This document proposes a new digital image encryption technique based on multi-scroll chaotic delay differential equations (DDEs). The technique uses a XOR operation between separated binary planes of a grayscale image and a shuffled attractor image from a DDE. Security keys include DDE parameters like initial conditions, time constants, and simulation time. Experimental results using a 512x512 Lena image in MATLAB demonstrate the DDE dynamics, encryption/decryption security through histograms, power spectrums, and image correlations. Wrong key decryption is also shown. The technique offers potential for simple yet secure image transmission applications.

1. A New Simple Digital Image Cryptography
Technique Based on Multi-Scroll Chaotic Delay
Differential Equation
S. Maksuanpan and W. San-Um
Intelligent Electronic Systems Research Laboratory (IES)
Faculty of Engineering, Thai-Nichi Institute of Technology (TNI)
Patthanakarn, Suanlaung, Bangkok, Thailand, 10250. Fax :(+662)-763-2700, Tel :(+662)-763-2600
E-mail addresses: sarun.maksuanpan@gmail.com, wimol@tni.ac.th
Abstract—A new simple digital image cryptography technique
based on multi-scroll chaotic Delay Differential Equation (DDE)
is presented. The proposed cryptography technique realizes a
XOR operation between separated planes of binary gray-scale
image and a shuffled multi-scroll DDE chaotic attractor image.
The security keys are parameters in DDE, including initial
conditions, time constants, and simulation time that sets final
states of an attractor. Experimental results are performed in
MATLAB. Nonlinear dynamics of DDE are described in terms of
equilibrium points and an infinite-dimensional system of
Ordinary Differential Equation (ODE) with demonstrations of 3-
scroll attractors in both time and phase-space domains.
Encryption and decryption security performances of a gray-scale
Lena image with 512x512 pixels are evaluated through
histograms, 2-dimensional power spectrums, image correlation
plots and coefficients. Demonstrations of wrong-key decrypted
image are also included. The proposed technique offers a
potential alternative to simple-but-highly-secured image
transmissions in information privacy protection applications.
Keywords-component; Cryptography, Attractor Image, Security
Keys, Delay Differential Equation, 2-dimensional power spectrums.
I. INTRODUCTION
Recent advances in communication technologies have led
to great demand for secured image transmissions through wired
and wireless networks in 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
cryptography, which is a technique of information privacy
protection under hostile conditions [1]. Image cryptography
may be classified into two categories, i.e. (1) pixel value
substitution 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 algorithms for such cryptography, for
example, Data Encryption Standard (DES), International Data
Encryption Algorithm (IDEA), Advanced Encryption Standard
(AES), and RSA algorithm may not applicable in real-time
image encryption due to large computational time and high
computing power, especially for the images with large data
capacity and high correlation among pixels [2].
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
conventional encryption algorithms, chaos-based encryptions
are sensitive to initial conditions and parameters whilst
conventional algorithms are sensitive to designated keys.
Furthermore, chaos-based encryptions spread the initial region
over the entire phase space, but cryptographic algorithms
shuffle and diffuse data by rounds of encryption [3]. 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 encryption aspects consequently offer
high flexibility in encryption design processes and acceptable
privacy due to vast numbers of chaotic system variants and
numerous possible encryption keys.
Chaos-based encryption algorithms are performed in two
stages, i.e. the confusion stage that permutes 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
parameters of 1-D, 2-D, and 3-D chaotic maps such as Baker
map [4,5], Arnold cat map [6,7], and Standard map [8, 9] for
secret key generations. Furthermore, the combinations of two
or three different maps have been suggested [10,11] in order to
achieve higher security levels. Despite the fact that such maps
offer satisfactory security levels, iterations of maps require
specific conditions of chaotic behaviors through a narrow
region of parameters and initial conditions. Consequently, the
use of iteration maps has become typical for most of proposed
ciphers and complicated techniques in pixel confusion and
diffusion are ultimately required.
The DDE has emerged in mathematical models of natural
systems whose time evolution depends explicitly on a past
state, and can be described by an infinite-dimensional system
that can exhibit complex chaotic behaviors with a relatively
2013 5th International Conference on Knowledge and Smart Technology (KST)

2. Grayscale Image
512x512 Pixels
8-Bit Binary Number
per Pixel
[a0,a1,a2,…,a7] [a0]
[a1]
1-Bit Binary Number
per Pixel
Dec2Bin Separate
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
[b0]
[b1]
[b7]
[b0,b1,b2,…,b7]
XOR
Operations
(a)Imageto
beencrypted
(b)DDE
Attractor
8-Bit Binary Number
per Pixel
1-Bit Binary Number
per Pixel
(c)Encrypted
Image
Combine
Bin2Dec
Keys
:[k0,k1,k2,k3,k4]
Figure 1. Proposed encryption and detection algorithms using XOR
operation between separated planes of binary gray-scale image and a shuffled
multi-scroll DDE chaotic attractor image.
simple first-order differential equation. Existing DDEs include
the prominent Mackey-Glass DDE [12], which models the
production of white blood cells, and the Ikeda DDE [13],
which models a passive optical resonator system. Recently, In
recent years, further chaotic DDEs [14-16] based on the
Mackey-Glass DDE have been reported through the use of
piecewise-linear nonlinearities corresponding to a complex
two-scroll and multi-scroll attractors. In addition, the simplest
DDE with a sinusoidal nonlinearity [17] based on the Ikeda
DDE has also been presented.
This paper introduces a new simple digital image
cryptography technique based on multi-scroll chaotic Delay
Differential Equation (DDE). The proposed technique
employs only a simple XOR operation between separated
planes of binary gray-scale image and a shuffled multi-scroll
DDE chaotic attractor image. The security keys are parameters
in DDE, including initial conditions, time constants, and
simulation time that sets final states of an attractor.
Experimental results are performed in MATLAB using a gray-
scale Lena image with 512x512 pixels. Nonlinear dynamics of
DDE and Encryption and decryption security performances are
demonstrated. This paper is organized as follows; Section II is
an analysis of a multi-scroll chaotic DDE, Section III presents a
new digital image encryption and decryption algorithms,
Section IV provide all simulation results and discussions,
conclusion is finally drawn in Section V.
II. REALIZATIONS OF MULTI-SCROLL CHAOTIC DELAY
DIFFERENATIAL EQUATION
The first-order multi-scroll chaotic DDE is expressed in a
simple first-order differential equation as follows [18];
)(  xbFaxx n 
where a and b are unity, and the nonlinear term Fn(xτ) is a
piecewise-linear nonlinear function described as
)))12((sgn())12((sgn()( 1   mxmxxF n
mn 

where n and m are positive integers. The nonlinear function in
(2) particularly exhibits a stair-shape positive 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
)1sgn()1sgn(   xxaxx 
The DDE in (3) possesses three equilibrium points at {-2,0,2}
and the corresponding characteristic equation of its linearized
form, i.e. τ=0, can be obtained by the partial derivative with
respect to x as follows;
)1()1(1  xx  
where δ(·) is a Dirac delta function. The eigenvalues 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
coincides with the dimension of the system. Therefore, the
DDE in (3) can be approximated by an infinite-dimensional
system of ODEs as

)(
)1sgn()1sgn(
1
0
ii
i
NNN
xxN
x
xxxx






(5)
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 t. 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 vicinity 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
attractors 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 dimension of the DDE systems.
III. PROPOSED IMAGE ENCRYPRION AND DECRYPTION
ALGORITHMS
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 encryption and detection algorithms. In the
encryption process, the original gray-scale image is initially
converted into binary matrix in which each pixel is represented
by 8-bit binary numbers. For example, the pixel p(1,1) contains

3. Time constant : τ
f(t)
Figure 2. Bifurcation diagram of the time constant τ over the region [1,2].
Time (s)
f(t)
f(t-τ)
f(t)
Time (s)
f(t)
f(t-τ)
f(t)
(a)
(b)
Figure 3. Time domain waveforms and corresponding chaotic attractor
images; (a) The case τ=2, (b) The case τ=3.
the binary number a0-a7. Each pixel will then be separated into
eight planes corresponding to binary bits a0 to a7. It can be
considered that such eight planes are all represented in matrix
forms with a single binary number in each pixel, which is ready
for further Excusive-OR (XOR) operations.
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
behaviors. 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 shuffled 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
obtained from such XOR operations 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 encryption
Original Image Histogram of Original Image 2D Power Spectrum of original Image
Decrypted Image Histogram of Decrypted Image 2D Power Spectrum of Decrypted Image
Encrypted Image Histogram of Encrypted Image 2D Power Spectrum of Encrypted Image
(b)
(a)
(c)
Histogram of Decrypted ImageDecrypted Image with wrong keys 2D Power Spectrum of Decrypted Image
with wrong keys
(d)
Figure 4. Histograms and 2D power spectrum tests; (a) Original image, (b)
Encrypted image, (c) Decrypted image, (d) Decrypted image with wrong keys.
process in a backward algorithm as long as the security keys
are known.
IV. EXPERIMENTAL RESULTS AND SECURITY ANALYSIS
Experimental results have been performed in a computer–
aid design tool MATLAB. Nonlinear dynamics of DDE were
initially simulated and Encryption and decryption security
performances were subsequently evaluated. The initial
condition of DDE was set at 0.1. The simulation time was run
to 0.2s. Fig. 2 shows a bifurcation diagram, showing a route to
chaos of the time constant τ over the region [1, 2]. The
bifurcation shows that chaotic behaviors appears from
approximately τ=1.57. The highly chaotic region appears from
τ=1.73 and sustains over the all range of time constants. As for
demonstration, the values of τ=2 and τ=3 were selected in this
paper. Fig.3 depicts the time domain waveforms and
corresponding chaotic attractor images in the cases τ=2 and
τ=3. It is shown in Fig.3 that the time domain waveforms are
apparently chaotic and the attractors resemble three-scroll in a
positive slope shape. Such two cases of τ=2 and τ=3 show
apparently different chaotic images where the case τ=3
provides more randomness. These two chaotic attractor images
have been employed for encryption process.
Verifications of effectiveness of the proposed encryption
and decryption algorithms have been performed based on three
methods including, intensity histograms, 2D power spectrum,
and correlation of adjacent pixels. First, the histogram is a

4. Pixels values on (x,y)
Pixelsvalueson(x+1,y)
Pixels values on (x,y)
Pixelsvalueson(x+1,y)
Original Image Encrypted Image
Pixels values on (x,y) Pixels values on (x,y)
Pixelsvalueson(x,y+1)
Pixelsvalueson(x,y+1)Original Image Encrypted Image
(a)
(b)
Figure 5. Image correlation tests in original and encrypted images; (a)
Horizontally adjacent pixels, (b) vertically adjacent pixels.
(a) Cameraman (b) Barbara
(c) Boat (d) Finger print
Figure 6. Additional test images; (a) cameraman, (b) barbara, (c) Boat, and
(d) Finger print
graph that displays the number of pixels in an image at
different intensity values. In particular, the histogram of an 8-
bit grayscale image has 256 different intensity levels,
graphically displaying 256 numbers with distribution of pixels
amongst these grayscale values. Second, the 2D power
spectrum can be obtained through a Discrete Fourier Transform
(DFT) analysis technique and the algorithm is given by [19]
)
2
exp()
2
exp(),(),(
1
0
1
0 N
jvq
M
jup
qpfvuF
M
p
M
p
 
 



 
TABLE I. SUMMARY OF CORRELATION COEFFICIENTS OF LENA IMAGE.
Image 1 Image 2 Correlation Coefficients
Original Lena Original Lena 1.0000
Original Lena Encrypted Lena 0.0001
Original Lena Decrypted Lena 1.0000
Original Lena
Decrypted Lena
with Wrong keys
0.0006
TABLE II. SUMMARY OF DIFFERENNT IMAGES WITH DIFEREENT SIZE.
Images Image Sizes
Performance Analyses
Correlation
Coefficients
Speed
(ms)
Barbara
256×256 0.0028 110.474
512×512 0.0009 442.739
Camera man
256×256 -0.0000 107.910
512×512 -0.0009 583.384
Boat
256×256 -0.0001 107.910
512×512 0.0029 566.886
Finger print
256×256 -0.0018 99.749
512×512 0.0024 454.761
where p and q are a coordinates pair of an image, M and N are
the size of image, f(p,q) is the image value at the pixel (p,q).
Last, the correlation is a measure of relationships between two
pixels intensities of two images. In this paper, the correlation
between two vertically adjacent pixels and two horizontally
adjacent pixels are investigated by
))())(((),cov( qEqpEpEqp  
where p and q are also a coordinates pair of an image, E(p) and
E(q) are summation of total coordinates pairs.
Fig. 4 shows the histograms and 2D power spectrum tests;
(a) Original image, (b) Encrypted image, (c) Decrypted image,
(d) Decrypted image with wrong keys. The set of security key
is as follows; {Initial condition=0.1, τ=2, time=0.2s}. It can be
seen from Fig.4 (a) that the intensities of Lena image in the
histogram are contributed with different values in a particular
shape. In addition, the power spectrum is not flat having a peak
of intensity in the middle. Nonetheless, the encrypted image in
Fig.4 (b) has a flat histogram and power spectrum, indicating
that the intensity values are equally contributed over all the
intensity range. These results qualitatively guarantee that the
encrypted image is secured since the original image is already
diffused and invisible. In Fig.4(c), both histogram and power
spectrum resemble those in Fig.4 (a), indicating that the
decrypted image is very closely similar to the original image.
In other words, the transmitted image is decrypted successfully.
It is also illustrated in Fig.4 (d) that the decrypted image with
wrong keys is could not be retrieved where histogram and
power spectrum are different from the original image.
Fig.5 shows the image correlation tests in original and
encrypted images. It is seen in Fig.5 (a) that the horizontally
two adjacent pixels are highly correlated while the correlation
of the encrypted image is negligible. This result is also similar
in Fig.5 (b) in the case of the correlation of the vertically two
adjacent pixels. It can be considered from Fig.5 that the
encrypted image and the decrypted image are not correlated,
indicating that the encryption is successfully achieved. In order
to quantify the correlation, the correlation coefficients are

5. summarized in Table I. In the case where two images are
identical, the value of correlation coefficient is “1”. However,
if the two images are uncorrelated, the value of correlation
coefficient is “0”. It is seen in Table I that the original and the
decrypted Lena images are identical since the value of
correlation coefficient is unity. The value of correlation
coefficients of encrypted Lena image and decrypted Lena
image with wrong keys are relatively close to zero, indicating
that these images are uncorrelated. In addition, Fig.6 shows
four additional test images, including (a) cameraman, (b)
Barbara, (c) Boat, and (d) Finger print. Table 2 summarizes
correlation coefficient and speed analysis. It is seen that the
correlation coefficients of all images are relatively close to
zero. It can also seen that the speed analysis using a 4-GB
1067-MHz computer notebook with DDR3 SDRAM. The
speed is fairly fast in the range of 100 ms to approximately
600 ms.
CONCLUSION
The digital image cryptography technique based on multi-
scroll chaotic Delay Differential Equation (DDE) has been
presented. As for an attempt to achieve simple-but-highly-
secured image encryption and decryption algorithms in a
category of chaos-based cryptosystems, the proposed
cryptography technique realizes a XOR operation between
separated planes of binary gray-scale image and a shuffled
multi-scroll DDE chaotic attractor image where security keys
can be achieved by parameters in DDE, including initial
conditions, time constants, and simulation time that sets final
states of an attractor. Experimental results performed in
MATLAB using a gray-scale Lena image with 512x512 pixels
have been investigated in terms of chaotic nonlinear dynamical
behaviors, histograms, 2-dimensional power spectrum, and
correlation performances. The results have shown that both
histogram and 2D power spectrum of the encrypted image are
flat with highly uncorrelated, indicating that the image has
successfully been encrypted. The characteristics of the
decrypted image resemble the original image, indicating the
encrypted image can be decrypted effectively. The proposed
encryption and decryption algorithms have therefore offered a
potential alternative to simple-but-highly-secured image
transmissions in information privacy protection applications.
ACKNOWLEDGEMENTS
The authors are grateful to Thai-Nichi Institute of
Technology for research fund supports.
REFERENCES
[1] M. Philip, “An Enhanced Chaotic Image Encryption” International
Journal of Computer Science, Vol. 1, No. 5, 2011.
[2] G.H. Karimian, B. Rashidi, and A.farmani, “A High Speed and Low
Power Image Encryption with 128- bit AES Algorithm”, International
Journal of Computer and Electrical Engineering, Vol. 4, No. 3, 2012.
[3] G. Chen, Y. Mao, C.K. Chui, “A symmetric image encryption scheme
based on 3D chaotic cat maps”, Chaos, Solitons and Fractals, Vol. 21,
pp. 749-761, 2004.
[4] X. Tong, M. Cui, “Image encryption scheme based on 3D baker with
dynamical compound chaotic sequence cipher generator”, Signal
Processing, Vol. 89, pp. 480-491, 2009.
[5] J.W. Yoon, H. Kim,“An image encryption scheme with a pseudorandom
permutation based on chaotic maps”, Commun Nonlinear Sci Number
Simulation, Vol. 15, pp. 3998–4006, 2010.
[6] X. Ma, C. Fu, W. Lei, S. Li, “A Novel Chaos-based Image Encryption
Scheme with an Improved Permutation Process”, International Journal
of Advancements in Computing Technology, Vol. 3, No. 5, 2011.
[7] K. Wang, W. Pei, L. Zou, A. Song, Z. He, “On the security of 3D Cat
map based symmetric image encryption scheme”, Physics Letters A,
Vol. 343, pp. 432–439, 2005.
[8] K. Wong, B.S. Kwok, and W. Law, “A Fast Image Encryption Scheme
based on Chaotic Standard Map”, Physics Letters A, Vol. 372, pp. 2645-
2652, 2008.
[9] S. Lian, J. Sun, Z. Wang, “A block cipher based on a suitable use of the
chaotic standard map”, Chaos, Solitons and Fractals, Vol. 26, pp. 117–
129, 2005.
[10] K. Gupta, S. Silakari, “New Approach for Fast Color Image Encryption
Using Chaotic Map”, Journal of Information Security, pp. 139-150, 2011
[11] F. Huang, Y. Feng, “Security analysis of image encryption based on
two-dimensional chaotic maps and improved algorithm”, Front.Electr.
Electron. Eng. China, Vol. 4, No. 1, pp. 5-9, 2009.
[12] M. C. Mackey, L. Glass, "Oscillation and chaos in physiological control
systems, Science, Vol. 197, pp. 287-289, 1977.
[13] K. Ikeda, K. Matsumoto, "High-dimensional chaotic behavior in systems
with time-delayed feedback", J.Phys. Nonlinear Phenom., Vol. 29, pp.
223-235, 1987.
[14] H. Lu, Z. He, "Chaotic behavior in firstrst-order autonomous
continuous-time systems with delay", IEEE Trans. Circuits Syst. I:
Fund. Th. Appl., Vo1. 43, pp. 700-702, 1996.
[15] A. Tamasevicius, G. Mykolaitis, S. Bumeliene, "Delayed feedback
chaotic oscillator with improved spectral characteristics", Electron.Lett.,
Vol. 42, pp. 736-737, 2006.
[16] L. Wang, X. Yang, "Generation of multi-scroll delayed chaotic
oscillator", Electron. Lett., Vo1. 42, pp. 1439-1441, 2006.
[17] JC.Sprott, "A simple chaotic delay differential equation", J. Physics Lett.
A, Vol. 366, pp. 397-402, 2007.
[18] W. San-Um,B. Srisuchinwong, “A Simple Multi-Scroll Chaotic Delay
Differential Equation”, Electrical Engineering/Electronics, Computer,
Telecommunications and Information Technology (ECTI) Association
of Thailand, pp. 137-140, 2011.
[19] F. Han, J. Hu, X. Yu, Y. Wang, ”Fingerprint images encryption via
multi-scroll chaotic attractors”, Applied Mathematics and Computation,
Vol. 185, pp. 931–939, 2007.