Improvement of chaotic secure communication scheme based on steganographic method and multimodal dynamic maps

Int. J. Systems, Control and Communications, Vol. 6, No. 4, 2015 305
Copyright © 2015 Inderscience Enterprises Ltd.
Improvement of chaotic secure communication
scheme based on steganographic method and
multimodal dynamic maps
Masoud Khodadadzadeh* and
H. Gholizadeh-Narm
Department of Electrical Engineering,
Shahrood University of Technology,
Shahrood, Iran
Email: masoud.khodadad@shahroodut.ac.ir
Email: gholizade@shahroodut.ac.ir
*Corresponding author
Abstract: In this paper a novel picture steganographic method using
multimodal chaotic maps for improvement of chaotic secure communication is
presented. Stego image is sent by modulating parameters of the transmitter. The
presented method includes a receiver system for asymptotic convergence which
estimates uncertain parameters of Rossler system. The gain of the receiver
system changes continuously by a high order sliding mode adaptive controller
(HOSMAC), so that system output errors converges to zero. The converged
parameters are used to determine the family of maps. After deciding map set,
grey level modification for hiding the message is selected and the stego image
is produced. Using the synchronisation and chaotic modulation method, the
proposed method is studied in the field of secure communications.
Keywords: chaotic synchronisation; multimodal chaotic maps; steganography;
grey level modification; GLM; secure communication.
Reference to this paper should be made as follows: Khodadadzadeh, M. and
Gholizadeh-Narm, H. (2015) ‘Improvement of chaotic secure communication
scheme based on steganographic method and multimodal dynamic maps’,
Int. J. Systems, Control and Communications, Vol. 6, No. 4, pp.305–320.
Biographical notes: Masoud Khodadadzadeh received his BSc in
Telecommunication Engineering from the Sadjad Institute of Higher Education,
Mashhad, Iran in 2011 and MSc in Control Engineering from Shahrood
University of Technology, Shahrood, Iran in 2014. His research interests
include control systems, chaotic systems, and image processing with particular
emphasis on special techniques for steganography methods.
H. Gholizadeh-Narm received his PhD in Electrical Engineering in 2010. He is
currently an Assistant Professor with the Department of Electrical and
Robotics, Shahrood University of Technology, Iran. His research interests
include chaotic systems, control of micro-grids, power systems and control.

306 M. Khodadadzadeh and H. Gholizadeh-Narm
1 Introduction
The second generation of chaotic secure communication systems was introduced between
1993 and 1995, known as chaotic modulation. This generation utilised two different
methods for modulating message into chaotic carrier. In the first one (chaotic parameter
modulation), message signal was used to change the parameters of chaotic transmitter
(Yang and Chua, 1996); while in the second one (non-automatic chaotic modulation),
message signal was used to change the phase space of the chaotic transmitter (Wu and
Chua, 1993). Unlike chaotic parameter modulation in which, transmitter switches among
different trajectories in different chaotic attractors, in non-automatic chaotic modulation,
transmitter switches among different trajectories of the same chaotic attractor.
Since the bifurcation of a chaotic system is an extremely complex, finding the way by
which parameters change is really difficult, even if an intruder has partial information
about the chaotic system structure in transmitter. In receiver side, an adaptive controller
is used to regulate chaotic system’s parameters adaptively so that synchronisation error
approaches zero (Chua et al., 1996).
The degree of system security can be increased by the second generation; however
such a security level is not satisfactory yet. For this reason, at first, we steganography
message signal using the grey level modification (GLM) method with multimodal chaotic
maps and send it using chaotic parameter modulation. Then with synchronisation in
receiver side and by determining system’s undefined parameters and also by specifying
map set, the message signal is retrieved.
Steganography is the art of hiding a message in a communicational channel or route
in such a way that no one, apart from the intended recipient, can be aware of the
existence of the message. GLM is a method for mapping information using GLM
and utilises the concept of odd and even numbers to map information in a picture
(Wu et al., 1996). This method is a one-to-one map between binary information and
selected pixel of a picture.
Multimodal chaotic maps are used to increase the security of chaotic secure
communication systems, in steganographic method, multimodal chaotic maps are used to
select pixels. We use a family of maps whose domain is partitioned according to the
maximal number of modals to be generated each of which consists of a logistic map. The
number of members of a set equals to the maximum number of modals.
In general, one of methods used to increase the security of secured chaotic
communication systems is to send encrypted information using such systems. Having a
semi-noisy nature, chaotic systems have various applications in this field. Normally
various generations of secured chaotic systems do not have a high security individually.
A solution to eliminate this problem is to combine various generations and use novel
encryption methods before sending information. Unlike encryption in which an attacker
knows the encrypted data that are sent but only is not able to recognise its crypt,
steganography sends hidden data without attackers even being informed of sending
procedure of such hidden data. Considering these characteristics, steganography is one of
the solutions that may be utilised.
The rest of this paper is organised as follows. In Section 2 synchronisation method is
introduced. Section 3 is devoted to the generation of multimodal chaotic maps based on
the logistic map. In Section 4, the steganographic method is discussed. In Section 5, we
develop our proposed method using multimodal chaotic maps. To assess the effectiveness

Improvement of chaotic secure communication scheme 307
of our method, we present some numerical simulations in Section 6. Finally, we present
the conclusions in Section 7.
2 System description and synchronisation method
In chaotic secure communication system both master and slave systems are chaotic.
Chaos synchronisation means, the trajectories of the slave system can track that of the
master system starting from arbitrary initial condition.
2.1 Transmitter
Rossler system’s dynamic is as follows:
( )
1 1 1
1 1 1
1 1 1
x y z
y x ay
z b z x c
= − −
= +
= + −
(1)
This system has a chaotic behaviour for vicinity of a = b = 0.2, c ∈ [3, 11], and a large set
of initial conditions.
2.2 Some algebraic properties
For the purpose of chaotic synchronisation of two Rossler systems, we introduce the
following definitions.
Definition 1: Consider a smooth nonlinear system, described by a state vector
1{ }i n n
iX x R=
= ∈ and by the output vector 1{ } ,i m m
iG g R=
= ∈ of the form:
( , ), ( )X f X P G h X= = (2)
where h(⋅) is a smooth vector function and p ∈ Rl
is a constant parameters vector, with
l < n. Let G(j)
denote the jth
time derivative of the vector G. We say that the vector state X
is algebraically observable, if it can be uniquely expressed as
( )(1) ( )
, ,..., j
X G G G= Φ (3)
for some integer j and for some smooth function Φ.
Definition 2: Under same conditions as in Definition 1. If the vector of parameters, P
satisfies the following relation
( ) ( )( ) ( )
1 2,..., ,...,j j
G G Y Y PΩ = Ω (4)
where Ω1(⋅)and Ω2(⋅)are, respectively, n × 1 and n × n smooth matrices, then P is said to
be algebraically identifiable with respect to the output vector G (Fliess and Sira-Ramírez,
2003).

To this end the following state x1, can be rewritten, as
1 1 1x g ag= − (5)
where the outputs is chosen such that g1 = y1 and g2 = z1. Moreover, substituting the
above expression into the third differential equation of (1), we have
1 2 2 1 2 2b g g g ag g cg+ − = + (6)
Hence, we conclude that Rossler system is algebraically observable and identifiable with
respect to the available outputs g1 = y1 and g2 = z1. it is possible to solve the
synchronisation problem of the uncertain Rossler system provided that the states y1 and z1
are always available and state x1 is non-available. Moreover the vector of parameters
p = (a, c) can be simultaneously recovered.
2.3 Receiver
Consider the uncertain Rossler system (1), referred as the transmitter system, with the
available output states y1 and z1. And let us propose the following receiver controlled
system:
( )
2 1 1 1
2 2 1 2
2 1 2 3
ˆ
ˆ
x y z u
y x ay u
z b z x c u
= − − +
= + +
= + − +
(7)
Then, the synchronisation objective is to find u = (u1, u2, u3) and ˆ ˆ ˆ( , )p a c= such that the
unknown Rossler system (7) follows the Rossler system (1) with different initial
condition and ˆp converging to the actual values of (a, c).
2.4 Transmission of message signals by chaotic parameter modulation
In this section we discuss the case when both parameters a and c of system (1) are used to
transmit message signals I1(t) and I2(t). We use modulation rules to modulate I1(t) and
I2(t). in parameters of the transmitter in (1).
The modulation rules are given by
1 1
2 2
ˆˆ ˆ( ) ( ), ( ) ( ),
ˆˆ ˆ( ) ( ), ( ) ( ),
a t a I t a t a I t
c t c I t c t c I t
= + = +
= + = +
(8)
where 1
ˆ ( )I t and 2
ˆ ( )I t are the recovered message signals.
Now let us introduce the following errors:
1 2 1 2 1 2; ; ;x y ze x x e y y e z z= − = − = − (9)
1 1 1 2 2 2
ˆ ˆ; ;
ˆ ˆ; ;
a a a c c c
I I I I I I
= − = −
= − = −
(10)

and according to them, we define the following vectors:
( ) ( ) ( )1 2, , ; , ; ,T T T
x y ze e e e p a c I I I= = = (11)
From equations (1) to (7), and taking into account the modulation rules (8) we have:
1
1 2
2 3
x
y
z x
ue
e e ex ay I y u
e ze cz I z u
−⎡ ⎤⎡ ⎤
⎢ ⎥⎢ ⎥= = + + −⎢ ⎥⎢ ⎥
⎢ ⎥⎢ ⎥ − − −⎣ ⎦ ⎣ ⎦
(12)
where for simplicity, we stand for y = y1 and x = x1. As we can see, the above system can
be considered as a control problem where the vector inputs u and p must be proposed
such that e asymptotically converges to zero.
2.5 Control design
In this section a high order sliding mode adaptive controller (HOSMAC) proposed in
Mata-Machuca et al. (2012) is used at the receiver to maintain synchronisation by
continuously tracking the changes in the modulated parameters. Then, I1(t) and I2(t) can
be recovered by using this controller.
Consider a Lyapunov function
1 1 1
2 2 2
T T T
V e e p p I I= + + (13)
The time derivative of V along the trajectories of (9) is then given by
1 1 2
2 3 1 1 2 2
x x y y y y
x z z z z
V aa cc e u e e aye I ye e u
ze e cze I ze e u I I I I
= + − + + + −
+ − − − + +
(14)
Now, in order to make V semi-definite negative, we propose ; ,p u and I as
1
2 1
3 2
( )
( )
y z
d
y y
d
z z
e zeu
u u k e sign e
u k e sign e
+⎡ ⎤⎡ ⎤
⎢ ⎥⎢ ⎥= = ×⎢ ⎥⎢ ⎥
⎢ ⎥⎢ ⎥ ×⎣ ⎦ ⎣ ⎦
(15)
y
z
yea
p
zec
⎡ ⎤ −⎡ ⎤
= =⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎣ ⎦⎣ ⎦
(16)
1
2
y
z
yeI
I
zeI
⎡ ⎤ −⎡ ⎤
⎢ ⎥= = ⎢ ⎥
⎢ ⎥ ⎣ ⎦⎣ ⎦
(17)
where k1 and k12are strictly positive constants and d is any positive even integer.

3 Multimodal chaotic maps
A unimodal map is a continuous 1D function ℜ → ℜ with a single critical point c0 and
monotonically increasing on one side of c0 and decreasing on the other (Campos-Cantón
et al., 2011). The dynamics of the system is governed by the function
1 ( , )n nx f x β+ = (18)
where xn is the system state after n iterations and β is the bifurcation parameter and initial
condition c0.
We use a family of maps whose domain is partitioned according to the maximal
number of modals to be generated. The theory of multimodal maps is studied in de Melo
and van Strien (1993). Here, we are interested in the definition given in Smania (2005)
for a particular type of multimodal maps.
A map fβ is k-modal, if it is continuous and has k critical points denoted by c0, c1, …,
ck–1 in I = [a, b] ⊂ R, monotonically increases on the left of each ci and monotonically
decreases on the right of each ci, (i = {0, 1, 2, …, k}).
We say that f is a k-modal map if it can be written as a composition of k unimodal
maps f1, f2, …, fk with the following properties:
• fi: Ii → I has a unique critical point (a maximum)
• f(ci) = f(cj), for i ≠ j
•
1
.
k
i
i
I I
=
=∪
The parameterised family F of maps fβ is defined by the following piecewise function
( )( ) [ )1 1( ) , for , ,r r r rf x d x x d x d d+ += − − ∈β β
where
}{( )/ 0,1,2,..., 1 .rd r k r k= = −
Note that
[ ] [ ) [ )
1 1
1 1
0 0
0, / , , and , .
k k
r r r r
r r
J k γ I d d d d
− −
+ +
= =
∈ = = =∪ ∩β α φ
4 Steganographic method
Steganography is an art of hiding information inside others. The main purpose of
steganography is to hide a message in another one in a way to prevent any attacker to
detect or notice the hidden message (Katzenbeisser and Petitcolas, 2000). GLM
Steganography is a technique to map data by modifying the grey level values of the
image. It is a one-to-one mapping between the binary data and the selected pixels in an
image (Al-Taani and Al-Issa, 2009). From a given image a set of pixels are selected
based on a mathematical function, for this purpose we use a multimodal chaotic map. The
grey level values of those pixels are compared with the bit stream that is to be mapped in
the image. Initially, the grey level values of the selected pixels (odd pixels) are made

even by changing the grey level by one unit. Once all the selected pixels have an even
grey level, it is compared with the bit stream, which has to be mapped. The first bit from
the bit stream is compared with the first selected pixel. If the first bit is even, then the
first pixel is not modified as all the selected pixels have an even grey level value. But if
the bit is odd, then the grey level value of the pixel is decremented by one unit to make its
value odd, which then would represent an odd bit mapping. This is carried out for all bits
in the bit stream and each and every bit is mapped by modifying the grey level values
accordingly.
5 Proposed method
To hide the information in different steganographic method, the proper pixel should be
selected so that they are completely random and cannot be identified and exert minimal
impact on the visual properties. To begin the process of embedding, we first select a set
of pixels, which would be used for hiding the data.
In the proposed method, we used multimodal chaotic map for choosing random
pixels. Domain of multimodal chaotic maps is partitioned according to the maximal
number of modals to be generated. To this end, by choosing system parameters, we can
consider the parameter ci = ki, i = 1, 2, 3, 4 as family of multimodal maps and the
parameter aj = rj, j = 1, 2, 3, 4 as member of the family. The correspondence between
parameters are given in Tables 1 and 2.
Table 1 Correspondence between Ki and ci
Ki ci
K1 = 1 c1 = 1
K2 = 2 c2 = 2
K3 = 3 c3 = 3
K4 = 4 c4 = 4
Table 2 Correspondence between ri and ai
rj aj
r1 = 1 a1 = 1
r2 = 2 a2 = 2
r3 = 3 a3 = 3
r4 = 4 a4 = 4
Selecting these parameters the monoparametric family F of multimodal chaotic maps fβ
can be described as
(1/ 4 ) for [0,1/ 4);
(1/ 2 )( 1/ 4) for [1/ 4,1/ 2);
( )
(3/ 4 )( 1/ 2) for [1/ 2,3/ 4);
(1 )( 3/ 4) for [3/ 4,1];
β
x x x
x x x
f x β
x x x
x x x
− ∈⎧
⎪ − − ∈⎪
= ⎨
− − ∈⎪
⎪ − − ∈⎩
(19)

where β ∈ J = [0, 64], this interval is determined by k = 4, γ = 0.25, and. Then, r = 0, 1, 2,
3 and the family F consists of the following four members:
1 the quadmodal map f64 for r = 0
2 the trimodal map f48 for r = 1
3 the bimodal map f32 for r = 2
4 the unimodal map f16 for r = 3.
Figure 1 shows the phase diagram of stretching and folding of structure for various
member of this family.
Figure 1 The block diagram of proposed method (see online version for colours)
By mapping the xn and xn+1 axis of the phase diagram showing stretching and folding of
the chaotic map to x and y axis of the cover image respectively, maximum of xn+2 are
selected pixels for candidate of embedding data. The selected points are shown in
Figure 2.
The uncertain parameters of Rossler systems are used to determine the family of
maps. After parameter adaptation and suitable synchronisation in receiver, the exact
parameters and family of maps will be available.
The secret message can be images, texts or sound. We put message into a bit stream
and choose the size of 32-bit for it. Using a function (multimodal maps) that takes two
numbers as keys; pixel of image is randomly selected. The grey level values of those
pixels are compared with the bit stream that is to be mapped in the image. The first bit
from the bit stream is compared with the first selected pixel. If the first bit is even, then
the first pixel is not modified as all the selected pixels have an even grey level value. But
if the bit is odd, then the grey level value of the pixel is decremented by one unit to make
its value odd, which then would represent an odd bit mapping. This is carried out for all
bits in the bit stream and each and every bit is mapped by modifying the grey level values
accordingly.

Figure 2 Three-dimensional phase diagrams showing stretching and folding structure of the
quadmodal chaotic map for k = 4, (a) r = 0 (b) r = 1 (c) r = 2 (d) r = 3 (see online
version for colours)
(a) (b)
(c) (d)
We use modulation rules to modulate I1(t) and I2(t) in parameters of the transmitter. The
modulation rules are as follows:
1 1
2 2
ˆˆ ˆ( ) ( ), ( ) ( ),
ˆˆ ˆ( ) ( ), ( ) ( ),
a t a I t a t a I t
c t c I t c t c I t
= + = +
= + = +
(20)
where 1
ˆ ( )I t and 2
ˆ ( )I t are the recovered message signals.
After adaptive synchronisation and parameters estimation in receiver, we can encrypt
the Stego image and find the message data. Figure 1 shows the block diagram of our
proposed method.

6 Experimental results
In our experiments, we used an image with size of 256 × 256. Computer simulations have
been carried out in order to test the effectiveness of the proposed method. We use
modulation rules to modulate I1(t) and I2(t) in parameters of the transmitter. The
modulation rules are given by (20). where 1
ˆ ( )I t and 2
ˆ ( )I t are the recovered message
signals.
We changed the image to vector so that it was transmitted as I1(t) in our secure
communication scheme. We can transmit other image with I2(t). Figure 3 shows the
original image and vector of the image used for simulation.
Figure 3 Selected pixels for candidate of embedding data corresponds to the quadmodal map
shown in Figure 1 (see online version for colours)
(a) (b)
(c) (d)

We can choose the transmitter system parameter p = (a, c); while the arbitrary
initial conditions were selected as x1(0) = 1, y1(0) = –1, z1(0) = 1. Figure 4 shows
the attractor and the behaviour of the whole state of the Rossler system for
p = (a = 0.2, c = 5.7).
Figure 4 The image using for steganography: (a) original image (b) vector of the pixels
(see online version for colours)
(a)
(b)
This system displays a chaotic behaviour for the parameters values in a neighbourhood
{a = 0.2, c = 5.7}, therefore by mapping c and a interval to ki = 1, 2, 3, 4 and
rj = 0, 1, 2, 3 respectively, these numbers can be used to specify multimodal dynamic
maps.
Therefore, by choosing system parameters, we can consider the parameter ci = ki,
i = 1, 2, 3, 4 as family of multimodal maps and the parameter aj = rj, j = 1, 2, 3, 4 as
member of the family. Selecting c2 = 4 → k2 = 2 and a1 = 0.2 → r1 =0, one biomodal map
of this family F of multimodal chaotic maps can be described as
16
(1/ 2 ) for [0,0.5);
( ) 16
(1 )( 1/ 2) for [0.5,1];
x x x
f x
x x x
− ∈⎧
= ⎨
− − ∈⎩
Figure 5 shows the phase diagram of stretching and folding of structure for one biomodal
map of this family.

Figure 5 (a) Rossler chaotic system attractor (b) Qualitative behaviour of the Rossler system
(a)
(b)
By mapping the xn and xn+1 axis of the phase diagram showing in Figure 5 to x and y axis
of the cover image respectively, maximum of xn+2 are selected pixels for candidate of
embedding data. The selected points are shown in Figure 6.
To show the performance of the proposed control strategy, we carried out simulation
using the set-up as above, and fixing the receiver system gains as k1 = k2 = 0.7 and m = 4;
with the receiver system initialised at 1 1 1 1
ˆˆ(0) (0) (0) 0, (0) 0, 2.x y z p I= = = = = − In
Figure 7, we can see that the synchronisation errors asymptotically converge to zero and
parameters estimate to real value in receiver system.
As we expected a better performance can be obtained as long as the time is increased.
Figure 8 shows secret message error between transmitter and receiver in our chaotic
secure communication scheme that converges to zero.

Figure 6 (a) Family F of biomodal logistic maps (b) The phase diagram of stretching and
folding of this structure (see online version for colours)
(a)
(b)
Figure 7 Selected pixels for candidate of embedding data corresponds to the one biomodal map
shown in Figure 5(b) (see online version for colours)

Figure 8 (a) Synchronisation errors (b) Parameters estimation, when the master system is
initialised at (1, 1, 1); and the actual parameters vector is fixed as p = (0.2, 4)
(a)
(b)
In this paper, we consider a text as a secret message data which is embed in cover image
using proposed steganographic method. Figure 9 shows the stego image after embedding
message. After using synchronisation and parameter estimation, the secret message can
be retrieved exactly.
Secret message is used in experimental results: This is a secret message …
7 Conclusions
In this paper, we proposed a steganographic method using multimodal maps for the fist
time that is imperceptible while a secret message is concealed in a cover image. In our
scheme, synchronisation and parameters identification of the constant unknown

parameters of Rossler system were used. Indeed, we improve the security of
chaotic secure communication approach via parameter modulation by using multimodal
maps. The experimental results show that the proposed steganographic method is
capable of achieving high quality stego images and high embedding capacity (especially,
when block embedding is performed). Compared with other methods like LSB1, LSB2,
MBNS, etc. selected pixels for candidate of embedding data is more random. Finally,
numerical simulations were carried out to evaluate the performance of the proposed
solution.
Figure 9 Rossler system for chaotic communication. Numerical results for message signal s1 and
Information recovery error 1 1
ˆ( )I I− (see online version for colours)
Figure 10 Stego image after embedding data
Note: This image shows that the message cannot be identified and exert minimal impact
on the visual properties.

References
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