50120140501009

International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-

6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME

68

IMAGE ENCRYPTION AND COMPRESSION BASED ON COMPRESSIVE

SENSING AND CHAOS

Prof. Maher K. Mahmood(1)

, Jinan N. Shehab(2)

1(Electrical Engineering Department/ University of Al – Mustansiriya, Bagdad, Iraq)

2(Computer and Software Department/ University of Diyala, Baquba, Iraq)

ABSTRACT

This paper presents image encryption based on Compressive Sensing (CS) and chaos. Image

compression and encryption are done based on CS, which is used due to many properties; greatly

reduces the signal sampling rate, power consumption, storage volume and computational complexity,

in additional to above; CS combined compression and encryption in the same step. Since CS-based

encryption method alone fails to resist against the chosen-plaintext attack. Hence, the output of CS is

again encrypted based on multi-chaotic system. This is used to enhance the security. Also, multi-

chaotic is used as key will increase key space, since multi-initial conditions and multi-parameters

make it very difficult to decrypt without knowing all those values, the structure of this system is

more complex than the low-dimensional chaotic systems and it is more difficult to forecast such

chaotic. The simulation results show that the cipher image has large key space, low storage and

transmitted requirement, high security and low encryption time requirement, incoherence, key

sensitivity and good statistical property. Also the recovered image has good quality (to human

perception) and preserves both the intelligibility and the characteristics of the image.

Keywords: Image Encryption and Compression, Image Encryption Based on Compressive Sensing

and Chaos, Multi-Chaotic Based CS, Multi-Chaos Based Image Encryption.

I. INTRODUCTION

Image encryption; is a technique that provides security to images by converting the original

image to another image which is difficult to understand, since billions of information such as image,

video,..etc are produced and processed per day and this information either are sent through the

channel insecure with limited capacity or are stored. In both cases, this information requires to be

minimized in order to get less number of data and contains the largest number of information. Hence,

there is an urgent need for compression and encryption at the same time. Unlike text messages,

INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING &

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ISSN 0976 – 6367(Print)

ISSN 0976 – 6375(Online)

Volume 5, Issue 1, January (2014), pp. 68-84

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69

image data have special features make text encryption algorithm cannot directly implemented to

images because image size is much greater than that of the text, and the other problem is that,

decryption text must be equal to the original text however this requirement is not necessary for image

data. The traditional encryption algorithms such as DES, AES, IDEA which are used for text or

binary data, appear not to be ideal for multimedia applications, the basic reasons; huge in size and

bulk capacity, high redundancy and a high correlation between pixels, then traditional encryption

methods are difficult to apply and slow to process[1]. In the classical secret communication's

approach, the messages encrypted and compressed, separately. Now, it is possible to directly

compress and encrypt in the same step. Compressive sensing is a novel technique built upon the path

breaking work by Candes, Romberg, Tao, and Baraniuk[2][3].

Chaos Theory; which was developed by Edward Lorenz, studying the behavior of dynamical

systems that are highly sensitive to initial conditions “The Butterfly Effect”[1][4]. This paper

presents an compressive sensing and chaotic system based image encryption and compression.

Section II describes the CS theory, in section III we describe in details multi-chaotic based image

encryption, while sections IV and V we describe the proposed algorithm about using CS and chaotic

system together in encryption and compression; while the performance of the algorithm and

simulation results with compression tests and encryption tests are presented in Section VI. Section V

concludes the paper with some remarks.

II. COMPRESSIVE SENSING THEORY

The basic concept of CS is to represent the original signal in a convenient basis Ψ. Then it

employs a non-adaptive linear projection onto observation matrix Φ that preserves the structure of

the signal and uncorrelated with the transform basis Ψ, and then the signal can be accurately

reconstructed by solving the convex optimization problem or greedy pursuit algorithm with a small

amount of measured values[5]. CS relies on two principles 1) sparsity: - which pertains to the signals

of interest, Sparsity expresses the idea that the information rate of signals can be much smaller than

suggested by its bandwidth. and 2) incoherence: - which pertains to the sensing modality,

Incoherence expresses idea that signals having sparse representation in representation basis Ψ must

be spread out in the sensing basis Φ[6]. CS framework that mainly consists of two crucial parts: -

sampling (encoding) and recovery (decoding).

a. SAMPLING (ENCODING)

Sampling mainly contains two parts: signal presented in sparsity and measurement matrix: -

1. SIGNAL PRESENTED IN SPARSITY The signal X is called a K-sparse/compressible if it can be represented as a linear

combination of only k basis vectors; only k elements of the vector S are non-zero [8][9]. For image

data consider a real value, finite length, two-dimensional discrete-time signal X, which can be

viewed as (N×N) pixels in RN×N

with elements x (n, n). The signal X � RN×N

which can be expanded

on the orthonormal basis (such as DCT, DWT) ψ = [ψ1 ψ2……ψN], a signal X can be expressed as :-

X ��S�,�

��Ψ�,�

�� or X � ΨS �1�

Where S is the (N×N) pixels of weighting coefficients Si,j =��, ��= ��,�� and T denotes

transpose, containing exactly K nonzero coefficient, K << N.



70

Ψ is a specific N×N dictionary (sparsifying basis matrix) that its columns are orthonormal and spans

X domain and S is the coefficient vector of X in basis Ψ= [ψ1 ψ2 ……..ψN].

Clearly X and S are equivalent representations of the signal X in the spacial domain and S in

the Ψ domain as shown in Fig.(1).

2. MEASUREMENT MATRIX (Φ) It is any random generated matrix such that the information in every S sparse signal is not

damaged by dimensionality reduction from N×N to M×N samples[7]. Consider a general linear

measurement process that computes (K<M<N) inner products between X and collection of vectors

��

:-

Y� � �x, �� 2�

then by substituting Ψ from (1) in Y we can write it as:-

! � "� � #$% � Θ% �3�

where Θ is called sensing matrix using only with compressible signal, Θ � RM×N

, Θ=Φ ψ is an

(M×N) matrix. And Φ is called measurement matrix, and if signal or image is sparse (don't need

transform domain) then Φ is called sensing matrix Θ=Φ, Φ= [φ1,..φM]T � R

M×N.

Y: is (M×N) measurement vector ( compressive sensing measurement) as shown in Fig.(1).

The measurement process is non-adaptive, meaning that Φ is fixed and does not depend on the signal

X[8] .This matrix is given by Candes, Romberg and Tao [2]. We begin with ill-conditioned problem

(M<N) and let X be a K-sparse and the K locations of the non-zero coefficients in S are known, this

problem can be solved provided M≥K by deleting all those columns and elements corresponding to

the zero-element using the following equation: -

Y=ΦK XK =ΦKΨKSK (4)

Where K is the support sets which is the collection of indices corresponding to the non-zero

elements of S.

1 ' () * ||Θ,||-||.||- * 1 / () �5�

The necessary and sufficient condition for (5) to be good condition is that for any K-sparse

vector V shares the same K non-zero entries as S. The sensing matrix should satisfy this condition,

for some 0<δ<1[12].

Θ is the sensing matrix can be seen as a transformation of the signal from the signal space to

the measurement space, where the measurement space is smaller than the signal space, Θ must

preserve the Euclidean length(||.||2) of these particular K-sparse vectors and δ is the isometry constant

(is the smallest value satisfying (5)) [9].Our aim is to get familiar with this inequality, this inequality

will be used repeatedly under the name Restricted Isometry Property (RIP). In practice however,

performance analyses based on RIP turns out to be challenging because of the difficulty of finding

() for a given specific measurement-matrix[10]. In a practical scenario, one can instead bound ()

with the mutual coherence. Another property of the measurement matrix is the mutual coherence.

The RIP requires incoherence that can be defined as follows: suppose we are given a pair (Φ,Ψ) of

orthobases of the R . The first basis Φ is used for sensing the object X as in (2) and the second basis

Ψ is used to represent X as in (3). The coherence between the Φ and Ψ is :-



71

µ(Φ,ψ) = √N max1≤K j≤N |�φk,ψj�| (6)

where 5 is coherence parameter measures the largest correlation between any two elements of Φ and

Ψ then if Φ and Ψ contain correlated elements the coherence is large, otherwise it is small as for how

large and how small, it follows from linear algebra that :-

µ(Φ,ψ) � [1,√N] (7)

CS is mainly concerned with low coherence pairs or incoherence that requires the row {φj} of

Φ cannot sparsely represent the columns {ψi} of Ψ and vice versa[9].

Figure (1) Compressive Sensing Diagram

b. RECOVERY (SIGNAL RECONSTRUCTION) Reconstruction of signal is nonlinear procedure with the aim to recover initial signal or its

sparse representation from M measurements and sensing matrix Θ. Based on the knowledge of

information measurements (Y,Φ,Ψ) the signal can be recovered by solving an underdetermined

linear system of equations.

III. MULTI-CHAOTIC BASED ON IMAGE ENCRYPTION

In the last years, an increasing attention has been devoted to the use of chaos theory to

implement the encryption process. In spite of much chaos-based on image encryption schemes have

been proposed, but the class of cryptosystem uses the confusion-diffusion architecture proposed by

Fridrich. The main advantage of these encryptions lies in the observation that a chaotic signal looks

like noise for non-authorized users ignoring the mechanism for generating it. Secondly, time

evolution of the chaotic signal strongly depends on the initial conditions and the control parameters

of the generating functions then slight variations in these quantities yield quite different time

evolutions[1].From above chaos-based image encryption appears a good combination of speed,

security and flexibility either a chaotic block cipher or chaotic stream cipher. Although the

application of a 1-D chaotic method such as (Logistic map, Cat map,...etc) based on image

encryption is convenient and quick but some weakness appeared such as, small key space, weak

security and complexity [20]. But the encryption sequences produced by using multidimensional like

(Lorenz, Rossler, .etc) have excellences; One is that the structure of this system is more complex

than the low-dimensional chaotic systems, It is more difficult to forecast such chaotic sequences. The

other is that the real value sequences of three system variables can be used separately or put together

to use, the design of encrypting sequence is more convenient[1][11]. In this paper, four types of

chaotic systems are used, These are :-

Insecure

channel

X Ψ Y=ΦX = ΦΨS = ΘS Recovery

Φ

Inverse

�6

�7 � �6%8 96



72

1. LORENZ SYSTEM Lorenz system is a classical chaotic system of differential equation arose from the work of

meteorologist mathematician Edward N. Lorenz, he published in 1963 [12]. The dynamic equation of

Lorenz system is as shown in Table (1). Among them, a, b and c are the system parameters. The

Lorenz attractor is shown in Fig.(2(A)).

Figure(2) Chaotic Attractor (A) Lorenz Attractor (B) Rossler Attractor (C) Chua Attractor(D)

Henon Attractor

2. ROSSLER SYSTEM Otto E. Rossler, tries to enhance the Lorenz attractor and designs his own model for

chaos in 1976 [13].The dynamic equation of Rossler system is shown in Table (1), where

a, b, and c are control parameters of the system. The Rossler is shown in Fig.(2(B)).

3. CHUA SYSTEM The first real physical dynamical system, capable of generating chaotic phenomena in the

laboratory, similar to those in the Lorenz system, was invented by Chua in 1992[14]. Because of its

simplicity, robustness, and low cost that Chua’s circuit has become a favorite tool for analytical,

numerical and experimental study of chaos. Chua’s circuit can be described by differential equations

as shown in Table(1).a, b, c, m0, m1 are parameters of the system. The Chua's circuit is shown in

Fig. (2(C)).

A B

C D

-4

-2

0

2

4

-1

-0.5

0

0.5

1

-6

-4

-2

0

2

4

6

-1.5 -1 -0.5 0 0.5 1 1.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-10

-5

0

5

10

15

-20

-10

0

10

0

10

20

30

40



73

4. HENON MAP A two-dimensional discrete-time nonlinear dynamical system proposed by the French

astronomer Michel Henon in 1976[15]. See equation in Table(1). The map depends on two

parameters a, and b. This map is shown in figure (2(D)).

IV. COMPRESSION AND ENCRYPTION PROCEDURE

The proposed algorithm for transmitter side is shown in Fig.(3) is elaborated in the

following :-

1. Dividing Original Image:- Usually, the size of a natural image will be considerably large, the

original image is resize into N×N pixels and then divide into four equal size blocks.

2. Discrete Wavelet Transform (DWT) Based CS:- Generally, the image itself is not sparse, but if

image is represented in certain transformation then it will be sparse. In this work, the DWT is used to

do the 4-level wavelet decomposition of the input block (each block of the original image

separately).When we apply DWT into blocks as shown in figure(4), then each block represents S is

the sparse/compressible image matrix with K-nonzero coefficients.

3. Chaos-Based Measurement Matrix:- Generation of the pseudo-random measurement matrix Φ

utilizing a cryptographic key, offers a natural method for encrypting the signal during CS. The

security of the encryption method relies on the fact that Φ is not known to an attacker that does not

have the pseudo-random key used to generate Φ. Consequently, finding a proper Φ satisfying RIP

and incoherence is one of the most important problem in CS. Here, in this work the chaotic sequence

is used to construct such a sensing matrix, called chaotic matrix. Based on sensitivity to initial

conditions and parameters, egodicity and statistical property of chaotic sequence, one can prove that

chaotic matrix can also have RIP with overwhelming probability, provided that S ≤ α (M/ log (N/S))

[2]. Unlike the one-dimensional separate system, the Lorenz system needs to make use of the

numerical solution of differential equation to obtain the real value chaotic sequences. We apply the

Range-Kutta method to solve the Lorenz system based on these qualifications: -

Once obtaining the X(n), Y(n) and Z(n) real value chaotic sequences, then, before using the

chaotic sequence to construct Φ, we need to transform chaotic sequence generated by using Eqs.

(8,9) into an integer sequence(0-255). Magnification and modulo transformation to the two chaos

types (Lorenz and Chua) are done as follows:-

XL (n) = mod (floor (XL (n) ×1015

), 256) , XC (n)= mod (floor (XC (n) ×1016

), 256)

YL (n) =mod (floor (YL (n) ×1015

), 256) , YC (n) =mod (floor (YC (n) ×1016

),256)

ZL (n) = mod (floor (ZL (n) ×1015

), 256) , ZC (n) = mod (floor (Z C (n) ×1016

),256)

To make proposed system more secure. The Lorenz and Chua sequences are combined

together by using XOR to get new chaotic sequence.

XLC (n) =BITXOR (XL (n) , YC (n))

YLC (n) =BITXOR (YL (n) , ZC (n) ) (12)

ZLC (n) =BITXOR (ZL (n) , XC (n) )

FLC (n) =BITXOR (XL (n) , XC (n))



74

To normalize, divide by M:-

XLC (n) =1/M × (XLC (n))

YLC (n) =1/M × (YLC (n)) (13)

ZLC (n) =1/M × (ZLC (n))

FLC (n) =1/M × (FLC (n))

where M is the number of measurements and represents the new number of rows for image after

reduction. This M which decides the compression ratio and also the reconstruction performance.To

make the values between positive and negative and this makes work more secure and have better

reconstruction :-

KΦ1 = XLC (n) - Xone

KΦ2 = YLC (n) - Xone (14)

KΦ3 = ZLC (n) - Xone

KΦ4 = FLC (n) - Xone

Xone =all 1's matrix of size (M×N/2)

Then the new Φ= :;<� ;<=;<> ;<?@ , the size of Φ equal 2M×N elements ,as shown in Fig.(5).

4. Compressive Sensing Measurement Y:- In this work, measurements Y are obtained by projecting the resultant from DWT into chaotic matrix

Φ to take important information with non-zero values without duplicating.

5. Lloyd's qantizer:- The resultant coefficients of Y will be large number (64bits/pixel). We must

quantize to give minimum bits/pixel(in this work, (8bits/pixel) was shown to be enough).

Quantization is implemented through the well-known Lloyd quantizer. Before using Lloyd all the

elements of Y matrix are divided by 100 to reduce the high values of Y. as well as there are negative

values in Y are difficult to apply the Lloyd values is withdrawn by the middle or minimum value in

Y. This step is applied to each block separately: -

YA�M, N/2� �Y DM, N2E ' g

100 �15�

where g represents the min value in Y(M,N/2).Then using Lloyd's algorithm in the new value

of HI.The resultant quantized compressive sensing measurement Yq(4-block). But the existing CS-

based encryption methods fail to resist against the chosen-plaintext attack. In this research we have

tried to find a simple, fast and secure algorithm for image encryption, Then a new symmetric image

cipher based on the widely used confusion–diffusion architecture is used. The proposed stream

cipher is based on the use of Lorenz, Chua, Rossler and Henon to construct keys used in confusion

and diffusion. Previously; we generated two types of Chaos(Lorenz and Chua). They are now

modified to be between values (0,255). In the same way to Lorenz, Rossler and Henon sequences are

generated and adjusted between (0,255). Most researchers used chaotic image encryption that

depends on one chaotic systems like Lorenz or Rossler systems. In this work, a new chaotic is

presented based on adding two variables from different 4-chaotic systems (Lorenz, Chua, Henon and

Rossler) to construct new variable chaotic sequences, as shown in Table (2).

International Journal of Computer Engineering and Technology (IJCET), ISSN 0976


Figure (3) The Proposed Compression and Encryption Image at Transmitted Side, D

Reconstruction at Receiver Side

Figure(4) Wavelet Transform Decompositions (4

Figure (5) Measurement Matrix


6375(Online), Volume 5, Issue 1, January (2014), © IAEME

75

The Proposed Compression and Encryption Image at Transmitted Side, D

Reconstruction at Receiver Side

Wavelet Transform Decompositions (4-Blocks)

Measurement Matrix Φ Using Chaotic System (Four Blocks)



The Proposed Compression and Encryption Image at Transmitted Side, Decryption and

Using Chaotic System (Four Blocks)



76

Table (2) The New Keys Generated that will be used in Encryption

The new keys for confusion mechanism The new keys for diffusion mechanism.

Kc1 =XL YH , Kc2 =YL XH

Kc3 =XC YR , Kc4 =YC ZR

Kc5 =ZC XR , Kc6 =XH YR

Kc7 =YH ZR , Kc8 =ZL XR

Kc9 = [ZL ZC ZR mod(XL+YH,255)]

Kc10 = [XL YC XH ZR]

Kc11 =XH XR , Kc12 =YH YR

Kd1=[XL XC XH XR] , Kd2=[YR YH YC YL]

Kd3=[ZL ZC YH ZR] , Kd4=[ZR XH ZC YL]

Kde1=reshape (Kd1, M, N/2)




Kde=(Kde1 Kde2) (Kde3 Kde4)

6. Confusion:- Unlike the text data that has only two neighbors, each pixel in the image is in

neighborhood with eight adjacent pixels. For this reason, it is important to disturb the high

correlation among image pixels to increase the security level of the encrypted images. This work

employs six-steps of confusion procedure:-

Step(1): - Conduct the function “Sort” on Kc9 for constructing scrambling index array SI9 with

dimension (2MN×1) arranged in ascending order. Accordingly, the scrambling array I9 can be

produced from chaotic key Kc9. Transform the quantized CS image from Yq(2M×N) into Yq(2MN×1),

and then rearrangement the pixels on Yq according to the sort of the chaotic key I9, we can get the

scrambling matrix Yc1(2MN×1). Transfer back the matrix Yc1(2MN×1) to Yc1(2M×N).

Step(2): - Dividing image result from step (1) Yc1(2M×N) into 4-equal block, and repeat the process by

which the first step was done. After the division of the image, we deal with each individual block and

transform it into a single row and the first step is restored for each block. But here, the sort of the key

(Kc1, Kc3, Kc5, Kc7) is used for the block (1,2,3,4), respectively. And then return every block from

(1-D) into (2-D) ,then get YC2(2M×N).

Step(3): -Working process of permuting between blocks, here the image is divided into 64 blocks

and use a sort of key construction from two types of chaos:-

Kb= YL XC

Step(4): -The resulting image from the previous step is treated like one block and we repeat the

process in the first step but here the image is transformed into a single column. In this step we use

the sort of the key Kc10 as shown in Table(2).Return back to 2-D then Yc4(1×2MN) become Yc4(2M×N).

Step(5): -The pixels are processed within each block after converting pixels within each block to a

single column. Here we use the sort of the key (Kc2, Kc4, Kc6, Kc8) for the block (1, 2, 3, 4)

respectively. Then return back every column in every block into (2-D) Yc5(M×N/2).

Step(6): -In the last step of confusion procedure image (Yc5(2M×N)) is treated like one (2-D) block and

permutation pixel positions start in all image such as the row according to the sort of the key Kc11

and column according to sort of the key Kc12.

7. Diffusion:- Although pixel positions of an image were scrambled in the previous steps, generally

the distribution of gray-scales of the image is still unchanged, i.e., the histogram of the plain image

(here the plain image is Yq) is about the same as that of the (Yc6). This leaves a door widely open for

statistical attack and chosen-plaintext attack. Thus, a diffusion process is necessary to make the

spread influence of each single pixel overall the image[4]. The goal of the diffusion step is to encrypt



images by changing the grey scale values to cr

mask and image will result in confusion step

adjacent pixels and this information is spread out in the backward direction over

The vertical diffusion(VD) considers t

last pixel of the last column in the image and then moves backward row

the last pixel is modified by XORing itself and the corresponding value in key stream

before last pixel is modified by XORing the last and before last pixels and the corresponding value in

Kde. The last pixel of each column (except the last column

first pixel of the previous column and itself

transmitter to the receiver through insecure channel.

V. RECOVERY AND DECRYPTION

Suppose that at the receiver side as shown in Fig.

g, Kci and Kde from a separate secured channel then depended on (Y

reconstruct image. And then, we proposed dividing an

applying GP (OMP (Orthogonal Matching Pursuit)

(Compressive Sampling Matching Pursuit

VI. NUMERICAL SIMULATION RESULTS

a. Data Set In these experiments, three grayscale images all of size 512x512 pixels are used to test the

proposed algorithm. The images used are shown in the T

The following subsections will review the evaluation measures for both image compression

and encryption: -

b. METRICS FOR IMAGE COMPRESSION (RECOVERY)The terminology “recovery”

measurement data. In measuring the quality of the reconstructed image, the peak signal

known as PSNR is used. PSNR of an a X a 8

calculated as: -

where X (n,n) represented the intensity of a pixel in the original image, while its reconstructed

counterpart is denoted by (n,n). PSNR is measured in decibels (dB), N: height of the image, N:

width of the image, The result shown is in Table(4 Besides measuring the image quality, we also measure the compression ratio (compression is

done for sub blocks N/2×N/2 to M×N/2):

Lena.bmp



77

changing the grey scale values to create an encrypted image. Hence, XORing the chaotic

age will result in confusion step. In the diffusion step, we mix the properties of vertical

adjacent pixels and this information is spread out in the backward direction over the whole image.

) considers the image obtained after step (6) as the input.

the image and then moves backward row-major order. In this process,

the last pixel is modified by XORing itself and the corresponding value in key stream

el is modified by XORing the last and before last pixels and the corresponding value in

Kde. The last pixel of each column (except the last column) is modified by XORing the modified

and itself, the resultant image encryption Yen is transmitted from

transmitter to the receiver through insecure channel.

. RECOVERY AND DECRYPTION USING GREEDY PURSUIT(GP)

er side as shown in Fig.(2), Yen is received, along with the keys K

separate secured channel then depended on (Yen, g, KΦi, Kci and

we proposed dividing an sparse image to a block of 256×256

(Orthogonal Matching Pursuit), SP (Subspace Pursuit)

(Compressive Sampling Matching Pursuit)) to each block.

. NUMERICAL SIMULATION RESULTS

In these experiments, three grayscale images all of size 512x512 pixels are used to test the

used are shown in the Table (3).

Table (3) Test Images


METRICS FOR IMAGE COMPRESSION (RECOVERY) ” refers to decrypt and then reconstruct the plain image from its

In measuring the quality of the reconstructed image, the peak signal

PSNR of an a X a 8-bit grayscale image X and its reconstruction

n,n) represented the intensity of a pixel in the original image, while its reconstructed

PSNR is measured in decibels (dB), N: height of the image, N:

image, The result shown is in Table(4).

Besides measuring the image quality, we also measure the compression ratio (compression is

done for sub blocks N/2×N/2 to M×N/2): -

Peppers.png



XORing the chaotic

ix the properties of vertical

the whole image.

It starts from the

major order. In this process,

the last pixel is modified by XORing itself and the corresponding value in key stream Kde, the pixel

el is modified by XORing the last and before last pixels and the corresponding value in

) is modified by XORing the modified

is transmitted from

is received, along with the keys KΦi,

, Kci and Kde) we can

256×256 pixels and

(Subspace Pursuit) and CoSaMP

In these experiments, three grayscale images all of size 512x512 pixels are used to test the


uct the plain image from its

In measuring the quality of the reconstructed image, the peak signal-to-noise ratio

and its reconstruction is

n,n) represented the intensity of a pixel in the original image, while its reconstructed

PSNR is measured in decibels (dB), N: height of the image, N:

Besides measuring the image quality, we also measure the compression ratio (compression is

Black.bmp



78

Compression ratio � uncompressed Vile sizecompressed Vile size � Y Z Y

2[ Z Y �17�

Rate of compression�RC� � 1compression ratio �

2[Y �18�

In general, the higher the compression ratio, the smaller is the size of the compressed file[1]

as shown in table(3).

Table (4) Results of PSNR and Rate of Compression ( Chaotic-Based Measurement Matrix)

Image

Name

Measurement

Reduction M

PSNR(dB) RC Time to reconstruct (Second)

OMP SP CoSaMP OMP SP CoSaMP

Lena 84

100

120

15.939

21.339

29.77

22.28

28.007

30.203

18.9775

27.3529

29.9581

0.3281

0.3906

0.4688

16.937

20.717

20.157

10.7

19.058

15.798

15.107

18.511

17.710

Black 12 ∞ ∞ ∞ 0.0469 0.6861 0.0679 0.1216

Peppers 84

100

120

19.549

28.242

31.244

23.677

29.50

31.22

22.5895

29.3015

31.1868

0.3281

0.3906

0.4688

14.782

18.475

22.136

11.198

16.573

16.842

14.359

16.833

18.137

Since a fixed 8-bit rate quantization was used, the compression ratio depends only on the

parameter M, from the results in Table(4), one notice that a large M means more coefficients to be

captured, this yields the high quality of reconstruction and high compression rate while small M

yields an aborted case. Also we discussed three reconstruction algorithms and compared the

advantages and disadvantages of them. PSNR is used to measure the quality of recovery. Higher

PSNR value gives better recovery performance. From Table (4) OMP algorithm can achieve very

high PSNR when the size of measurements is large, it is not accurate any more if the M is small.

Since the algorithm picks the optimal entries one by one, it is very slow. So OMP is not an ideal

algorithm in reality. CoSaMP algorithm is faster than OMP as shown in Table(4). The PSNR is

acceptable if we have a large size of measurements. SP algorithm is fastest among these algorithm as

shown in Table(4). It is not difficult to tell SP algorithm can offer a robust recovery by using fewer

measurements comparing to OMP and CoSaMP. By testing the images, algorithms can provide

satisfactory results when the images are smooth(Black image). But when the images are rough or

have a lot of details (Lena and Peppers images), the recovery results are not good. This kind of

images needs more measurements to reconstruct the images.

c. EVALUATION OF ENCRYPTION PROCESS

To prove that proposed technique has high security and can resist all kinds of known attacks.

Here, some security analysis results are carried out on the scheme :-

1. Key Analysis A good image encryption algorithm must be sensitive to the cipher key, and the key space should be

large enough to make brute-force attacks infeasible.



79

A. Key Space Analysis Key space size is the total number of different keys that are used in the encryption. The

chaotic system used in this work is highly sensitive to the initial values of the system, the key space

size is = 10168

, when we compare the key space of encrypted images obtained from the CS-based

encryption methods in [5][16][17] then key space is 3.4028×1038

. This will provide more sufficient

security against the brute force attack than methods in [7][10][16].

B. Key Sensitivity Test Sensitivity which is a basic criterion for an encryption method requires that a slight change

results in a completely different output

• At Transmitter Side The first test verified the key sensitivity of the proposed image encryption using multi-

chaotic algorithm at transmitter side.

Figure (6) The Comparison between Encrypted Images by

A) Original Key (Q), B) Neighbored Key (Q ̂) and C) Different between A) and B)

Fig.(6(A)&(B)) depicted the corresponding two encrypted images. The difference image

between these two encrypted images was shown in figure(4 (C)) for perceptual observation, All the

percentage values exceeded 99%, which indicated that the tiny change in key brought great changes

in the encrypted image.

• At The Receiver Side The encryption system should be sensitive to the small changes on the decrypted keys. And,

generate a wrong decrypted image, if there is a small difference in the decryption keys[15]. Only the

same keys, should give the same image at the receiver side, as shown in figure (7(B)&(C)). Also, one

can use this test to see sensitivity Φ, if the initial conditions are changed by any part of the keys that

was used to generate Φ as shown in Fig.(7(E)).

Figure (7) Sensitivity Tests of Keys

From Fig.(7) we can see that the original image cannot be restored even if a tiny difference of

the key due to the extreme initial condition sensitive property of a chaos system.



80

2. Statistical Analysis An ideal cipher should be robust against any statistical attack. So, this work performs

statistical analysis by calculating the histograms and correlations coefficients.

A. Correlation Coefficients Analysis: - Correlation coefficient is the measure of extent and direction of linear combination of two

random variables[12]. This metric can be calculated as follows: -

Corrab � |cov�x, y�|eD�x� Z eD�y� �19�

Where x and y are the gray-scale values of two pixels at the same indices in the plain and cipher

images, while cov(.,.) and D(.) were computed as follows: -

E�x� � 1N �x� �20�

��

D�x� � 1N ��x� ' E�x��= �21�

��

cov�x, y� � 1N �ix� ' E�x�jiy� ' E�y�j �22�

��

To test the correlation between two (vertically, horizontally and diagonally) adjacent pixels in

a original and cipher image, are used respectively. First, randomly select 2000 pairs of adjacent

pixels (Vertical, Horizontal, Diagonal) from image (original and then encrypted). Then, calculate the

Corrxy of each pair by using the formulas (19),(20),(21) and (22). The results are shown in Table (5).

Table (5) Correlation Coefficients of Adjacent Pixels

Image

name

Direction Original

image

Encrypted

image in

[7,10]

Encrypted

image in [16]

Encrypted

by CS only

Encrypted

image in this

work

Lena Horizontal

Vertical

Diagonal

0.9719

0.9850

0.9593

0.0215

0.0808

0.0176

0.0033

0.0009

0.0058

0.77

0.02

0.0104

0.0000509

0.0033

0.0094

Black Horizontal

Vertical

Diagonal

∞

∞

∞

0.0214

0.0837

0.0187

0.0008

0.0046

0.0039

∞

∞

∞

0.00028

0.0021

0.0002199

Peppers Horizontal

Vertical

Diagonal

0.9894

0.9921

0.9829

0.0205

0.0737

0.0174

0.0007

0.0007

0.0035

0.869

0.1045

0.117

0.000107

0.000109

0.000025



81

The results show that the correlation coefficient is very close to zero in ciphered image, and

thus the proposed encryption algorithm is less predictable and more secure. This quantitative

evaluation demonstrated that the proposed method reduced the correlation by one order of magnitude

compared with the methods in[7][10][16].

B. Correlation Distribution(Similarity) of The Adjacent Pixels The correlation distribution test for horizontal, vertical, and diagonal adjacent pixels have

been performed for the proposed encryption algorithm and the results are gathered in Fig.(8).

Figure(8) Correlation Distribution, Shows the Test Results of Encrypted Images Obtained from the

Chaos and CS-Based Encryption Methods

c. Histograms Analysis To prevent the leakage of information to an opponent, it is also advantageous if the cipher

image bears little or no statistical similarity to the plain image. An image-histogram illustrates how

pixels in an image are distributed by graphing the number of pixels at each color intensity level.

(a)Original histogram (b)After CS Yq (c)After confusion Yc6 (d)After diffusion Yen

Figure (9) Histogram Test.

The histogram of the encrypted image by CS only Fig.(9b) is totally different from that of the

original image. But proposed system is still weak against statistical attacks. From the observation of

the Fig.(9b,c), the cipher values inherit the Gaussian distribution property introduced by the

measurement matrix, Gaussian distribution property of the measurement matrix leads to a

nonuniform distribution cipher image, which leaks statistic information to analysts. The proposed

method completely dissipates this Gaussian distribution replacing by fairly uniform distribution as

shown in Fig.(9d).

0 50 100 150 200 2500

50

100

150

200

250

0

2000

4000

6000

8000

10000

12000

0 50 100 150 200 250

0

500

1000

1500

2000

2500

3000

0 50 100 150 200 2500

2000

4000

6000

8000

10000

12000

0 50 100 150 200 250

0

2000

4000

6000

8000

10000

12000

0 50 100 150 200 250

0

200

400

600

800

1000

1200

0 50 100 150 200 250

0

0.5

1

1.5

2

2.5

3

3.5

4

x 104

0 50 100 150 200 250

0

2000

4000

6000

8000

10000

12000

0 50 100 150 200 250

0

2000

4000

6000

8000

10000

12000

0 50 100 150 200 250

0

200

400

600

800

0 50 100 150 200 250

Lena

Black

Pixel gray value on location Pixel gray value on location Pixel gray value on location Pixel gray value on location (X,Y)(X,Y)(X,Y)(X,Y) Pixel gray value on location Pixel gray value on location Pixel gray value on location Pixel gray value on location (X,Y)(X,Y)(X,Y)(X,Y)

Pixel Pixel Pixel Pixel gray gray gray gray valvalvalvalue ue ue ue on on on on location location location location (X+1,Y+1(X+1,Y+1(X+1,Y+1(X+1,Y+1))))

Pixel Pixel Pixel Pixel gray gray gray gray value value value value on on on on location location location location (X+1,Y+1)(X+1,Y+1)(X+1,Y+1)(X+1,Y+1)

Original image (Lena) Encryption image Original image (Black) Encryption image



82

3. DIFFERENTIAL ANALYSIS Generally, an opponent may make a slight change (modify only one pixel) of the encrypted

image so as to observe the change in the result. In this way, he may be able to find out a meaningful

relationship between the original image and the cipher image. This is known as the differential

attack. Since compressive sensing plays a role in sampling data, in this test, Yq are regarded as the

“plaintext image”. We argue that this test routine is impartial because the security strength of the

second encryption stage must not be stronger than that of the entire cryptosystem. We modified one

pixel of Yq, and then iteratively performed the second stage stream cipher. To test the influence of

one pixel change on the whole cipher-image, two most common measures NPCR (Number of Pixel

Change Rate) and UACI (Unified Average Changing Intensity) are used. Let the two cipher images

be C1 and C2, whose corresponding plain images have only one pixel difference. Label the gray

values of the pixels at grid (i, j) in C1 and C2 by C1(i, j) and C2(i, j), respectively. Define a bipolar

array D with the same size as image C1 or C2, namely, if C1(i, j) = C2(i, j) then D(i, j) = 0, otherwise

D(i, j) = 1. The NPCR and UACI are defined by: -

NPCR � ∑ D(i, j)�,�2M Z N Z 100% (23)

UACI � 12M Z N s� |c�(i, j) ' c=(i, j)|255�,� t Z 100% (24)

The higher the values of NPCR and UACI are the better the encryption[16]. The results of

these two tests are shown in Table (5).

From Table (6) the proposed method achieved the optimal diffusion, and outperformed

method in [16]. Then the proposed algorithm has a good ability against known plain text attack.

4. INFORMATION ENTROPY Information entropy can be used to characterize the confusion, and is calculated by :-

E(C) � ∑ PiC�,�jlog=(1 p(C�,�)) (25)⁄

Where P (ci,j) represents the probability of symbol ci,j for a grayscale image.

Table (6) NPCR and UACI Performance for Measuring the Plaintext Sensitivity

Image

name

Method in [16] (results are

given for Lena image only)

CS Only Proposed method

NPCR UACI NPCR UACI NPCR UACI

Lena 0.0038 0.0013 0.0031

1.24×10-5

0.9992 0.2512

Black 0.0054 9.72×10-5

0.9977 0.2505

Peppers 0.0041 1.6×10-5

0.9984 0.2506



83

From the observation of Table(7), the proposed method achieved outstanding confusion, and

outperformed the CS-based methods [7][10][16], in the sense that the corresponding entropy E(C)

was more close to the maximum value of 8 bits.

5. THE AVALANCHE EFFECT METRIC The avalanche effect metric can be used to test the efficiency of the diffusion mechanism. A

single bit change can be made in the image P to give a modified image Pw. Both P and Pw are encrypted

to give C and Cx. The avalanche effect metric is the percentage of different bits between C and Cx . If C

and Cx differ from each other in half of their bits, we can say that the encryption algorithm possesses

good diffusion characteristics[16].

The results listed in Table (8) showed that the change rate achieved by proposed method was

extremely close to the ideal case.

VII. CONCLUSION

Different techniques are used in this paper to implement image encryption and compression

such as 2-D wavelet transform based sparse representation, 2-types of chaotic sequence(Lorenz and

Chua)combined together based 4-blocks measurement matrix and used different key for each block.

(OMP, SP and CoSaMP) based reconstruction image, 6-kinds of confusion mechanism all depends

on chaotic sequence generated from combined 4-different chaotic types and 4-chaotic types XORing

with confused image to get diffusion image. With usage of CS based compression, we get first level

of security since; an original image is encrypted as a set of coefficients by a secret orthogonal

transform. Without knowing fixed M, quantized level and seed used to generate the exactly

measurement matrix, will be impossible to reconstruct original image. Compression ratio as well as

the reconstruction performance decided by the factor M and quantization level, increase M and

quantization level enhance reconstruction quality. For the weakness of using CS only-based image

encryption, we proposed a new quantized encryption algorithm based on multi-chaotic system. Then

this system gives the second level of security by using chaos based image encryption that has many

merits; It has a large enough key space to resist all kinds of brute-force attacks, since the key space

of the proposed system = 10168

. The cipher-image has a good statistical property, the histogram of

the encrypted image is fairly uniform, the correlation coefficient of two adjacent pixel are very small yzero, and the entropy is y8bits. The encryption algorithm is very sensitive to the secret keys and

plain-image, the NPCR and UACI of cipher image(Lena) are 0.9992 and 0.25. Then proposed

Table (7) Entropy Test

Image

name

Entropy for original

image

Entropy after

CS

Method in

[7,16]

Method in

[10]

Proposed

method

Lena 7.5707 2.3514 6.8048 7.9973 7.9987

Black 0 0 7.2538 7.9969 7.9980

Peppers 6.9911 2.1264 6.8567 7.9974 7.9986

Table (8) Avalanche Effect Test

Image name Method in [16](result are given for Lena) CS Only Proposed method

Lena 0.0018 0.0018

0.4456

Black 0.0027

0.4978

Peppers 0.0017 0.4965



84

algorithm has good ability against known plain text attack and encryption image has a highly

confidential security. From all above Chaos and CS-based image encryption and compression

appears a good combination of compression, security and flexibility.

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Technology

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