50120140501009

International Journal of Computer Engineering and Technology (IJCET), ISSN 0976INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING &
6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME

TECHNOLOGY (IJCET)

ISSN 0976 – 6367(Print)
ISSN 0976 – 6375(Online)
Volume 5, Issue 1, January (2014), pp. 68-84
© IAEME: www.iaeme.com/ijcet.asp
Journal Impact Factor (2013): 6.1302 (Calculated by GISI)
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IJCET
©IAEME

IMAGE ENCRYPTION AND COMPRESSION BASED ON COMPRESSIVE
SENSING AND CHAOS
Prof. Maher K. Mahmood(1),
1

Jinan N. Shehab(2)

(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, multichaotic 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,
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International Journal of Computer Engineering and Technology (IJCET), ISSN 09766367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME
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 ‫ א‬RN×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 ൌ ෍ ෍ S୧,୨ Ψ୧,୨
୧ୀଵ ୨ୀଵ

or

X ൌ ΨS

்
Where S is the (N×N) pixels of weighting coefficients Si,j =‫ߖ =ۄߖ ,ܺۃ‬௜,௝ ܺ and
transpose, containing exactly K nonzero coefficient, K << N.

69

ሺ1ሻ
T

denotes

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
ே

൛߮௝ ൟ

௝ୀଵ

:Y୧ ൌ ‫ۃ‬x, Ԅ‫ۄ‬

ሺ2ሻ

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

ሺ3ሻ

where Θ is called sensing matrix using only with compressible signal, Θ ‫ א‬RM×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 ‫ א‬RM×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 െ ߜ௄ ൑

||Θ୚||మ
||௏||మ

൑ 1 ൅ ߜ௄

ሺ5ሻ

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
ߜ௄ for a given specific measurement-matrix[10]. In a practical scenario, one can instead bound ߜ௄
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 :70

µ(Φ,ψ) = √N max1≤K j≤N |‫ۃ‬φk, ψj‫|ۄ‬

(6)

where ࣆ 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[ א‬N]

(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].

X

Ψ

Y=ΦX = ΦΨS = ΘS

Φ
Insecure
channel

෡
ࡿ

Recovery

Inverse
෡
ߖ

෠
෡መ
ܺ ൌ ߖܵ

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 :-

71

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)).

40

30

20

10

0
10
15

0

10
5

-10

0
-5
-20

A

-10

B

0.6

0.4
6
4

0.2

2
0
0
-2

-0.2

-4
-0.4

-6
1
0.5

4

-0.6

2

0
0
-0.5

-2
-1

-0.8
-1.5

-4

C

-1

-0.5

0

0.5

1

1.5

D

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)).
72

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) ,
YL (n) =mod (floor (YL (n) ×1015), 256) ,
ZL (n) = mod (floor (ZL (n) ×1015), 256) ,

XC (n)= mod (floor (XC (n) ×1016), 256)
YC (n) =mod (floor (YC (n) ×1016),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) )
ZLC (n) =BITXOR (ZL (n) , XC (n) )
FLC (n) =BITXOR (XL (n) , XC (n))

73

(12)

To normalize, divide by M:XLC (n) =1/M × (XLC (n))
YLC (n) =1/M × (YLC (n))
ZLC (n) =1/M × (ZLC (n))
FLC (n) =1/M × (FLC (n))

(13)

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
KΦ3 = ZLC (n) - Xone
KΦ4 = FLC (n) - Xone

(14)

Xone =all 1's matrix of size (M×N/2)
‫ܭ‬
Then the new Φ= ൤ ఃଵ
‫ܭ‬ఃଷ

‫ܭ‬ఃଶ
൨ , the size of Φ equal 2M×N elements ,as shown in Fig.(5).
‫ܭ‬ఃସ

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: Y୥ ሺM, N/2ሻ ൌ

N
Y ቀM, 2 ቁ െ g
100

ሺ15ሻ

where g represents the min value in Y(M,N/2).Then using Lloyd's algorithm in the new value
of ࢅࢍ .The resultant quantized compressive sensing measurement Yq(4-block). But the existing CSbased 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).
74

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

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

Figure(4) Wavelet Transform Decompositions (4-Blocks)
(4

Figure (5) Measurement Matrix Φ Using Chaotic System (Four Blocks)

75

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
Kd1=[XL XC XH XR] , Kd2=[YR YH YC YL]
Kc3 =XC
YR , Kc4 =YC
ZR
Kd3=[ZL ZC YH ZR] , Kd4=[ZR XH ZC YL]
XR , Kc6 =XH
YR
Kc5 =ZC
Kde1=reshape (Kd1, M, N/2)
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
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
76

International Journal of Computer Engineering and Technology (IJCET), ISSN 0976
images by changing the grey scale values to create an encrypted image. Hence, XORing the chaotic
cr
mask and image will result in confusion step. In the diffusion step, we mix the properties of vertical
age
step
ix
adjacent pixels and this information is spread out in the backward direction over the whole image.
The vertical diffusion(VD) considers t image obtained after step (6) as the input. It starts from the
)
the
last pixel of the last column in the image and then moves backward row-major order. In this process,
row major
the last pixel is modified by XORing itself and the corresponding value in key stream Kde, the pixel
before last pixel is modified by XORing the last and before last pixels and the corresponding value in
el
Kde. The last pixel of each column (except the last column) is modified by XORing the modified
column)
first pixel of the previous column and itself, the resultant image encryption Yen is transmitted from
itself
transmitter to the receiver through insecure channel.
V. RECOVERY AND DECRYPTION USING GREEDY PURSUIT(GP)
.
Suppose that at the receiver side as shown in Fig.(2), Yen is received, along with the keys KΦi,
er
Fig.
g, Kci and Kde from a separate secured channel then depended on (Yen, g, KΦi, Kci and Kde) we can
reconstruct image. And then, we proposed dividing an sparse image to a block of 256×256 pixels and
applying GP (OMP (Orthogonal Matching Pursuit) SP (Subspace Pursuit) and CoSaMP
Pursuit),
(Compressive Sampling Matching Pursuit to each block.
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 Table (3).
T
Table (3) Test Images
Lena.bmp

Peppers.png

Black.bmp

The following subsections will review the evaluation measures for both image compression
and encryption: b. METRICS FOR IMAGE COMPRESSION (RECOVERY)
The terminology “recovery” refers to decrypt and then reconstruct the plain image from its
”
uct
measurement data. In measuring the quality of the reconstructed image, the peak signal
signal-to-noise ratio
known as PSNR is used. PSNR of an a X a 8-bit grayscale image X and its reconstruction is
8
calculated as: -

where X (n,n) represented the intensity of a pixel in the original image, while its reconstructed
n,n)
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
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): 77


Compression ratio ൌ

uncompressed ϐile size
ܰൈܰ
ൌ
compressed ϐile size
2‫ ܯ‬ൈ ܰ

Rate of compressionሺRCሻ ൌ

ሺ17ሻ

1
2‫ܯ‬
ൌ
compression ratio
ܰ

ሺ18ሻ

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

OMP

SP

CoSaMP

Time to reconstruct (Second)
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.

78

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 multichaotic 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.

79

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: Corr୶୷ ൌ

|covሺx, yሻ|

ሺ19ሻ

ඥDሺxሻ ൈ ඥDሺyሻ

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: ୒

1
Eሺxሻ ൌ ෍ x୧
N

ሺ20ሻ

୧ୀଵ

୒

1
Dሺxሻ ൌ ෍ሺx୧ െ Eሺxሻሻଶ
N

ሺ21ሻ

୧ୀଵ

୒

1
covሺx, yሻ ൌ ෍൫x୧ െ Eሺxሻ൯൫y୧ െ Eሺyሻ൯
N

ሺ22ሻ

୧ୀଵ

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
Encrypted
image
in image in [16]
[7,10]
0.0215
0.0033
0.0808
0.0009
0.0176
0.0058

Lena

Horizontal
Vertical
Diagonal

0.9719
0.9850
0.9593

Black

Horizontal
Vertical
Diagonal

∞
∞
∞

Peppers

Horizontal
Vertical
Diagonal

0.9894
0.9921
0.9829

80

∞
∞
∞

0.00028
0.0021
0.0002199

0.0007
0.0007
0.0035

0.0205
0.0737
0.0174

0.77
0.02
0.0104

Encrypted
image in this
work
0.0000509
0.0033
0.0094

0.0008
0.0046
0.0039

0.0214
0.0837
0.0187

Encrypted
by CS only

0.869
0.1045
0.117

0.000107
0.000109
0.000025

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).
Encryption image

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

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

250

200

150

100

50

0
0

50

100

150

200

250

Pixel gray value on location
(X,Y)

Original image (Black)

Encryption image

12000

10000

8000

6000

4000

2000

0
0

50

100

150

200

250

Pixel gray value on location
(X,Y)

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
1200

3000

12000

12000

10000

10000

8000

8000

6000

6000

4000

4000

2000

2000

1000

2500

2000

800

1500

600
1000

400

500

0
0

50

4

x 10
4

Lena
100

150

200

250

0
50

100

150

200

250

0
0

50

100

150

200

250

0

50

100

150

50

100

150

200

250

800
12000

10000

8000

8000

6000

6000

4000

4000

2000

3

12000

10000

3.5

200

0
0

2000

600

2.5
2
1.5

400

1
0.5
0
0

50

Black
100

150

200
200

250

0

0

0
0

50

100

150

200

250

0

50

100

150

200

250

0

200

250

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).

81

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 ൌ
UACI ൌ

∑୧,୨ D(i, j)
2M ൈ N

ൈ 100%

(23)

1
|cଵ (i, j) െ cଶ (i, j)|
቎෍
቏ ൈ 100%
2M ൈ N
255

(24)

୧,୨

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

Lena
Black
Peppers

Method in [16] (results are
given for Lena image only)

NPCR
0.0038

UACI
0.0013

CS Only

Proposed method

NPCR
0.0031
0.0054

UACI
1.24×10-5
9.72×10-5

NPCR
0.9992
0.9977

UACI
0.2512
0.2505

0.0041

1.6×10-5

0.9984

0.2506

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) ൌ ∑ P൫C୧,୨ ൯logଶ (1⁄p(C୧,୨ ))
Where P (ci,j) represents the probability of symbol ci,j for a grayscale image.

82

(25)


Image
name
Lena

Entropy for original
image
7.5707

Black
Peppers

0
6.9911

Table (7) Entropy Test
Entropy after
Method in
CS
[7,16]
2.3514
6.8048
0
2.1264

7.2538
6.8567

Method in
[10]
7.9973

Proposed
method
7.9987

7.9969
7.9974

7.9980
7.9986

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
ഥ
ഥ
single bit change can be made in the image P to give a modified image P. Both P and P are encrypted
ത . The avalanche effect metric is the percentage of different bits between C and C . If C
ത
to give C and C
ത
and C differ from each other in half of their bits, we can say that the encryption algorithm possesses
good diffusion characteristics[16].

Image name
Lena
Black
Peppers

Table (8) Avalanche Effect Test
Method in [16](result are given for Lena) CS Only
0.0018
0.0018
0.0027
0.0017

Proposed method
0.4456
0.4978
0.4965

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
ൎzero, and the entropy is ൎ8bits. 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
83

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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[5]

[6]
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[8]
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[10]

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[12]
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[14]
[15]
[16]
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