Baboon.tif Airplane.tif

Vertical Correlation

Other chaos based techniques Ismail

et al (2007) 0.00054724 -0.00096242 -0.00005729

Our present technique 0.00007429 -0.000008235 0.001656900

Horizontal Correlation

Other chaos based technique

Ismail et al (2007)-0.0003686 -0.004313800 -0.00082698

Our present technique 0.00056377 -0.000774680 0.000563420

Diagonal Correlation

Other chaos based technique

Ismail et al (2007)0.0026055 0.001244600 0.003412100

Our present technique 0.0030529 0.002380700 0.000932380

84

Key Sensitivity Test with Several Slightly Different Keys : The key

sensitivity (Patidar et al 2009, Chang et al 2001, Kocarev and Jakimovski

2001) test have been performed through the following key sensitivity tests:

Decryption key sensitivity, Encryption key sensitivity and Key space analysis.

In decrypted lena.tif image is as shown in Figure 4.7(a) and 4.7(b)

with a wrong decryption key differing from the original private key by one

bit. In the present case the novel method has high degree of key sensitivity as

shown in the Figure 4.8 (a) and 4.8 (b). This confirms that an adversary

cannot retrieve the plaintext with a wrong key.

Figure 4.7 (a) Plain Image and Histogram of Plain Image

Figure 4.7(b) Cipher Image and Histogram of Cipher image

Figure 4.8 (a) Decrypted Image and Histogram of Cipher Image with Wrong Key

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

85

Encryption key sensitivity: The parametric changes between

encrypted image of lena.tif with two keys differing by 1 bit and found the

resultant as in Table 4.8. The present method provides the difficulty of a

related key attack.

Table 4.8 Quantitative Analysis of Sensitivity Tests

Analysis CAB NPCR(%) UACI(%)

Encryption key sensitivity

0.0011 99.6124 18.4749

Plain text sensitivity 0.0015 99.3347 22.4449

Key Space Analysis: The key space for a good cryptosystem should be

sufficiently large to make the brute-force attack infeasible. Key spaces imply

the total number of different keys which can be used for the purpose of

encryption and decryption. With respect to the speed of computers today, the

key space should be more than 2100 = 1030 in order to avoid brute-force attacks

(Pisarchik and Zanin 2008). The present novel method uses a 256 bit key as

the result the key space would become 2256 -1. Consequently, the sensitivity

of the algorithm to the key is significantly improved.

The present algorithm is highly sensitive to changes even in the

least significant bit of the key as demonstrated in the process of decryption

using wrong key varying by only one bit from the original key as in the

Figure 4.7(a) and 4.7(b).

PSNR and Speed Analysis : PSNR (Krishnamoorthi and Sheba Kezia

Malarchelvi 2008, Jiri and Karel 2009) of encrypted image and original image

are obtained as given by equation (3.8).

86

Table 4.9 Comparison of PSNR’s for the PS based, BFTCGH based, Integrated Imaging PS based and Present Novel Method with 85% Data Loss as Index

PS-based dB

BFT CGH-based

dB

Integral imaging and PS – based

dB

presentmethod

dBPSNR 5.58 9.45 17.11 16.98

Apart from the security consideration, running time and speed is

also an important on image encryption. The simulator for the present novel

method is implemented using the MATLAB 7.6 and the performance has

been measured on a 2.2 GHz Pentium IV with 1 GB of RAM running in

Windows XP.

Table 4.10 shows the performance evaluation (Marwa et al 2008) of

the present scheme. The analysis has been done by selecting 12 images each

one is of different sizes (1.5, 9, 24, 44 and 72 MB etc) having different

contents and using 10 randomly chosen secret keys for each image. Some of

the performance evaluation to present novel method and other methods are

shown in the Table 4.10

Table 4.10 Time of Image Encryption and Algorithms

Algorithm Lena(seconds)Combinational Permutation (Mitra et al 2006)

0.33

CKBA (Yen and Guo 2000) 1.05Encryption using SCAN pattern (Maniccam and Bourbakis 2004)

2.54

ECKBA (Socek et al 2005) 1.84Present Technique 1.72

87

Average Moving Distance of Scrambling :The average moving distance of

scrambling (Xiangdong et al 2008) is defined as

| | = ( ) + ( ) (4.4)

where , represents the pixel coordinate of a point in original image and

( , ) represents the pixel coordinate of that point in scramble image. The

statistical results are evaluated for moving distance of pixel. The results from

other chaos based method and present novel method is shown in the

Table 4.11. From the table, the larger the average moving distance of the

scrambling method less relation between the original image and the scramble

image and the increases the efficiency of the method.

Table 4.11 Statistical Results of Average Moving Distance of Present Algorithm

Image Average moving distance of pixel

Other chaos based techniques (Xiangdong et al 2008)

Our present technique

Lena.tif 85.2564 135.0582

Peppers.tif 85.0564 135.5139

Entropy: Entropy (h) (Mohammad and Jantan 2003) is calculated as given by

equation (3.7). The maximum h an 8–bit image can attain is 8. The average of

present novel method gives 7.99. Hence a statistical attack is too difficult to

make.

Sensitivity Analysis: The sensitivity analysis (Ahmed et al 2007) could be

performed through net pixel change rate and unified average change in

intensity as given by equation (3.9) and (3.10). The sensitivity analysis is

obtained as above for LENA image. The results are shown in the Table 4.12.

88

Table 4.12 Key Sensitive Analysis on LENA Image

Analysis CAB NPCR(%) UACI(%)

Encryption key sensitivity 0.0011 99.6124 18.4749

Plaintext sensitivity 0.0015 99.3347 22.4449

Ciphertext sensitivity 0.0064 99.9298 19.2178

4.3 A CHAOTIC MAP BASED IMAGE ENCRYPTION USING

INTEGRATED PIXEL SCRAMBLING WITH DIFFUSION

4.3.1 Introduction

In the previous section CSDP based image scrambling technique is

presented. The CSDP based techniques take more time for encryption of the

image but it has very high key space. Hence, always there is a scope for

improvement in the speed and security measures. Therefore, a new chaotic

image encryption based on simple scrambling and logical operation is

introduced.

The present novel method is analyzed and its performances are

evaluated by statistical measures such as histogram, cross correlation,

entropy, PSNR, scrambling distance, speed and key sensitivity analysis.

4.3.2 Chaotic Map

The chaos (Parker and Chua 1995, Wu and Rulkov 1993) can be

generated by using various chaotic maps. In the present method chaotic map

is used to produce the chaotic sequence which is used to control the

encryption process.

89

Logistic Map

A simple and well-studied example from (Dachselt and Schwarz

2001, Parker and Chua 1995) of a 1D map that exhibits complicated behavior

is the logistic map from the interval [0,1] in to [0,1], parameterised by :

( ) = ( ) (4.5)

where 0 4. This map constitutes a discrete-time dynamical system in

the sense that the map : [0,1] [0,1] generates a semi-group through the

operation of composition of functions.

The state evolution is described by ( ) = ( ) and denote

( ) = ( ) (4.6)

For all [0,1], a “discrete-time” trajectory{ } , where

= ( ), can be generated. The set of points { , … . } [0,1] is

called the (forward) orbit of . A periodic point of is a point [0,1]

such that = ( ) for some positive integer . The least positive integer n

is called the period of . A periodic point of period 1 is called a fixed point.

For differentiable g, a periodic point x with period n is stable if

( ) < 1 and unstable if ( ) > 1 ,

where = ( ). In the logistic map, as is varied from 0 to 4, a period-

doubling bifurcation occurs. In the region [0,3], the map possesses

one stable fixed point. As is increased, the stable fixed point becomes

unstable and two new stable periodic points of period 2 are created.

90

As is further increased, these stable periodic points in turn

become unstable and each spawns two new stable periodic points of period 4.

Thus the period of the stable periodic points is doubled at each bifurcation

point.

Each period-doubling episode occurs in a shorter “parameter”

interval, decreasing at a geometric rate each time. Moreover, at a finite , the

period-doubling episode converges to an infinite number of period doublings

at which point chaos is observed.

4.3.3 The Present Novel Method for Image Security System to the

Chaotic Map based Image Encryption using Pixel Scrambling

with Diffusion

In the present novel method, the security system is discussed

through key stream generator and design of encryption, decryption. For the

same algorithm, IISPD technique for image encryption analysis of IISPD is

also performed.

4.3.3.1 Key Stream Generator

The random sequence from the logistic map with secret key has

been generated as mentioned in equation (3.3).

For [0,1] and ( , 3.987653210001,4), and are the system control

parameter and initial condition. A secret key value is whose typical value

is 0.9876543219991. Depending on the value of , the dynamics of the

system can change dramatically. The choice of in the equation above guarantees

the system is in chaotic state and output chaotic sequences have perfect

randomness (Abdulkarim et al 2010, Raynel et al 2009). There are two

91

logistic maps generated for the above purpose based on one integer number

and two floating point numbers.

The integer number is height / width of the image. The first chaotic

logistic map is said to be and second chaotic logistic map is said to be .

For the first logistic map two floating point numbers are secret keys, and

integer number, which is size of the image. Similarly, second logistic map is

generated based on two floating point number are secret key (which is passed

as parameter obtained from the first chaotic logistic map), and integer

number, which is size of the image. Then the values generated by both the

maps are converted in to decimal.

4.3.3.2 Design of Encryption and Decryption Model

The encryption is simple. First the adjacent pixels of an image in a

row is bitwise XOR’ ed with its neighbor pixels and based on chaotic key

the pixels are scrambled. This process is repeated for all the rows until it

creates a row scrambling. Similarly the adjacent pixel of an image in a

column is bitwise are XOR’ed with its neighbor pixels and based on chaotic

key the pixels are scrambled as process.

This process is repeated for all the columns until it creates a column

scrambling. The combination of both row and column scrambling would form

a cipher image1. Further, the diffusion process is carried out by bitwise

XOR‘ing cipher image1 and chaotic key , generates a cipher image2 as

shown in the Figure 4.9.

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Figure 4.9 The Present Novel Method of Encryption

Cipher 1

Secret Key Initialization

Logistic Map XK / Row

Logistic Map Y/Col

Yk/col

For all row’s

For all Column’s

Column Scrambling

RowScrambling

Adjacent pixels in rows Bit XOR’ed

Adjacent pixels in column’s Bit XOR’ed

Pixel values are XOR’ed with

Chaotic sub key

Original image

Transmit through

unsecured channel

Cipher 2

93

Decryption operation is performed in a similar manner as to the

encryption. The differences to are that the key is traversed in the reverse

direction and the rotations based on the key bits. The key bits are used to

rotate the pixel in the opposite direction to the one used in encryption. For

example, in the encryption the row was rotated right-ward, in for decryption it

will be rotated left-ward and in order to retain the correct sequence of

rotation, the key is traversed in the reverse direction in all the rotation loops.

4.3.3.3 The Present Algorithm used in the Present Novel Method for Reading of (plain) Original Image (OI)

The original image is converted to gray scale if it is color image.

= , , where and , and are height and width of

the original image in pixel respectively.

The Secret Key

The secret key in the present method of encryption technique is a

set of two floating point numbers and one integer = ( , , ), where mu

is whose typical value is 3.9876543210001, is initial value of the

chaotic map i.e key where typical value is 0.9876543219991 and is width

of the image.

= ( , ( ), )

where mu is whose typical value is 3.9876543210001, ( ) is last value of

map and column is Width of the image. = the logistic map is generated

with the value said above and X is multiplied with number of rows and fixed

as row.

Ykey = the logistic map is generated with the value said above and Y

is multiplied with the number of columns and fixed as Column.

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Row Scrambling

Step 1: First the pixel values are XOR’ ed with its adjacent pixel values

XOR with Adjacent Pixels

FOR i = 1 to row

FOR j=1 to column-1

OI (i,j+1) =OI(I ,j) OI(i,j+1)

END

END

Step 2:

The ROW Scrambling is done as follows

FOR i = 1 to ROW

v = OI(x Key(i),All column);

OI(Xkey(i),All column) = OI(i,All Column);

OI(i,All Column) = v;

END

Step 3:

Similarly for column scrambling the pixel values of the adjacent

pixel values are XOR’ed.

XOR with Adjacent Pixels

FOR i = 1 to column

FOR j=1 to row-1

OI(j+1,i) = OI(j,i) OI(j+1,i)

END

END

95

Step 4: The Scrambling of Columns is done as follows

FOR i = 1 to Y key

v = OI (All rows,Y Key(i));

OI(All rows,Y Key(i)) = OI(All rows,i);

OI(All rows,i) = v; END

Step 5: Then chaotic key value Y key is XOR’ed with image.

FOR i=1 to row y=(mu,x(i),col)

y = y * column;

Y key = integer(y) FOR j=1 to column

OI(i,j) = OI(i,j) Y key( j)

END END

Step 6: The current pixel value of the Original image is XOR’ed with its

neighbors as follows :

XOR with Next pixel FOR i = 1to row

FOR j to 1:col-1

OI(i,j+1) = OI(i,j) OI(i,j+1) END

END

96

4.3.3.4 The Various Transformation of Pixel in the Plain Image during

Scrambling and Flow Diagram of Present Image Encryption -

IISPD

The image has been represented as 2D box and pixels of the plain

image are demonstrated by blue color shading. The scrambled image pixels

are represented by red color. After scrambling the image is XORéd with

Logistic map, it is represented by grey color.

Stage1: Adjacent pixels of values are XOR’ ed with its adjacent pixel values

before Row scrambling for all individual pixel in the image.

Stage 2: Applying Logistic map to scramble Column for every individual

pixel in the image.

Logistic Map equation: 1 ( )n nx g xwhere,

: [0,1] [0,1]g0 4.

thjColum nOI 1

t h

jC o l u m nO I

[ ] t h

k e y c o l u m niX [ ] t ha l l c o l u m nX i

97

Stage 3: Adjacent pixels of values are XOR’ ed with its adjacent pixel values

before Column Scrambling for all individual pixel in the image.

Stage4: Applying Logistic map to scramble Row for every individual pixel in the

image. Logistic Map equation:

1 ( ( ))n keyy g nX where, : [0,1] [0,1]g

0 4.

Stage5: The Scrambled Row and Column of the image XOR ‘ed with

Logistic maps every individual pixel in the image and called Cipher image -I.

th

irowO I

1t hi r o wO I

[ ] t hA l l r o w siY

[ ] t h

k e y r o wiY

98

Logistic Map equation: = ( ( ))

Stage 6: Finally, the current pixel value of the Cipher image-I is XOR’ed

with its neighbors as follows

4.3.4 Analysis and Evaluation of Security Problem for Chaotic Map

based Image Encryption using Integrated Pixel Scrambling

with Diffusion

Histogram Analysis: An image histogram illustrates (Shubo et al 2009,

Patidar et al 2009, Jiankun and Fengling 2009, Fishawy and Zaid 2007) how

pixels in an image are distributed by graphing the number of pixels at each

color intensity level as presented in section 3.1.4. One typical example among

them is shown in Figure 4.10(b).

EX-OR

CIPHER IMAGE-I

thj Colum nOI 1

t h

j C o l u m nO I

CIPHER IMAGE-I

CIPHER IMAGE-II

LogisticMap

99

(a) (b) (c)

Figure 4.10 (a) Original Image (b) Histogram of Original Image (c) Histogram of Cipher Images

The histogram of a plain image contains large spikes. These spikes

correspond to gray values that appear more often in the plain image. The

histogram of the cipher image is shown in Figure 4.10(c), is uniform significantly

different from that of the original image and bears no statistical resemblance

to the plain image. Therefore, it does not provide any clue to employ any

statistical attack on the present image novel method of encryption procedure.

Correlation Coefficient Analysis: For a plain image having definite visual

scene, each pixel is highly correlated with its adjacent pixels either in

horizontal, vertical direction and diagonal direction. In ideal case an image

encryption scheme should produce a cipher image with no such correlation in

the adjacent pixels as shown in the Figure 4.11(b).

Figure 4.11 (a) The Correlation of Original Image (b) The Correlation of Cipher Image

100

The horizontal, vertical and diagonal correlations of the adjacent

pixels in plain image and cipher images (Shubo et al 2009, Patidar et al 2009,

Jiankun and Fengling 2009, Fishawy and Zaid 2007, Ahmed et al 2007,

Krishnamoorthi and Sheba Kezia Malarchelvi 2008, Zhang et al 2007) as

shown in the Table 4.13.The correlation coefficients for the original and

encrypted images are calculated by using the equation (3.4)-(3.6) as

mentioned in section 3.1.4 and shown in the Table 4.14. It is clear that the

two adjacent pixels in the original image are highly correlated, but there is

negligible correlation between the two adjacent pixels in the encrypted image.

Table 4.13 The Average Values of Various Cross Correlation Values between Plain Images and their Corresponding Cipher Images Produced by using 100 Randomly Chosen Secret Keys

S.No Image Size Direction of Adjacent Pixels CC V CCH CCD

1 Lena 512×512 -0.0000137 0.0014838 0.00193042 Man 1024× 1024 0.0001455 -0.0020267 0.00191733 Truck 512×512 0.0002895 -0.0002960 0.00579024 Girl 256× 256 0.0009585 -0.0530100 0.00859725 House 512× 512 0.0039037 -0.0078371 0.01279306 Tree 256×256 -0.0019082 0.0056540 0.00184377 Jelly beans1 256×256 0.0096442 -0.0099263 -0.00138508 Jelly beans2 256×256 0.0002329 -0.0119130 0.02032909 Splash 512×512 0.0042977 -0.0037865 0.000743710 Girl (Tiffany) 512×512 0.0026670 -0.0118100 0.004214411 Mandrill 512×512 -0.0023210 -0.0010998 0.006324212 Airplane (F-16) 512×512 -0.0012490 -0.00097192 0.006207613 Sailboat on lake 512×512 0.0048180 -0.0022361 0.006014314 Peppers 512×512 0.0003603 0.0017060 0.006155815 Aerial 256×256 -0.0039220 0.0118920 -0.002707516 Airplane 256×256 0.0042030 -0.0096231 0.016641017 Clock 256×256 -0.0041880 0.0106830 0.014376018 Resolution chart 256×256 -0.0008446 -0.0048259 0.011594019 Chemical plant 256×256 -0.0021289 -0.0042305 0.016563020 Couple 512×512 0.0005150 -0.0031021 0.0027036

101

The schematic diagrams as 3D bar chart is shown below to compare

the present method to the existing method (Ismail et al 2007) in Horizontal

and vertical correlations in Figures 4.12 and 4.13.

.Figure 4.12 Compares Present Method with AES Image Encryption

Method in Terms of Horizontal Correlation Co efficient

Figure 4.13 Compares Present Method with AES Image Encryption Method in Terms of Vertical Correlation Co efficient

102

Table 4.14 Correlation Co efficient of the Cipher Image for Encryption Quality (Ismail et al 2007)

Image AES

(Buchholz Jörg 2001)

Ismail et al Gun et al Chen et alPresent method

Lena 0.0029048 -0.0001046 0.0090000 0.0089000 -0.00070495

Ship 0.0049048 0.0000425 0.0042000 0.0022000 -0.00008723

Penguin 0.0099048 0.0005917 0.0114000 0.0100000 0.000106790

PSNR and Speed Analysis: PSNR (Ramana Reddy et al 2009) of encrypted

image and original image are obtained by using the equation (3.8) as

presented in section 3.1.4. From the statistics in Table 4.16, from the table the

higher the visual quality of the encrypted image, the lesser the number of

changed pixels will be, and the larger the values of PSNR with less time cost

are indentified. The PSNR for PS based, BFTCGH based, Integrated imaging

PS based and present novel method with 85% data losed are calculated. The

calculated values are compared with PSNR obtained for the present novel

method. The differences are shown in the Table 4.16.

Table 4.15 Comparison of PSNR’s for the PS based, BFTCGH based, Integrated Imaging PS based and Present Novel Method with 85% Data Loss As Index

PS-based BFT CGH-based

Integral Imaging and PS – based

Present method

PSNR(dB) 5.58 9.45 17.11 12.8

Apart from the security consideration, running time and speed

(Marwa et al 2008) are also important on image encryption.

103

In Table 4.17 shows the performance evaluation of the present

scheme. The analysis has been done by selecting 12 images each one is of

different sizes (1.5, 9, 24, 44 and 72 MB etc) having different contents and

using 10 randomly chosen secret keys for each image. Some of performance

evaluation to present novel method and (Marwa et al 2008) other methods are

shown in the Table 4.17

Table 4.16 Time of Image Encryption and Algorithms (seconds)

Algorithm Lena Goldhill

Combinational Permutation (Mitra et al 2006) 0.3300 0.9800

CKBA (Yen and Guo 2000) 1.0500 2.2700

Encryption using SCAN pattern

(Maniccam and Bourbakis 2004)

2.5400 4.7700

ECKBA (Socek et al 2005) 1.8400 2.8600

Present Technique 0.2926 0.4277

Key Sensitivity Test and Evaluation Analysis with Several Slightly different Keys

The key sensitivity test (Ahmed et al 2007) has been performed

through the following key sensitivity tests. The encryption scheme should be

key-sensitive meaning that a smallest change in the key will cause a

significant change in the output. In the present method, used the fixed initial

value = 0.123456789, = 0.265200000, changing the system parameter

with a single bit (Shubo et al 2009, Patidar et al 2009, Etemadi Borujeni et al

2009, Ismail et al 2007).

104

The system parameter can be any value in the finite area

3.569945< 4, thus and can provide (1) and (2) with the same value.

The key sensitivity test is performed in detail according to the

following steps:

(1) First, a 256×256 image is encrypted by using the test

key1 = 0.123456789 and its corresponding encrypted image is

referred as encrypted image A as shown in Figure 4.14(a).

(2) The least significant bit of the key is changed, so that the

original key becomes key2. Key2 = 0.123456788, which is

used to encrypt the same image, and its corresponding

encrypted image is referred as encrypted image B as shown in

Figure 4.14(b).

(3) Again, the same image is encrypted by the key3 = 0.123456787

and its corresponding encrypted image is referred as encrypted

image C as shown in Figure. 4.15(c).

Image A: cipher image of key1; Image B: cipher image of key2; Image C:

cipher image of key3.

Figure 4.14 (a) Cipher Image with Key 1 and its Histogram

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Figure 4.14 (b) Cipher Image with key 2 and its Histogram

Figure 4.15 (a) Cipher Image with Key1 (b) Cipher Image with Key 2 (c) Cipher Images with Key3 and (d) Difference Image with Key1 and KEY2

(4) Finally, the above three encrypted images A, B and C,

encrypted by the three slightly different keys, are compared as

given below in Table 4.15.

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Table 4.17 Key Sensitivity test with Several Slightly different Keys

Image Encrypted Image1

Encrypted image2

Pixel Difference

Present MethodCorrelation Coefficient

Man Image A

Image C

Image B

Image B

Image A

Image C

99.60

99.61

99.61

-0.00030842

-0.00020265

-0.00011947

Tree Image A

Image C

Image B

Image B

Image A

Image C

99.60

99.59

99.62

0.00460000

-0.00330000

0.00110000

Airplane

(F-16)

Image B

Image C

Image B

Image A

Image A

Image C

99.62

99.61

99.59

0.00190000

0.00070425

0.00290000

Average Moving Distance of Scrambling: The average moving distance of

scrambling (Xiangdong et al 2008) is calculated as mentioned in equation (4.4).

The statistical results are evaluated for moving distance of pixel. The results

from other chaos based method and present novel method is shown in the

Table 4.18. From the table, the larger the average moving distance of the

scrambling method less relation between the original image and the scramble

image and the increases the efficiency of the method.

Table 4.18 Statistical Results of Average Moving Distance of Present row Scrambling Algorithm with Randomly Choosing 1000 Keys for a 256×256 Image

Average Moving Distance Maximum Minimum Average

LENA 141.2014 118.1848 130.4608

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Entropy: The Entropy (Fishawy and Zaid 2007) is calculated as mentioned in

equation (3.7). The average of present novel method gives 7.99. Hence a

statistical attack is too difficult to make. In Table 4.19 compares Cross

Correlation and Entropy for various images, it shows that the present model is

highly secured.

Table 4.19 Comparison of Cross Correlation Efficient (CC) and Entropy

S.No Image Size CC Entropy

1 Lena 512 x 512 -0.00070495 7.9993

2 Man 1024 x 1024 -0.00070225 7.9998

3 Truck 512 x 512 -0.00118770 7.9993

4 Girl 256 x 256 0.00219600 7.9970

5 House 512 x 512 -0.00033701 7.9973

6 Tree 256 x 256 0.00051871 7.9970

7 Jelly beans1 256 x 256 0.00295000 7.9969

8 Jelly beans2 256 x 256 -0.00056896 7.9978

9 Splash 512x512 -0.00260560 7.9993

10 Girl (Tiffany) 512x512 0.00158940 7.9992

11 Mandrill 512x512 0.00008723 7.9993

12 Sailboat on lake 512x512 0.00010679 7.9992

13 Peppers 512x512 -0.00058102 7.9993

14 Aerial 256x256 0.00105870 7.9974

15 Airplane 256x256 0.00184570 7.9972

16 Resolution chart 256x256 -0.00055520 7.9973

17 Chemical plant 256x256 0.00102370 7.9974

18 Couple 512x512 0.00241990 7.9993

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Sensitivity Analysis: The present encryption novel method uses three

different chaotic maps, with different initial values and they are used in row

scrambling, column scrambling and XOR operation. Therefore, the present

method provides a choice of using three different keys.

Hence, larger keys space of iterations (logistic map) to skip before

the actual encryption/decryption starts. The key key-sensitive analysis is

tabulated in Table 4.16, the complete key space for the present encryption

/decryption technique is ~1045, i.e., the effective key of log 2[(10.558)3

X 1045] ~ 157 bits, which is sufficient enough to resist the brute-force attack.

The sensitivity analysis (Ahmed et al 2007) could be performed

through net pixel change rate and unified average change in intensity as

mentioned in equation (3.9) and (3.10). The NPCR (Fishawy and Zaid 2007)

value of the difference image is as follows: NPCR of AB = 99.6002, NPCR

of BC = 99.6239 and NPCR of CA = 99.6140. The present novel method

gives encryption scheme reaches an average UACI of 15-22%.

The Chaos theory has already proved that it is an excellent

alternative to provide a fast, simple, and reliable image encryption scheme

and has a high enough degree of security. From an engineer’s perspective,

chaos-based image encryption technology is very promising for real-time

security of a still image and video communications in military, industrial, and

commercial applications. In the first present work, an image encryption

scheme is present based on CSDP based scrambling technique using two

chaotic maps have been described in detail.

The system is a block cipher based architecture and its

effectiveness is tested. In the second present work, a new image encryption

algorithm with a large pseudorandom permutation which is computed from

chaos logistic maps and PMMLCG generators. The initial condition of chaos

109

logistic map and PMMLCG can be selected easily. In the third present work,

a chaos based image encryption with integrated scrambling technique has

been discussed. A detailed statistical analysis is given above and the

experimental results shows that it outperforms the existing techniques, both in

terms of speed and security.

Analysis of the statistical information of encrypted images in the

experimental tests, shows that the present algorithm provides reasonable

security against statistical cryptanalysis. In the next chapter a design of a new

image encryption technique and effect of diffusion technique in an image

encryption technique is explored.