Journal of Signal and Information Processing, 2011, 2, 238-244
doi:10.4236/jsip.2011.23033 Published Online August 2011 (http://www.SciRP.org/journal/jsip)
Copyright © 2011 SciRes. JSIP
Image Encryption Based on the General Approach
for Multiple Chaotic Systems
Qais H. Alsafasfeh, Aouda A. Arfoa
Electrical Engineering Department, Tafila Technical University, Tafila, Jordan.
Email: qsafasfeh@ttu.edu.jo, aouda@ttu.edu.jo
Received April 10th, 2011; revised July 5th, 2011; accepted July 13th, 2011.
ABSTRACT
In the recent years, researchers developed image encryption methods based on chaotic systems. This paper proposed
new image encryption technique based on new chaotic system by adding two chaotic systems: the Lorenz chaotic system
and the Rössler chaotic system. The main advantage of this technique is stronger security, as is shown in the encryption
tests.
Keywords: Lorenz Equations, Rossler Syste m s, Image Encryption, Chaos
1. Introduction
Recent researches of nonlinear dynamical systems have
been increasingly based on Chaos [1]. Image encryption
schemes have been increasingly studied to meet the de-
mand for real-time secure image transmission over pri-
vate or public networks. Conventional image encryption
algorithm such as data encryption standard (DES), is not
suitable for image encryption because of the special
storage characteristics of an image [2] and weakness of
low-level efficiency when the image is large [3].The
chaos-based encryption has suggested a new and efficient
way to deal with the intractable problem of fast and
highly secure, image encryption.
Chaotic systems have many important properties, such
as the sensitive d ependence on initial conditions and sys-
tem parameters, the density of the set of all periodic
points and topological transitivity, etc. Most properties
are related to some requirements such as mixing and dif-
fusion in the sense of cryptography [4]. Matthews pro-
posed a chaotic encryption algorithm in 1989 [5], since
then, increasing researches of image encryption technol-
ogy are based on chaotic systems [4,6-12]. These meth-
ods have high-level efficiency but also weakness, such as
small key space and weak security and complexity to
overcome these drawbacks. This paper proposes a image
encryption scheme based on combining two chaotic sys-
tems (Lorenz and Rosslere) and compares the results
with the other techniques. The paper is organized as fol-
lows. Section 2 discusses the proposed scheme. Section 3
presents experimental results. Section 4 to Section 7 dis-
cuss and examine the new scheme by applying statistical
tests test also the security of the chaotic encryption is
discussed. Conclusions ar e draw n in Section 8.
1.1. Lorenz Chaotic System
The Lorenz equations are a fairly simple mod el in which
to study chaos.
xyx
yrxyxz
zxy z



(1)
The arbitrary parameters
, r and b > 0 and for this
example are
= 10, r = 28, b = 8/3. This system of
differential equations is then solved numerically using
Matlab’s ode 45 Runge Kutta routine. The results are
plotted in Figure 1 [12] .
1.2. Rossler Chaotic System
A system of three differential equations has a simpler
strange attractor than Lorenz’s. That the Rossler system
has only one quadratic nonlinear xz numerical integra-
tion shows this system has a strange attractor for a = b=
0.2, c = 5.7 as shown in Figure 2 [12] .

xyz
yxay
zbzxc
 


(2)
2. The Proposed Scheme
First, most researchers used chaotic image encryption
Image Encryption Based on the General Approach for Multiple Chaotic Systems239
Figure 1. Chaotic system (time series and chaotic attractor
for Lorenz system).
Figure 2. Chaotic system (Chaotic attractor for rossler sys-
tem).
depending on only one chaotic system like Lorenz and
Rossler systems. A chaotic was presented in [13], which
is based on adding two chaotic systems (Lorenz and
Rossler), a new chaotic system was generated as shown
in Equation (3).
()
20
5(
xyxyz
yrxyxzxay
zxybzsxzc

 

)
(3)
Most of researchers agree on the following definition
“chaos is aperiodic long term behavior in a deterministic
system that exhibits dependence on initial condition”
[12]. So to examine theses conditions we note in Figure
3 the new system has two attractors, thus satisfying the
above definition.
The security of Lorenz and Rossler encryption meth-
ods depend on three parameters but in the proposed
scheme in this paper the security level was increased to
six paramete rs.
Encryption Based Multiple Chaotic
The goal of the step is to encrypt images by shuffling
pixel values and then changing the grey scale values to
create an encrypted image. The pixel values are rear-
ranged using the XOR and then the grey scale values are
changed using a multiple chaotic systems, and therefore
after a set number of iterations (which depends on the
size of the image), generated elements have been stored
Figure 3. The phase plane for new system (Chaotic attrac-
tor projected onto xy-plane).
Copyright © 2011 SciRes. JSIP
Image Encryption Based on the General Approach for Multiple Chaotic Systems
240
within the chaotic matrix of size the same as the original
image’s size. As with the other algorithms that made use
of the XOR operation to encrypt, decryption is a simple
matter of recreating the matrix of chaotic elements and
XOR with the decrypted image matrix and therefore after
a set number of iterations the original image will return.
The actual procedure for encryption is shown in Fig-
ure 4.
Step 1. Set the key (initial condition and parameters
in the acceptable intervals.
000
,,,,,,, ,rabcyx z

Step 2. Generate the first mask with the same size of
image.
Step 3. Perform the XOR between the chaotic mask
and original image.
Step 4. If the image is not encrypted generate the sec-
ond mask with the same size of image.
Step 5. Perform the XOR between the chaotic mask
and image in Step 3.
Step 6. If the image is not encrypted generate the third
mask with the same size of image.
Step 7. Perform the XOR between the chaotic mask
and image in Step 5.
Step 8. Pass the encrypted image
Step 9. Pass the encryption key.
Step 10. End.
3. Experimental Results
Simulation results and performance analysis of the pro-
posed image encryption scheme are provided in this
Figure 4. Flow chart of encryption based multiple chaotic
procedure.
section. We take 256 × 256 size 8 bits Lena image an
example Figure 5(a) its encrypted image with the en-
cryption initial parameters are
= (16, 45.6, 2, 0.2, 0.2, 5.7) but about initial condition we
divided initial condition for each generated mask

000
,,,,,,, ,rabcy x z

10 ,
x
20 30
,
x
x = (0.832345676543451, 0.523456765475432,
0.6345677654345643) as we can see, the encrypted im-
age is rough-and-tu mble, unknowable and 100% obscur e
of the image Figure 5(b) is the decrypted image by use
the same encryption key. It can be seen that the de-
crypted image is clear and correct without any distortion
see Figure 5(c) to see no error between original image
and decrypted image.
In Figure 6 their encrypted image with sharp edge with
the same encryption initial parameters and initial condi-
tions. As we can see, the encrypted image is rough-and-
tumble, unknowable and 100% obs cure of the image Fig-
ure 6(c) is the decrypted image by use the same encryp-
tion key. It can be seen that the decrypted image is clear
and correct without any distortion see Figure 6(c) to see
no error between original image and decrypted image.
4. Statistical Analysis
Use Statistical analysis has been performed on the pro-
posed image encryption algorithm, demonstrating its
superior confusion and diffusion properties which strongly
resist statistical attacks. This is shown by a test on the
histograms of the unencrypted images and on the corre-
lations of adjacen t pixels in the encrypted image.
(a) (b)
(c) (d)
Figure 5. (a) Original image (b) encrypted image, (c) de-
crypted image (d) error between original and encrypted
image.
Copyright © 2011 SciRes. JSIP
Image Encryption Based on the General Approach for Multiple Chaotic Systems241
(a) (b)
(c) (d)
Figure 6. (a) Original Image (b) encrypted image, (c) de-
Histograms of encrypted images: one typical example
am
extensive study of the correlation
be
crypted image (d) error between original and encrypted
image.
ong them is shown in Figures 7 and 8; one can see
that the histogram of the encrypted image for Lena image
is fairly uniform and is significantly different from that
of the original im a ge.
We have also done
tween image and its corresponding encrypted image by
using the proposed en cryption algorithm. The correlation
between two vertically adjacent pixels, two horizontally
adjacent pixels, and two diagonally adjacent pixels, We
have depicted the distributions of two horizontally, ver-
tically and diagonally adjacent pixels in the original and
encrypted images respectively see Figure 9 and Table 1
for Lena image and Figure 10 and Table 2 for sharp
edge image, we note the two adjacent pixels in the origi-
nal image are highly correlated, and the two adjacent
pixels in the encrypted image are highly not correlated.

1
1N
i
i
Ex x
N
5. Differential and Sensitivity Analysis
pixels Another important test is measured the number of
change rate (NPCR) to see the influence of changing a
single pixel in the or iginal image on the encryp ted image
by the proposed algorithm defined by Equation (7). The
NPCR measure the percentage of different pixel numbers
between the two images and the UACI (unified average
changing intensity) de fined by Equation (8). We take two
Figure 7. Histogram of original image and encrypted for
lena image.
Figure 8. Histogram of original image and encrypted for
sharp edge image.
Copyright © 2011 SciRes. JSIP
Image Encryption Based on the General Approach for Multiple Chaotic Systems
242
Figure 9. Correlations of two diagonal, horizontally and
able 1. Correlations coefficient of two diagonal, horizon-
Plain image Encrypted image
vertical adjacent pixels in the plain image and in the en-
crypted-image.
T
tally and vertical adjacent pixels in the plain image and the
encrypted-image.
Horizontal 0.9681 0.0483
Vertical 0.9434 0.1078
Diagonal 0.–0. 9238 0283
ncrypted images, C1 and C2, whose corresponding e
original images have only one-pixel difference. We de-
fine a two-dimensional array D, having the same size as
the image C1 and C2. If
 
11
,,CijCij then

,1Dij, otherwise D

,0ij

,
,
100%
ij
Dij
NPCR WH

(4)
 
12
,
,,
1100%
255
ij
CijCij
UACI WH




(5)
where W and H are the width and height of encrypted
image. We obtained NPCR for a large number of images
by using our encryption scheme and found it to be over
99% showing thereby that the encryption scheme is very
Figure 10. Correlations of two diagonal, horizontally an
able 2. Correlations coefficient of two diagonal, horizon-
Pla in image Encrypted image
d
vertical adjacent pixels in the plain image and in the en-
crypted-image.
T
tally and vertical adjacent pixels in the plain image and the
encrypted-image.
Horintal zo0.9804 0.0219
Vertical 0.9434 0.1668
Diagonal 0.9351 –0.0024
nsitive with respect to small changes in the plaintext
Space Analysis
duce a completely different
se
[14,15].
6. Key
Another secret key should pro
encrypted image. For testing the key sensitivity of the
proposed image encryption procedure, we use the wrong
key, initial parameters to decrypted the original image for
example if we encrypt the Lena image using
,,
000
,,,,, ,rabcyx z = (16, 45.6, 2, 0.2, 0.2, 5.7 ) and
10 2030
,,
x
xx = (0
0.634567765 .832345676543451, 0.523456765475432,
4345643) now we will try to decrypt the
encrypted image using wrong key for example
,,,r
000
,,,,,abcy x z = (16, 45.6, 2, 0.2, 0.2, 5.7) and
10,
x
20 30
,
x
x = (0.8
0.634567 32345676543451, 0.523456765475432,
7654345642) we note the decrypted image still
rough-and-tumble and unknowable see Figure 11(a) and
if we chose another key initial parameters are
,,,r
000
,,,,,,, ,rabcyx z

= (16, 45.6, 2, 0.2, 0.2, 5. 7) and
10 2030
,,
x
xx = (0.832345676543451, 0.523456765
Copyright © 2011 SciRes. JSIP
Image Encryption Based on the General Approach for Multiple Chaotic Systems
Copyright © 2011 SciRes. JSIP
243
osed method and recent methods.
` Lorenz [8] Rossler [9]Logistic Map [10] New Logistic Map [7]The 3D cat map [11]One D based [8] Encryption Based
Table 3. Comparison between prop
Multiple chaotic
Key Space 1.
158
2 16
10 2 ×
24
10 45
10 36
2 53
22 75
10
Time 10.84 s 0.5 0.4 12.27 s 2 s
Obscur100% <100% <100% 100%
0.33 s s
e 100% 100% 100%
754329, 0.6345677654345643) the result is tumble and
space should be
la
ty consideration, running speed of
4
unknowable as sho wn in Figure 11(b).
For a secure image encrypted, the key
rge enough to make the brute force attack infeasible
[16]. The key of the new algorithm consists of three
floating-point numbers. We note if we encrypt using 15
digits and just change the last digit the decrypted still
unknown that mean we use the first 15 digits of a
floating-point number, then there are 15 + 15 + 15 = 75
uncertain digits. So the possible key number is 45
10 .
Moreover the parameters

,,,,,rabc

are also
as the secret key. An imagith such a long
key space is sufficient for reliable practical use.
7. Time Analysis
used
e encrypted w
Apart from the securi
the algorithm is also an important aspect for a good en-
cryption algorithm. We have measured the encryption/
decryption rate of on 256 grey-s cale images of size 256 ×
256 by using the proposed image encryption scheme. The
time analysis has been done on Pentium-4 with 512 MB
RAM computer. The average encryption/decryption time
(a)
(b)
Figure 11. Decrypted using wrong key for lena image.
is 2 s which is less than those of the previous algo rithms.
a new nonlinear chaotic algorithm,
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