improvement of chaotic secure communication scheme based on steganographic method and multimodal...

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: [email protected] Email: [email protected] *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 zy x ayz 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 miG 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) ( ), ,..., jX 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 jG 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

ˆˆ

x y z uy x ay uz 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 p̂ 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 1̂( )I t and 2̂ ( )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 Tx 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

uee 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 12 2 2

T T TV 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

dy y

dz z

e zeuu u k e sign e

u k e sign e

+⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥= = ×⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ×⎣ ⎦ ⎣ ⎦

(15)

y

z

yeap

zec

⎡ ⎤ −⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

(16)

1

2

y

z

yeII

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

ii

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 10 0

0, / , , and , .k k

r r r rr 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 xx x x

f x βx 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 1̂( )I t and 2̂ ( )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 1̂( )I t and 2̂ ( )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 xf 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 (see online version for colours)

(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 1̂ˆ(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) (see online version for colours)

(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 1̂( )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 Al-Taani, A.T. and Al-Issa, A.M. (2009) ‘A novel steganographic method for gray-level images’,

International Journal of Computer, Information & Systems Science & Engineering, Vol. 3, No. 4, pp.613–618.

Campos-Cantón, E., Femat, R. and Pisarchik, A. (2011) ‘A family of multimodal dynamic maps’, Communications in Nonlinear Science and Numerical Simulation, Vol. 16, No. 9, pp.3457–3462.

Chua, L.O., Yang, T., Zhong, G-Q. and Wu, C.W. (1996) ‘Adaptive synchronization of Chua's oscillators’, International Journal of Bifurcation and Chaos, Vol. 6, No. 1, pp.189–201.

de Melo, W. and van Strien, S. (1993) ‘One-dimensional dynamics’, Vol. 25 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin.

Fliess, M. and Sira-Ramírez, H. (2003) ‘An algebraic framework for linear identification’, ESAIM: Control, Optimisation and Calculus of Variations, Vol. 9, No. 2, pp.151–168.

Katzenbeisser, S. and Petitcolas, F.A. (2000) ‘Information hiding techniques for steganography and digital watermarking’, DNA, Vol. 28, No. 2.

Mata-Machuca, J.L., Martínez-Guerra, R., Aguilar-López, R. and Aguilar-Ibañez, C. (2012) ‘A chaotic system in synchronization and secure communications’, Communications in Nonlinear Science and Numerical Simulation, Vol. 17, No. 8, pp.1706–1713.

Smania, D. (2005) ‘Phase space universality for multimodal maps’, Bulletin of the Brazilian Mathematical Society, Vol. 36, pp.225–274.

Wu, C.W. and Chua, L.O. (1993) ‘A simple way to synchronize chaotic systems with applications to secure communication systems’, International Journal of Bifurcation and Chaos, Vol. 3, No. 6, pp.1619–1627.

Wu, C.W., Yang, T. and Chua, L.O. (1996) ‘On adaptive synchronization and control of nonlinear dynamical systems’, International Journal of Bifurcation and Chaos, Vol. 6, No. 6, pp.455–471.

Yang, T. and Chua, L. (1996) ‘Secure communication via chaotic parameter modulation’, Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on, Vol. 43, No. 9, pp.817–819.

improvement of chaotic secure communication scheme based on steganographic method and multimodal...

Documents