1. 46 INTERNATIONNAL JOURNAL OF APPLIED BIOMEDICAL ENGINEERING VOL.5, NO.1 2012
Digital Medical Image Cryptosystem Based
on Infinite-Dimensional Chaotic Delay
Differential Equation For Secure
Telemedication Applications
Sarun Maksuanpan 1
and Wimol San-Um 1
,
ABSTRACT
Digital medical image cryptosystem based on
infinite-dimensional multi-scroll chaotic Delay Differ-
ential Equation (DDE) for secure telemedication ap-
plications is presented. The proposed cryptography
technique realizes XOR operations between separated
planes of binary gray-scale image and a shuffled multi-
scroll DDE chaotic attractor image. The security
keys are initial condition and time constant in DDE
represented by 56-floting-point number. Simulation
results are performed in MATLAB. Nonlinear dy-
namics of DDE are described in terms of equilibrium
points, time-domain waveforms, and 3-scroll attrac-
tor in phase-space domain. Encryption and decryp-
tion performances of three gray-scale human body
Computerized Axial Tomography (CAT) scan images
with 256256 pixels are evaluated through pixel den-
sity histograms, 2-dimensional power spectral den-
sity, key space analysis, correlation coefficients, and
key sensitivity. Demonstrations of wrong-key de-
crypted image are also included. The proposed tech-
nique offers a potential alternative to simple-but-
highly-secured image storage and transmissions in
telemedication applications.
Keywords: Medical image cryptosystem, Delay dif-
ferential equation, Chaos-based encryption.
1. INTRODUCTION
Recent advances in communication technologies
have led to great demand for secured image trans-
missions through internet networks for a variety of
particular applications such as in medical, industrial
and military imaging systems. The secured image
transmissions greatly require reliable, fast and robust
security systems, and can be achieved through cryp-
tography, which is a technique of information privacy
protection under hostile conditions [1]. Of particular
interest in telemedication in which distributed medi-
cation resources can be achieved anyplace, real-time
Manuscript received on July 29, 2012 ; revised on October
15, 2012.
1 The authors are with Intelligent Electronic Systems Re-
search Laboratory Faculty of Engineering, Thai-Nichi Institute
of Technology Pattanakarn, Suanluang, Bangkok, Thailand,
10250. Tel: (+662)763-2600 Ext.2926. Fax: (+662) 763-2700,
E-mail: wimol@tni.ac.th
telemammography examinations and digital medical
images will be diagnosed by distributed medical ex-
perts[2]. Consequently, medical treatment processes
that deal with patients confidential data are supposed
to strictly and only be accessible to authorized per-
sons. Most recent telemedication technologies trans-
port and storage medical images such as magnetic
resonance images (MRIs) and computed tomography
(CT) through Picture Archiving and Communication
Systems (PACS) as well as Digital Imaging and Com-
munications in Medicine (DICOM) [3], leading to the
need for cryptosystems that protect the confidential-
ity in terms of legal and ethical reasons.
Typically, image cryptography may be classified
into two categories, i.e. (1) pixel value substitu-
tion which focuses on the change in pixel values so
that original pixel information cannot be read, and
(2) pixel location scrambling which focuses on the
change in pixel position. Conventional encryption al-
gorithms for such cryptography, for example, Data
Encryption Standard (DES), International Data En-
cryption Algorithm (IDEA), Advanced Encryption
Standard (AES), and RSA algorithm may not ap-
plicable in real-time image encryption due to large
computational time and high computing power, espe-
cially for the images with large data capacity and high
correlation among pixels [4]. Recently, the utilization
of chaotic systems has extensively been suggested as
one of a potential alternative cryptography in secured
image transmissions. As compared with those of con-
ventional encryption algorithms, chaos-based encryp-
tions are sensitive to initial conditions and parame-
ters whilst conventional algorithms are sensitive to
designated keys. Furthermore, chaos-based encryp-
tions spread the initial region over the entire phase
space, but cryptographic algorithms shuffle and dif-
fuse data by rounds of encryption [5]. Therefore, the
security of chaos-based encryptions is defined on real
numbers through mathematical models of nonlinear
dynamics while conventional encryption operations
are defined on finite sets. Such chaos-based encryp-
tion aspects consequently offer high flexibility in en-
cryption design processes and acceptable privacy due
to vast numbers of chaotic system variants and nu-
merous possible encryption keys.
Chaos-based encryption algorithms are performed
in two stages, i.e. the confusion stage that permutes

2. Sarun Maksuanpan and Wimol San-Um 47
the image pixels and the diffusion stage that spreads
out pixels over the entire space. Most existing chaos-
based encryptions based on such two-stage operations
employ both initial conditions and control parame-
ters of 1-D, 2-D, and 3-D chaotic maps such as Baker
map [6,7], Arnold cat map [8,9], and Standard map
[10, 11] for secret key generations. Furthermore, the
combinations of two or three different maps have been
suggested [12, 13] in order to achieve higher security
levels. Despite the fact that such maps offer satis-
factory security levels, iterations of maps require spe-
cific conditions of chaotic behaviors through a narrow
region of parameters and initial conditions. Conse-
quently, the use of iteration maps has become typical
for most of proposed ciphers and complicated tech-
niques in pixel confusion and diffusion are ultimately
required.
The DDE has emerged in mathematical models of
natural systems whose time evolution depends ex-
plicitly on a past state, and can be described by
an infinite-dimensional system that can exhibit com-
plex chaotic behaviors with a relatively simple first-
order differential equation. Existing DDEs include
the prominent Mackey-Glass DDE [14] which mod-
els the production of white blood cells and the Ikeda
DDE [15] which models a passive optical resonator
system. In recent years, further chaotic DDEs [16-17]
based on the Mackey-Glass DDE have been reported
through the use of piecewise-linear nonlinearities cor-
responding to a complex two-scroll and multi-scroll
attractors. In addition, the simplest DDE with a si-
nusoidal nonlinearity [18] based on the Ikeda DDE
has also been presented.
This paper introduces a new digital medical image
cryptosystem based on infinite-dimensional multi-
scroll chaotic Delay Differential Equation (DDE) for
secure telemedication applications is presented. The
proposed cryptography technique realizes a XOR op-
eration between separated planes of binary gray-scale
image and a shuffled multi-scroll DDE chaotic attrac-
tor image. The security keys are initial condition and
time constant in DDE represented by 56-floting-point
number. Nonlinear dynamics of DDE will be de-
scribed in terms of equilibrium points, time-domain
waveforms, and 3-scroll attractor in phase-space do-
main. Encryption and decryption security perfor-
mances of three gray-scale human body CAT scan im-
ages with 256256 pixels are evaluated through density
histograms, 2-dimensional power spectral density, key
space analysis, image correlation coefficients, and key
sensitivity.
2. REALIZATIONS OF MULTI-SCROLL
CHAOTIC DELAY DIFFERENTIAL EQUA-
TION
The first-order multi-scroll chaotic DDE is ex-
pressed in a simple first-order differential equation
as follows [18];
˙x = −axτ + bFn (xτ ) (1)
where a and b are unity, and the nonlinear term
Fn (xτ ) is a piecewise-linear nonlinear function de-
scribed as
Fnxτ =
n∑
m=1
(sgn(xτ − (2m − 1)) (2)
+(sgn(xτ + (2m − 1)))
where n and m are positive integers. The nonlinear
function in (2) particularly exhibits a stair-shape pos-
itive slope, and offers 2n+1 scroll chaotic attractors
with complex dynamic behaviors depending on the
setting of the delay time τ. In this paper, the case
of three scroll with n=1 is realized. Consequently,
the resulting DDE obtained from (1) and (2) can be
expressed as
˙x = −axτ + sgn(xτ − 1) + sgn(xτ + 1) (3)
The DDE in (3) possesses three equilibrium points at
-2,0,2 and the corresponding characteristic equation
of its linearized form, i.e. =0, can be obtained by the
partial derivative with respect to x as follows;
−1 + δ(x − 1) + δ(x + 1) (4)
where δ(˙) is a Dirac delta function. The eigenval-
ues evaluated at each fixed point are all equal at -1,
which are negative real values, indicating that the
three equilibrium points are all stable nodes when
τ = 0. In the case where τ > 0, the characteristic
equation of DDE generally has infinitely many roots
while the number of characteristic roots of ODEs co-
incides with the dimension of the system. Therefore,
the DDE in (3) can be approximated by an infinite-
dimensional system of ODEs as
˙x0 = −xN + sgn(xN − 1) + sgn(xN + 1) (5)
˙xi =
N(xi−1 − xi)
τ
where 1 < i < N and the values of N approaches
infinity. The equation xi advances N discrete-time
lags of x0 over the interval t − τ to τ. It can be
considered that the term sgn(xN − 1) + sgn(xN + 1
in (5) provides five constants in a set of k, i.e. k =
{−2, −1, 0, 1, 2}, at any values of N.
The eigenvalues of (5) for the flow in the vicin-
ity of the stable equilibrium for N approaches infinity
are given by the solutions of λ = −exp (−λτ), which
can be expressed in terms of the Lambert function W
as λ = −W (−τ) /τ. The resulting eigenvalues are
always in the form of a pair of complex conjugates,
indicating that the equilibria are saddle focus points
when the DDE exhibit chaotic behaviors. It can be
considered that the values of the delay time τ sets the
chaotic behaviors with a specific topology of attrac-
tors based on the nonlinearity. Therefore, the use of
DDE as a resource of complex attractor images can
be employed for image encryption with a high degree
of complexity can be achieved through an infinite di-
mension of the DDE systems.

3. 48 INTERNATIONNAL JOURNAL OF APPLIED BIOMEDICAL ENGINEERING VOL.5, NO.1 2012
Fig.1:: Proposed encryption and detection al-
gorithms using XOR operation between separated
planes of binary gray-scale image and a shuffled multi-
scroll DDE chaotic attractor image.
3. PROPOSED MECHANISMS OF IMAGE
ENCRYPTION ALGORITHM
The proposed cryptography technique attempts to
achieve simple-but-highly-secured image encryption
and decryption algorithms in a category of chaos-
based cryptosystems. Fig. 1 shows the proposed en-
cryption and detection algorithms. In the encryption
process, the original gray-scale image is initially con-
verted into binary matrix in which each pixel is rep-
resented by 8-bit binary numbers. For example, the
pixel p(1,1) contains the binary number a0-a7. Each
pixel will then be separated into eight planes corre-
sponding to binary bits a0 to a7. It can be consid-
ered that such eight planes are all represented in ma-
trix forms with a single binary number in each pixel,
which is ready for further Excusive-OR (XOR) oper-
ations. Meanwhile, the chaotic DDE attractor image
is generated from Eqn. (2). This image is unique
since chaos is sensitive to initial conditions, i.e. an
extremely small change in the initial conditions or in
the time constant will result in largely chaotic be-
haviors. Therefore, the setting of initial conditions,
time constants, and simulation time of DDE equation
can be exploited as security keys in both encryption
and decryption processes. It is seen in Fig.1 that the
chaotic DDE attractor image in a matrix form is shuf-
fled prior to XOR operations. As a particular case,
this paper divides the chaotic DDE attractor image
into sixteen sections before shuffling. It should be
noted that the attractor image can also be shuffled
with more divided sections if desired.
The XOR operations diffuse the shuffled DDE
chaotic image and the eight binary images in parallel
process. The XOR operation yields bit ”1” if the two
input bits are different, but yields bits ”0” if the two
inputs are similar. The results from such XOR opera-
tions are eight matrices with single binary number in
each pixel. All the eight matrices are combined into a
single 8-bit matrix in which each pixel is represented
by [b0-b7]. As a result, the encrypted image can be
achieved. The decryption process also follows the en-
cryption process in backward algorithms as long as
the security keys are known.
4. SIMULATION RESULTS
Experimental results have been performed in a
computeraid design tool MATLAB. Nonlinear dy-
namics of DDE were initially simulated. As for
verification of effectiveness of the proposed encryp-
tion and decryption algorithms security performances
were subsequently evaluated. Three examples of dig-
ital medical images have been selected from [20-22]
which are CT scan images of human brain, spine and
heel with the 256 × 256 image size.
4.1 Multi-Scroll DDE Dynamical Behaviors
Fig. 2 shows the bifurcation diagram of the time
constant τ where chaotic regions are indicated by
dense area. The highly chaotic region appears from
τ=1.73 and are boundlessly sustained over all range
of time approaches infinity. In order to guaran-
tee chaotic behaviors of the DDE, the values of τ
must be any real numbers greater than 1.73. Ini-
tial conditions are not crucial and can be selected
from any numbers in the basin of attractors ex-
cept the equilibrium points. Fig.3 illustrates chaotic
attractor images and corresponding time domain
waveforms within 0.2 ms. Four different cases of
τ and x(0) were selected arbitrarily, including (a)
=1.821357 and x(0)=0.000001, (b) τ=2.239473 and
x(0)=0.000002, (c) τ=2.671521 and x(0)=0.000003,
and (d) τ=3.000001 and x(0)=0.000004. It is ap-
parent in Fig. 3 that the time domain waveforms are
chaotic and the chaotic attractors resemble the three-
scroll topology as described in Equ. (2) and (3). Such
four cases show distinctive chaotic regimes in terms of
dynamical behaviors. In other words, the increase in
the values of τ provides more randomness in time-
domain and more complicated attractor images in
Fig.2:: The bifurcation diagram of the time constant
τ where chaotic region is indicated by a dense area,
initializing from approximately τ > 1.5.

4. Sarun Maksuanpan and Wimol San-Um 49
Fig.3:: Chaotic attractor images and time do-
main waveforms for four different cases ofand x(0)
within 0.2ms, including (a) τ= 1.821357 and x(0)
= 0.000001, (b) τ= 2.239473 and x(0) = 0.000002,
(c) τ= 2.671521 and x(0) = 0.000003, and (d) τ=
3.000001 and x(0) = 0.000004.
phase-space domain. It can also be considered that
such chaotic attractor images are unique determined
by two particular parameters, i.e. time constant and
initial condition, which will be used as security keys
in this paper.
4.2 Multi-Scroll DDE Dynamical Behaviors
The security keys of the proposed encryption and
decryption algorithms are represented by floating-
point numbers, i.e. S×2E where S is a significand
and E is an exponent, throughout encryption and
decryption processes. In this work, the secret key are
given by
τ = 3.0012946528743651987234688167 (6)
x (0) = 0.0000012654982346587193581368 (7)
It can be seen that the secret keys are represented by
28 digits of a floating-point number ( 7.2058×1016),
resulting in 56 uncertain digits, which is a minimum
requirement of the 56-bit data encryption standard
(DES) algorithm [23]. It should be noted that the key
space can be designated longer while chaos from the
multi-scroll DDE is robust, but the longer key space
require longer time for simulations. With the secret
keys determined in (6) and (7), the proposed digital
medical image encryption and decryption algorithm
is certainly protected from the brute-force attack.
4.3 Histograms and 2D Power Spectral Anal-
ysis
The image histogram is a graph that illustrates the
number of pixels in an image at different intensity
values. In particular, the histogram of an 8-bit gray
scale image has 256 different intensity levels, graphi-
cally displaying 256 numbers with distribution of pix-
els amongst these gray scale values. In addition, the
2D power spectrum can be obtained through a Dis-
crete Fourier Transform (DFT) analysis and the algo-
rithm is given by [24] where x and y are a coordinates
pair of an image, M and N are the size of image, f(x,y)
is the image value at the pixel (x,y). Fig. 4 (a) to (d)
shows the histograms and the 2D power spectrum
tests of the brain image, the encrypted brain image,
decrypted brain image, and the decrypted brain im-
age with wrong keys, respectively. Fig. 5 (a) to (d)
shows the histograms and the 2D power spectrum
tests ofthe spine image, the encrypted spine image,
decrypted spine image, andthe decrypted spine image
with wrong keys, respectively. In addition, Fig. 6 (a)
to (d) shows the histograms and the 2D power spec-
trum tests of the heel image, the encrypted heel im-
age, decrypted heel image, and the decrypted heel
image with wrong keys, respectively.
It can be seen from Figs. 4 to 6 that the intensi-
ties of all original images in the histogram are con-
tributed with different values in a particular shape
and the power spectrum is not flat having a peak of
intensity in the middle. The encrypted image has a
flat histogram and power spectrum, indicating that
the intensity values are equally contributed over all
the intensity range and the original images are com-
pletely diffused and invisible. One can notice that
the histograms of all original are relatively flat with
some spikes due to the characteristics of medical im-
ages that generally contain black colors more than
the white colors. The decrypted images with right
keys provide similar characteristics of the original im-
ages while the decrypted images with wrong keys are
still diffused and the original images cannot be seen.
These results qualitatively guarantee that the image
is secured.
4.4 Correlation and Key Sensitivity Analysis
In order to quantify the encryption performance
and key sensitivity analysis, correlation between im-
age pairs, which is a measure of relationships between
two pixels intensities of two images, of the three real-
ized images have been analyzed. The covariance (Cv)
and the correlation coefficient (γxy) can be obtained
as follows [25];

5. 50 INTERNATIONNAL JOURNAL OF APPLIED BIOMEDICAL ENGINEERING VOL.5, NO.1 2012
(a)
(b)
(c)
(d)
Fig.4:: Histograms and 2D power spectrum tests;
(a) Brain image, (b) Encrypted Brain image, (c) De-
crypted Brain image, (d) Decrypted Brain image with
wrong keys.
Cv (x, y) =
1
N
N∑
i=1
(xi − E (x)) (yi − E (y)) (8)
γxy =
cov (x, y)
√
D(x)
√
D(y)
(9)
where the functions E(x) and D(x) are expressed as
E(x) =
1
N
N∑
i=1
xi (10)
E(x) =
1
N
N∑
i=1
(xi − E(x))2
(11)
and the variables x and y are grey-scale values of
pixels in corresponding pixels in different images or
two adjacent pixels in the same image. Typically, the
(a)
(b)
(c)
(d)
Fig.5:: Histograms and 2D power spectrum tests;
(a) Spine image, (b) Encrypted Spine image, (c) De-
crypted Spine image, (d) Decrypted Spine image with
wrong keys.
values of γxy are in the region [- 1, 1]. The values of
γxy in the region (-1,0) and (0,1) respectively indicate
positive and negative relationships, while the larger
number close to 1 or -1 have stronger relationships.
Two images are identical if γxy are precisely equal to
1 and -1. Using a random selection of 2,048 pairs of
pixels, Figs. (7), (8), and (9) show correlation of hor-
izontally, vertically, and diagonally adjacent pixels of
original and encrypted brain image, original and en-
crypted spine image, and original and encrypted heel
image, respectively. It can qualitatively be consid-
ered from Figs. (7), (8), and (9) that the adjacent
pixels of all encrypted images are highly uncorrelated
as depicted by scatters plots of correlations.
For the quantitative measures, Table 1 summarizes
correlation coefficients of 2,048 pixels of each image
pair. First, the correlations between all original and
encrypted images with correct keys are equal to unity,
indicating that the images are completely decrypted.
The original and encrypted brain, spine, and heel im-
ages respectively have the correlation coefficients of
-0.0038, -0.0025, and -0.0119, indicating that the im-
ages are uncorrelated as the values are closely equal

6. Sarun Maksuanpan and Wimol San-Um 51
(a)
(b)
(c)
(d)
Fig.6:: Histograms and 2D power spectrum tests; (a)
Heel image, (b) Encrypted Heel image, (c) Decrypted
Heel image, (d) Decrypted Heel image with wrong
keys.
(a) Correlation of adjacent pixels of original brain image
(b) Correlation of adjacent pixels of encrypted brain image
Fig.7:: Correlation of horizontally, vertically, and
diagonally adjacent pixels of (a) original brain image,
and (b) the encrypted brain image
(a) Correlation of adjacent pixels of original heel image
(b) Correlation of adjacent pixels of encrypted heel image
Fig.8:: Correlation of horizontally, vertically, and
diagonally adjacent pixels of (a) original Heel image,
and (b) the encrypted Heel image
(a) Correlation of adjacent pixels of original spine image
(b) Correlation of adjacent pixels of encrypted spine image
Fig.9:: Correlation of horizontally, vertically, and
diagonally adjacent pixels of (a) original Spine image,
and (b) the encrypted Spine image
to zero. In other words, the encrypted images are
secured. In order to analyze key sensitivity, two dif-
ferent cases of wrong keys were also investigated. The
key set 1 and set 2 were the changes in the lease sig-
nificant number and the most significant number of
the given key in (6). The results shows that the cor-
relation coefficients of all three images are still closely
equal to zero, indicating that the images are protected
even an extremely small changes of the security keys.
5. CONCLUSIONS
Since great demand for secured image storage and
transmissions through internet networks have been
increasing, especially for Telemedication application
in which distributed medication resources can be
achieved anyplace. This paper has presented the dig-
ital medical image cryptosystem based on infinite-

7. 52 INTERNATIONNAL JOURNAL OF APPLIED BIOMEDICAL ENGINEERING VOL.5, NO.1 2012
Table 1:: Summary of correlation coefficients of
2,048 pixels of each image pair.
Test
Image 1 Image 2 γxy
Images
Original Original 1
(1) Original Encrypted -0.0038
Brain Original Decrypted with correct keys 1
Image Original Decrypted with wrong keys Set 1 0.0121
Original Decrypted with wrong keys Set 2 -0.0046
Original Original 1
(2) Original Encrypted -0.0025
Spine Original Decrypted with correct keys 1
Image Original Decrypted with wrong keys Set 1 0.0049
Original Decrypted with wrong keys Set 2 -0.0070
Original Original 1
(3) Original Encrypted -0.0119
Heel Original Decrypted with correct keys 1
Image Original Decrypted with wrong keys Set 1 0.0055
Original Decrypted with wrong keys Set 2 -0.0026
dimensional multi-scroll chaotic DDE. The proposed
cryptography technique realizes a XOR operation be-
tween separated planes of binary gray-scale image
and a shuffled multi-scroll DDE chaotic attractor im-
age. The security keys have been assigned through
initial condition and time constant in DDE repre-
sented by 56-floating-point number. Nonlinear dy-
namics of DDE have been described in terms of equi-
librium points, time-domain waveforms, and 3-scroll
attractor in phase-space domain. Encryption and de-
cryption security performances of three gray-scale hu-
man body CAT scan images with 256×256 pixels are
evaluated through density histograms, 2-dimensional
power spectral density, key space analysis, image cor-
relation coefficients, and key sensitivity. Demonstra-
tions of wrong-key decrypted image are also included.
The proposed technique has offered a potential al-
ternative to simple-but-highly-secured image storage
and transmissions in telemedication applications.
6. ACKNOWLEDGEMENT
The authors are grateful to Thai-Nichi Institute of
Technology for research fund supports. The authors
would also like to thank Assist. Prof. Dr.Adisorn
Leelasantitham for his useful suggestions.
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Mr.Sarun Maksuanpan was born
in Samutsakorn Province, Thailand in
1991. He is a 4th-year student pur-
suing B.Eng. in Computer Engineer-
ing from Computer Engineering Depart-
ment, Faculty of Engineering, Thai-
Nichi Institute of Technology (TNI).
Currently, he is also a research assistant
at Intelligent Electronic Research Lab-
oratory. His research interests include
information security systems, cryptosys-
tems, artificial neural networks, and dig-
ital image processing.
Wimol San-Um was born in Nan
Province, Thailand in 1981. He received
B.Eng. Degree in Electrical Engineer-
ing and M.Sc. Degree in Telecommuni-
cations in 2003 and 2006, respectively,
from Sirindhorn International Institute
of Technology (SIIT), Thammasat Uni-
versity in Thailand. In 2007, he was
a research student at University of Ap-
plied Science Ravensburg-Weingarten in
Germany. He received Ph.D. in mixed-
signal very large-scaled integrated cir-
cuit designs in 2010 from the Department of Electronic and
Photonic System Engineering, Kochi University of Technology
(KUT) in Japan. He is currently with Computer Engineering
Department, Faculty of Engineering, Thai-Nichi Institute of
Technology (TNI). He is also the head of Intelligent Electronic
Systems (IES) Research Laboratory. His areas of research in-
terests are artificial neural networks, control automations, dig-
ital image processing, secure communications, and nonlinear
dynamics of chaotic circuits and systems.