Message Embedded Chaotic Masking Synchronization Scheme

July 30, 2016 11:25 WSPC/S0218-1274 1650140

International Journal of Bifurcation and Chaos, Vol. 26, No. 8 (2016) 1650140 (15 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218127416501406

Message Embedded Chaotic Masking SynchronizationScheme Based on the Generalized Lorenz System

and Its Security Analysis*

Sergej Celikovsky† and Volodymyr Lynnyk‡Institute of Information Theory and Automation

of the Czech Academy of Sciences,P.O. Box 18, Prague 8, 182 08, Czech Republic

†[email protected]‡[email protected]

Received March 2, 2016

This paper focuses on the design of the novel chaotic masking scheme via message embeddedsynchronization. A general class of the systems allowing the message embedded synchronizationis presented here, moreover, it is shown that the generalized Lorenz system belongs to this class.Furthermore, the secure encryption scheme based on the message embedded synchronization isproposed. This scheme injects the embedded message into the dynamics of the transmitter aswell, ensuring thereby synchronization with theoretically zero synchronization error. To ensurethe security, the embedded message is a sum of the message and arbitrary bounded function ofthe internal transmitter states that is independent of the scalar synchronization signal. The hex-adecimal alphabet will be used to form a ciphertext making chaotic dynamics of the transmittereven more complicated in comparison with the transmitter influenced just by the binary step-like function. All mentioned results and their security are tested and demonstrated by numericalexperiments.

Keywords : Chaotic masking; generalized Lorenz system; message embedded synchronization.

1. Introduction

During recent decades, the idea to use chaotic sys-tems in communication has been attracting manyresearchers [Pecora & Carroll, 1990; Parlitz et al.,1992; Kocarev et al., 1992; Cuomo & Oppenheim,1993; Wu & Chua, 1993; Kocarev & Parlitz, 1995;Milanovic & Zaghloul, 1996; Lian & Liu, 2000; Frad-kov et al., 2000; Chen et al., 2009; Kaddoum &Shokraneh, 2015; Fradkov et al., 2015; Chen &Dong, 1998]. Famous features of the chaotic sys-tems like strong dependence on the initial condi-tions, topological transitivity, wide spread spectrumof their signals, etc., directly suggested the idea ofusing chaotic systems to build up a new generation

of the communication methods and to design thenovel secure chaos-based communication schemes.Most of them are based on the synchronization of thechaotic systems [Materassi & Basso, 2008; Kocarevet al., 1992; Pecora & Carroll, 1990; Kolumbanet al., 1998]. Unfortunately the majority of the pro-posed methods are introduced without any crypt-analysis, they have rather weak security and theywould be very likely broken later on [Li et al.,2005; Li et al., 2006; Perez & Cerdeira, 1995;Alvarez et al., 2004; Yang et al., 1998a; Short, 1994;Hu & Guo, 2008]. The use of the continuous-timechaotic systems for the encryption of the digitaldata has been studied briefly [Alvarez & Li, 2006;

∗This work was supported by the Czech Science Foundation through the research Grant No. 13-20433S.

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S. Celikovsky & V. Lynnyk

Fig. 1. Block diagram of the conventional (classical) chaotic masking scheme. Here, m(t) is the message; Hx(t) is the outputchaotic signal of the transmitter; s(t) is the transmitted signal; Hx(t) is the output chaotic signal of the receiver; m(t) is therecovered message.

Dachselt & Schwarz, 2001; Kocarev, 2001] dueto the prevalently used analogue chaotic masking[Alvarez-Ramirez et al., 2002; Lian & Liu, 2000].

The purpose of this paper is to introduce yetanother version of the chaotic masking scheme. It isbased on the so-called message embedded synchro-nization of the continuous-time chaotic systems andit will be then used to transmit securely the digi-tal data represented by the hexadecimal waveform.To explain the novelty of this concept, let us firstclassify the existing and newly introduced chaoticmasking schemes:

• Conventional (Classical) Chaotic Mask-ing Schemes. These schemes have been developedprimarily for the analog communication [Kocarevet al., 1992; Cuomo et al., 1993; Cuomo & Oppen-heim, 1993; Lian & Liu, 2000; Liang et al., 2012].Their main idea is to mask a low-level messagesignal adding to it a suitable chaotic signal. Thediagram of this kind of communication schemes isshown in Fig. 1.

• Improved Chaotic Masking Schemes. Theseschemes [Wu & Chua, 1993; Milanovic & Zaghloul,1996; Kocarev & Parlitz, 1995; Liao & Huang, 1999;Chen, 2010] use a message signal to modulate thedynamics of the transmitter. More precisely, theycombine a conventional chaotic masking algorithmand a chaotic modulation algorithm. The transmit-ted signal is the sum of the information-bearingsignal and the output of the chaotic transmitter.

Dynamics of the transmitter is influenced by thesame information-bearing signal. The diagram ofthis kind of the communication schemes is shownin Fig. 2.

• Message Embedded Chaotic MaskingScheme (MECHMS). This scheme is introducedin the current paper and it uses the embeddedmessage signal to modulate the dynamics of thetransmitter equations to achieve the so-called mes-sage embedded synchronization. The diagram ofthis communication scheme is shown in Fig. 3.

As already noted, the main goal of the currentpaper is to introduce the chaotic masking schemebased on the message embedded synchronization.Furthermore, the class of systems where the abovescheme is possible is characterized and it is shownthat the well-known generalized Lorenz system(GLS) [Celikovsky & Vanecek, 1994; Celikovsky &Chen, 2002, 2005a, 2005b] belongs to that class.Thanks to that, the MECHMS will be implementedin detail for the GLS and its security aspects will beanalyzed. Finally, it will be shown that it is possibleto use this newly proposed chaotic masking schemeto encrypt the digital information.

Digital information secure encryption usingcontinuous-time chaotic systems was already stud-ied in [Celikovsky & Lynnyk, 2012; Lynnyk &Celikovsky, 2010], where the so-called Desynchro-nization Chaos Shift Keying (DECSK) methodwas introduced and its security analyzed in detail.

Fig. 2. Block diagram of the improved chaotic masking scheme. Here, m(t) is the message; Hx(t) is the output chaotic signalof the transmitter. Internal dynamics of the chaotic transmitter is influenced by m(t); s(t) is the transmitted signal; Hx(t) isthe output chaotic signal of the receiver; m(t) is the recovered message.

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Message Embedded Chaotic Masking Synchronization Scheme

Fig. 3. Block diagram of the MECHMS. Here, m(t) is the message; Hx(t) is the output chaotic signal of the transmitter;M(x(t)) is an arbitrary bounded function of the transmitter internal states x(t). Internal dynamics of the chaotic transmitteris influenced by the embedded message m(t); s(t) is the transmitted signal; Hx(t) is the output chaotic signal of the receiver;M(x(t)) is an arbitrary bounded function of the receiver internal states x(t); m(t) is the recovered message.

DECSK method uses the transmitter which con-sists of the chaotic system affected by the selectedparameter varying dependingly on the bit to beencrypted (“0” or “1”). The receiver consists oftwo perfectly synchronized transmitter copies cor-responding to those different values of that selectedparameter. The transmitter sends a short signalsegment generated by one of those parameter val-ues corresponding to the bit value to be sent. Itturns out that even a quite short time segmentis sufficient to have one of the receiver systemsquickly desynchronized, which enables to detect theencrypted bit. To make that method realistic, thesecond derivative error detection was used ensuringthat even two iterations only are sufficient to detectthe desynchronization. All these ideas were imple-mented in detail for the GLS. Later on, an intro-ductory study on how to use the GLS to encryptthe digital data using the chaotic masking was pre-sented in [Celikovsky & Lynnyk, 2013]. Neverthe-less, as it will be discussed later on, the security ofthis method is subject to some drawbacks.

The present paper thereby continues this lineof the research and uses the GLS to design the mes-sage embedded chaotic masking encryption schemecapable to transmit the digital information. To doso, the embedded message signal m(t) is added tothe chaotic output generated by the chaotic oscilla-tor whose dynamics depends on the same embeddedmessage m(t) = m(t) + M(x(t)). Here, M(x(t))is arbitrary bounded function of the transmitterinternal state x(t), which should be independentof the transmitted scalar synchronizing signal. Fur-thermore, it will be shown that it is possible to usethis property to implement the chaotic masking forthe digital signal modulation (information-bearingwaveform). Suitable waveform is to be selected,namely, the hexadecimal digital waveform will beused. This waveform is smoothed by the integra-tion procedure as the nonsmooth message visibly

affects the second derivative of the transmitted sig-nal being thereby easily decryptable. Such a sig-nal step-like function may have a small amplitudewhich makes the integrated signal even more unde-tectable, unless one has at his/her disposal the cor-rect carrying chaotic signal to subtract it from theencrypted message. All these properties provide afirm basis for the successful security analysis of thenewly proposed method.

The rest of the paper is organized as follows.The next section briefly repeats some known factsabout the GLS. Section 3 introduces the MECHMSfor the GLS, while its security analysis is providedby Sec. 4. Numerical experiments are collected inSec. 5. The final section draws the conclusions andgives some outlooks for future research.

2. The GLS and Its Synchronization

Definition 2.1. The following nonlinear system inR3 is called as the generalized Lorenz system (GLS):

x =

[A 0

0 λ3

]x +

0

−x1x3

x1x2

, A =

[a11 a12

a21 a22

].

(1)

Here, x = [x1, x2, x3]�, λ3 ∈ R, and A has eigenval-ues λ1, λ2 ∈ R, such that

−λ2 > λ1 > −λ3 > 0. (2)

The inequality (2) goes back to the well-knownShilnikov’s chaos analysis near the homoclinicityand it can be viewed as the necessary condition forthe chaos existence, see a more detailed discussionin [Celikovsky & Chen, 2002]. GLS is said to benontrivial if it has at least one solution that goesneither to zero, nor to infinity, nor to a limit cycle.The following result, enabling the efficient synthe-sis of a rich variety of the chaotic behaviors for theGLS, was obtained in [Celikovsky & Chen, 2002].

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Theorem 2.2 [Celikovsky & Chen, 2002]. For thenontrivial GLS (1)–(2), there exists a nonsingu-lar linear change of coordinates z = Tx, whichtakes (1) into the following generalized Lorenzcanonical form:

z =

λ1 0 0

0 λ2 0

0 0 λ3

z + cz

0 0 −1

0 0 −1

1 τ 0

z, (3)

where z = [z1, z2, z3]�, c= [1,−1, 0] and τ ∈ (−1,∞).

Actually, the parameter τ plays an importantrole being the single scalar “nonlinear bifurcationparameter”, while the remaining parameters haveonly a qualitative influence being the eigenvaluesof the approximate linearization of the GLS at theorigin. These qualitative parameters are required tosatisfy the robust condition (2) only, so that a finetuning may be done using the single scalar param-eter τ only. In [Celikovsky, 2004], the GLS wasfurther extended to the so-called hyperbolic-typegeneralized Lorenz system (HGLS) which has thesame canonical form as (4) but with τ ∈ (∞,−1).In such a way, the parameter range to be usedin the encryption can be extended even more. In[Celikovsky & Chen, 2005a], the complete and niceclassification of all related systems is given showingthat many previously introduced classes are actu-ally particular cases of the GLS or the HGLS.

Synchronization of two or more copies of theGLS is based on yet another canonical form, theso-called observer canonical form of the GLS, pro-vided by the following theorem.

Theorem 2.3 [Celikovsky & Chen, 2005b]. Both thenontrivial GLS (1) and its canonical form (3) arestate equivalent to the following form, called in thesequel as the observer canonical form of the GLS :

dη

dt=

(λ1 + λ2)η1 + η2

−η1

[λ1λ2 + (λ1 − λ2)η3 +

(τ + 1)η21

2

]

λ3η3 + K(τ)η21

,

K(τ) =λ3(τ + 1) − 2τλ1 − 2λ2

2(λ1 − λ2),

(4)

where η = [η1, η2, η3]� and the state equivalencecoordinate change and its inverse are, correspond-ingly :

η =[z1 − z2, λ1z2 −λ2z1, z3 − (τ + 1)(z1 − z2)2

2(λ1 −λ2)

]�,

(5)

z =[λ1η1 + η2

λ1 − λ2,λ2η1 + η2

λ1 − λ2, η3 +

(τ + 1)η21

2(λ1 − λ2)

]�. (6)

Indeed, the above observer canonical form, whenviewing η1 = x1 = z1 − z2 as the output, is verysimilar to the form linearizable by output injection.As a consequence, the following observer-based syn-chronization of two copies of the GLS is possible.

Theorem 2.4 [Celikovsky & Chen, 2005b]. Con-sider system (4) with the output η1 and its uni-formly bounded trajectory η(t), t ≥ t0. Further,consider the following system having the input ηm

1and the state η = (η1, η2, η3)�:

dη

dt=

l1 1 0

l2 0 0

0 0 λ3

η +

λ1 + λ2 − l1

−λ1λ2 − l2

0

ηm

1

+

0

−(λ1 − λ2)ηm1 η3 −

(12

)(τ + 1)(ηm

1 )3

K(τ)(ηm1 )2

,

(7)

where l1,2 < 0. For all ε ≥ 0, assume |η1(t) −ηm

1 (t)| ≤ ε. Then, it holds exponentially in time that

limt→∞ ‖η(t) − η(t)‖ ≤ Cε,

for a constant C > 0. In particular, for ηm1 ≡ η1,

(7) is the global exponential observer of system (4).

Proofs of Theorems 2.3–2.4 may be found in[Celikovsky & Chen, 2005b]. In the sequel, sys-tem (4) will be called as the master while (7) asthe slave. Theorem 2.4 can be used for the conven-tional chaotic masking in the sense of [Cuomo &Oppenheim, 1993] (see Fig. 1). Here, the messageto be masked is added to the synchronizing signalwhich corrupts the synchronization, as claimed byTheorem 2.4, where synchronization error does notgo to zero completely. Namely, ηm

1 = η1 + m(t),where m(t) is the signal representing message to bemasked, i.e.

limt→∞ ‖η(t) − η(t)‖ ≤ C‖m(t)‖.

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This drawback will be removed in the next sectionusing the message embedded synchronization.

3. Message EmbeddedSynchronization and Its Usefor Chaotic Masking

The message embedded synchronization is proposedin this section. It will be used for the chaotic mask-ing later on. Characterize first a more general classof the systems where the message embedded syn-chronization is possible. Namely, consider the non-linear system of the form[

x1

x2

]=

[F1 0

0 F2

] [x1

x2

]+

[ϕ1(Hx1, x2)

ϕ2(Hx1)

], (8)

where[x1

x2

]= x ∈ Rn, x1 ∈ Rn1 , x2 ∈ Rn2 ,

n1 + n2 = n, F is (n × n) matrix, H is (1 × n1)matrix, F1 is (n1 × n1) matrix, F2 is (n2 × n2)matrix. Further, suppose that (F1,H) is adetectable pair, F2 is Hurwitz and nonlinearfunctions ϕ1, ϕ2 are such that ϕ1 : Rn2+1 → Rn1 ,ϕ2 : R → Rn2. The synchronized copy of (8) canbe obtained using the scalar synchronizing signalHx(t) as follows[

y1

y2

]=

[F1 0

0 F2

][y1

y2

]+

[ϕ1(Hx1, y2)

ϕ2(Hx1)

]

+

[L1H(y1 − x1)

0

]. (9)

Here, L1 is (n1 × 1) matrix such that F1 + L1H isHurwitz and y1 ∈ Rn1 , y2 ∈ Rn2. Indeed, definee = (e1, e2)� = (y1 − x1, y2 − x2). Subtracting (8)from (9) gives[

e1

e2

]=

[F1 + L1H 0

0 F2

][e1

e2

]

+

[ϕ1(Hx1, y2) − ϕ1(Hx1, x2)

0

]. (10)

Notice, that e2 → 0 exponentially since F2 isHurwitz. Assuming that the synchronization signalHx(t) of (8) is bounded, one concludes easily that

ϕ1(H(x(t)), y2(t)) − ϕ1(H(x(t)), x2(t)) → 0

exponentially as t → ∞. Therefore, by Lemma 9.1of [Khalil, 2002] e1(t) → 0 exponentially as t → ∞,since F1 + L1H is Hurwitz. Summarizing, e(t) → 0exponentially as t → ∞ and therefore (8) and (9)are exponentially synchronized.

3.1. Message embedded chaoticmasking scheme

The previously introduced synchronization schemeis adapted here for the case when synchronizingscalar channel is to carry the additive message tobe masked as well. Let m(t) denote the masked andthereby securely encrypted message to be transmit-ted. More precisely, the signal Hx1 + m(t) will betransmitted through the public scalar channel. Todo so, consider the following system[

x1

x2

]=

[F1 0

0 F2

] [x1

x2

]+

[ϕ1(Hx1 + m(t), x2)

ϕ2(Hx1 + m(t))

]−

[L1m(t)

0

](11)

and its synchronized copy[y1

y2

]=

[F1 0

0 F2

][y1

y2

]+

[ϕ1(Hx1 + m(t), y2)

ϕ2(Hx1 + m(t))

]+

[L1H

0

]y1 −

[L1(Hx1 + m(t))

0

]. (12)

Indeed, define

e = (e1, e2)� = (y1 − x1, y2 − x2).

Subtracting (11) from (12) gives[e1

e2

]=

[F1 + L1H 0

0 F2

][e1

e2

]+

[ϕ1(Hx1 + m(t), y2) − ϕ1(Hx1 + m(t), x2)

0

]. (13)

Assuming that the synchronization signal Hx + m(t) is bounded, one has again that e → 0 exponentiallyas t → ∞.

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The message embedded chaotic maskingscheme (MECHMS) can be described as follows.Let m(t) be the message to be encrypted and trans-mitted, e.g. the plaintext waveform representation.Then m(t) = m(t) + M(x(t)) is the encryptedmessage to be embedded and transmitted. Here,M(x(t)) is an arbitrary bounded function of theinternal components of the state x(t), which shouldbe independent from the scalar synchronizing sig-nal Hx1. Using (11), one generates the transmittedsignal as

s(t) = m(t) + Hx1(t) = m(t) + Hx1 + M(x(t)).

Recovered message m(t) will be taken as m(t) =s(t) − Hy1(t) − M(y(t)). Note, that m(t) −m(t) = H(x1(t) − y1(t)) + M(x(t)) −M(y(t)), i.e.m(t) − m(t) → 0 exponentially as t → ∞.

Remark 3.1. Notice, that the observer canonicalform of the GLS (4) is the system in the form (8),

with

F1 =

[0 1

0 0

],

F2 = [λ3], H = [1, 0],

x1 =

[η1

η2

], x2 = [η3],

ϕ1 =

(λ1 + λ2)η1

−λ1λ2η1 − (λ1 − λ2)η1η3 − (τ + 1)η31

2

,

ϕ2 = K(τ)η21.

As a consequence, the GLS in its observer canoni-cal form can be used to implement the MECHMS.To be more specific, the transmitter of the GLSimplementation of the MECHMS has the followingform

dη

dt=

0 1 0

0 0 0

0 0 λ3

η −

l1

l2

0

m(t) +

(λ1 + λ2)(η1 + m(t))

−(η1 + m(t))[λ1λ2 + (λ1 − λ2)η3 +

(τ + 1)(η1 + m(t))2

2

]

K(τ)(η1 + m(t))2

, (14)

K(τ) =λ3(τ + 1) − 2τλ1 − 2λ2

2(λ1 − λ2), η = [η1, η2, η3]�. (15)

The corresponding receiver can be constructed as follows

dη

dt=

0 1 0

0 0 0

0 0 λ3

η +

l1

l2

0

η1 −

l1

l2

0

(η1 + m(t))

+

(λ1 + λ2)(η1 + m(t))

−(η1 + m(t))[λ1λ2 + (λ1 − λ2)η3 +

(τ + 1)(η1 + m(t))2

2

]

K(τ)(η1 + m(t))2

, (16)

where η = [η1, η2, η3]�, K(τ) is described byEq. (15) and η1 + m(t) is the synchronization sig-nal received from the transmitter via the publicchannel. Notice, that the amplitude of the mes-sage embedded signal m(t) is much smaller thanthe magnitude of η1(t) and in our numerical exper-iments1 it is equal to ≈ ±10−2.

4. Security Analysis of theMECHMS

Following [Alvarez & Li, 2006], we will analyze herewhether the proposed GLS implementation of theMECHMS used as a cryptosystem can be broken bythe chaos-specific attacks.

1MATLAB-SIMULINK ode4 Runge–Kutta procedure with the fixed step size equal to 10−3 is being used throughout thepaper.

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4.1. Message signal extraction

As a matter of fact, the MECHMS is the digitalchaotic cipher implemented via the GLS to trans-mit securely the digital information. Some kindsof the chaos-specific attacks proposed in the liter-ature have broken many analogue chaotic ciphers,but they are not useful enough against the digi-tal chaotic ciphers. Message signal extraction meth-ods, like the power spectral analysis proposed in[Alvarez et al., 2004; Yang et al., 1998a] and thereturn-map analysis introduced in [Li et al., 2006;Perez & Cerdeira, 1995; Yang et al., 1998b], are usu-ally performed without any knowledge of the spe-cific structure of the chaotic system used for themessage encryption. Furthermore, the power analy-sis attack is more efficient, since it does not requireany knowledge about the structure of the chaoticsystem used in the transmitter of the cryptosystem.Both these methods can be applied to extract themessage signal if the latter is the periodic signalor it consists of periodic frames within the dura-tion long enough. The MECHMS for the GLS isresistive against both kinds of the above attacksas its ciphertext is a digital signal, whose wave-form is dependent on the plaintext which affectsa ciphertext during each iteration of the compu-tation of the trajectory of the chaotic system inthe transmitter. Even if the GLS has some periodicframes, these frames are useless for the intruder,because a single periodic frame has ≈ 2 sec length2

and each point on the ciphertext waveform repre-sents a bit of the encrypted plaintext. Celikovskyand Lynnyk [2012] provides the security analy-sis of the DECSK encryption method [Celikovskyet al., 2006] based on the GLS via the power andthe return-map analysis. Both these methods areuneffective against the DECSK digital cipher andsince the MECHMS has a similar structure of thechaotic generator based on the GLS, one can assumethat the MECHMS is cryptographically resistantagainst the power and the return-map analysisas well.

4.2. Key space

The GLS has three initial values and four controlparameters. Every algorithm used for the secureencryption must have the key space cardinality ofat least 2128 to oppose the brute force attack. As the

secret key, the combination of the initial values andthe bifurcation parameter τ , both with the doubleprecision is proposed here. The key is sensitive todifferences equal to or larger than 10−14. Therefore,the total key size is equal to 1014 ·1014 ·1014 ≈ 2140,that makes this secure method resistive against thebrute force attack. The key space may be extendedusing the parameters λ1, λ2, λ3 and the period-ical switching of the bifurcation parameter τ , ifnecessary.

4.3. Parameter and stateestimation of the MECHMS

As noticed in [Wang et al., 2004], many secure com-munication schemes based on the chaotic dynamicsare not sensitive enough to the parameter mis-match, therefore the intruder can estimate thebifurcation parameter used in the transmitter viaapproximation of the parameter values. Many meth-ods of the parameter estimation were introducedduring the last two decades. Direct calculation ofthe parameters from the short piece of cipher-text was introduced for the Lorenz chaotic systemin [Vaidya & Angadi, 2003] and studied in detailfor Chua’s circuit in [Liu et al., 2004]. Evolution-ary algorithms [Zelinka et al., 2010] were inten-sively studied both for the chaos synthesis andthe parameter identification in the chaotic systems[Chang et al., 2008]. Estimation of the parametersof the transmitter in the Chaos Shift Keying (CSK)scheme [Dedieu et al., 1993] via the adaptationalgorithm was provided by the standard gradient(or speed-gradient) techniques in [Fradkov et al.,2000]. State and parameter estimation based on theadaptive observers can be found in [Zhang, 2002;Tyukin et al., 2013; Besancon, 2000]. It was shownin [Liang et al., 2008] that the states of the GLSwith unknown bifurcation parameter, used in thesecure synchronization proposed in [Celikovsky &Chen, 2005b], can be estimated by the adaptiveobserver introduced in [Zhang, 2002]. This adap-tive observer can be therefore used for the state andbifurcation parameter estimation of the improvedchaotic masking synchronization scheme using theGLS. Nevertheless, the quite long time period of theconstant unknown parameter value is necessary, sothat one can avoid such an attack by reasonablefrequent change of the value of τ .

2That corresponds to 2000 iterations, since the fixed step size is equal to 10−3 in the presented numerical simulations.

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To be specific, rewrite first system (4) asfollows:

dη

dt=

λ1 + λ2 1 0

−λ1λ2 0 −(λ1 − λ2)η1

0 0 λ3

η

+

0

−12η3

1

K1η21

+

0

−12η3

1

K2η21

τ, (17)

where K1 = λ3−2λ22(λ1−λ2)

, K2 = λ3−2λ12(λ1−λ2) , η = [η1, η2,

η3]�. Adaptive observer of [Zhang, 2002] for sys-tem (17) can be then constructed in the followingway:

dη

dt=

λ1 + λ2 1 0

−λ1λ2 0 −(λ1 − λ2)η1

0 0 λ3

η

+

l1 0 0

l2 0 0

0 0 0

(η − η) +

0

−12η3

1

K1η21

+

0

−12η3

1

K2η21

τ + Υ(t) ˙τ , (18)

˙τ = −Υ�(t)C�C(η − η), (19)

Υ(t) =

λ1 + λ2 + l1 1 0

−λ1λ2 + l2 0 −(λ1 − λ2)η1

0 0 λ3

Υ(t)

+

0

−12η3

1

K2η21

, (20)

where η = [η1, η2, η3]�, K1 = λ3−2λ22(λ1−λ2) , K2 =

λ3−2λ12(λ1−λ2) , C = [1, 0, 0], and li < 0, i = 1, 2, are suchthat the matrix

L =

[λ1 + λ2 + l1 1

−λ1λ2 + l2 0

](21)

is Hurwitz. The synchronization between (17)and (18) is achieved if limt→∞ |η(t) − η(t)| = 0,which means that (18) is the asymptotically stableobserver of (17).

The observer error e := (η − η) has thedynamics

e =

λ1 + λ2 + l1 1 0

−λ1λ2 + l2 0 −(λ1 − λ2)η1

0 0 λ3

e

+

0

−12η3

1

K2η21

ε − Υ(t)Υ�(t)

1

0

0

e1. (22)

In the sequel, denote the error of the bifurcationparameter as follows:

ε = τ − τ. (23)

Obviously, it holds

ε = ˙τ = −Υ1(t)e1. (24)

Further, introduce the combined error variable e asfollows

e = e − Υ(t)ε, (25)

then

e =

e1

e2

e3

=

e1

e2

e3

+

Υ1(t)

Υ2(t)

Υ3(t)

ε (26)

and

e1 = e1 + Υ1(t)ε. (27)

Using the system equation (17) and observer equa-tions (18)–(20) gives

e = e − Υ(t)ε − Υ(t)ε

=

λ1 + λ2 + l1 1 0

−λ1λ2 + l2 0 −(λ1 − λ2)η1

0 0 λ3

e. (28)

Moreover, (24) becomes

ε = −Υ1(t)(e1 + Υ1ε) = −Υ21(t)ε − Υ1(t)e1. (29)

Summarizing, by (28) one has that e → 0 as t → ∞due to the fact that the matrix (21) is Hurwitz andλ3 < 0. Moreover, if Υ2

1(t) is persistently excitingthen ε → 0 as t → ∞, too. Therefore by (26) e → 0as t → ∞ and one has both that η → η and τ → τ as

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t → ∞. Adaptive observer (18) can be used for theestimation of the bifurcation parameter τ and, con-sequently, to synchronize two GLSs without param-eter τ knowledge provided unknown τ is constantfor a suitably long time period. The correspond-ing numerical results will be described in Sec. 5.4later on.

5. Numerical Experiments

In this section, the numerical simulations of thevarious chaotic masking secure communicationschemes based on the GLS are demonstrated. Themain goal of this section is to demonstrate the effec-tiveness of the MECHMS (see Fig. 3) in compari-son to the improved masking scheme (see Fig. 2)applied to the secure communication. Notice, thatthe constant bifurcation parameter τ and the initialconditions of the transmitter in improved chaoticmasking scheme can be estimated after a certaintime period by the adaptive observer (18). There-fore, for more security of the MECHMS, the valueof the bifurcation parameter τ will be changed every1024 iterations (≈ 1 sec). Moreover, the ciphertextof the chaotic masking embedded synchronizationmethod consists of reduced values of the scalar sig-nal η1 with only eight digits after the decimal point(the numerical experiments with seven digits afterthe decimal point work quite well, too) and the mes-sage embedded signal depends on the reduced valueof η3.

5.1. Hexadecimal numeral system

Each hexadecimal digit represents four binary digits(bits) and the primary use of hexadecimal notationis a human-friendly representation of binary-codedvalues in computing. Hexadecimal digits are alsocommonly used to represent the computer memoryaddresses [Borowik et al., 2012]. One hexadecimaldigit represents a nibble, which is half of a byte (8bits). Earlier, in [Celikovsky & Lynnyk, 2006] it wasshown that using the multialphabet, it is possible toincrease the security and the bit rate in the DECSKcommunication scheme. Another goal of using the

hexadecimal numerical system is that the behav-ior of the transmitter will be changing accordingto 16 different chaotic systems in comparison tobinary digits use relying on two different chaoticsystems only. Table 1 describes the values of thelow-level signal with their binary and hexadecimalcode representations used in the numerical simula-tions within the paper.

5.2. Relation between security ofthe MECHMS and the signalform of the encrypted message

Celikovsky and Lynnyk [2013] illustrated the appli-cation of the chaotic masking scheme based onthe GLS for the encryption and the decryption ofthe information represented by the binary step-likefunction. The amplitude of the used binary step-likefunction expressing the original message m(t) wasmuch smaller than the amplitude of the signal gen-erated by the chaotic system used in the transmitter(“0” = 0, “1” = 0.00001). During the preparationof this paper the detailed analysis of the GLS-based improved chaotic masking scheme shown inFig. 2 has been performed. It has been revealedthat the low-level binary message can be decrypteddirectly from the openly transmitted ciphertextby the calculation of its second derivation. In theMECHMS the transformation of the original mes-sage by its integration before the injection into thetransmitter chaotic system was used. Therefore, thetransformed message is smoother than the origi-nal message and it eliminates the possibility of thedecryption of the original message by the secondderivation calculation. After the decryption of theoriginal message in the receiver, the decrypted mes-sage can be transformed into the original messageby the differentiation with respect to time.

5.3. Illustrative example of theMECHMS

Numerical simulations demonstrate the efficiency ofthe proposed MECHMS. Its diagram is presented

Table 1. Construction of the low-level signal from the hexadecimal information.

Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F

Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111Low-level signal −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8

(×10−3)

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in Fig. 3. Here, m(t) is the original transmit-ted message and m(t) is the recovered message.Finally, m(t) = m(t) + η3(t) × 10−2 is the embed-ded message. The transmitter generates the outputsignal m(t) + η1(t) which depends on the mes-sage m(t). Figure 4 illustrates the MECHMS GLSimplementation for the encryption and the decryp-tion of the digital information. Namely, Fig. 4(a)shows the original message to be encrypted, whileFig. 4(b) shows its integral. The dynamics of thetransmitter is affected by the mentioned integraland this integral is added to the chaotic outputsignal to be sent via communication channel aswell. The amplitude of the original message ismuch smaller than the chaotic signal generated bythe transmitter. Figure 4(c) illustrates the signalwhich is transmitted via the communication chan-nel. Finally, Fig. 4(d) shows the decrypted messagewhile the error of the synchronization is illustrated

in Fig. 4(e). Decryption scheme of the MECHMSrequires the mutual initial synchronization betweenthe chaotic transmitter and receiver up to the avail-able numerical precision, called in the sequel as the“numerical zero”. Therefore, the initial condition isthe immediate candidate for the secret key. As our“numerical zero” is 10−14, this key space is naturallydiscretized in the sense that two initial conditionsthat are closer to each other than the numericalzero should be represented by the same key. In[Celikovsky & Lynnyk, 2012; Lynnyk & Celikovsky,2010] the secure encryption system based on theGLS was described by the DECSK method. It useda similar principle for the formation of the secretkey. At the same time, the DECSK secure encryp-tion scheme cannot use the full carrier of the gen-erated signal and for the carrier signal at valuesclose to zero the use of more than one iteration perbit is required. The MECHMS does not face the

1000 1020 1040 1060 1080 1100Iterations

-0.01

0

0.01

m(t

)

1000 1020 1040 1060 1080 1100Iterations

-3.5

-3

-2.5

m(t

)dt

×10 -4

(a) (b)

1000 1020 1040 1060 1080 1100Iterations

0.4

0.6

0.8

m( t

)+η 1

( t)

1000 1020 1040 1060 1080 1100Iterations

-0.01

0

0.01

m(t

)

(c) (d)

1000 1020 1040 1060 1080 1100Iterations

-4

-3

-2

-1

e m=

m(t

) −m

(t) ×10 -5

(e)

Fig. 4. Time histories related to the encryption and the decryption of the plaintext “93D76A73304466B91E52A151ADDB34CADA95023E29BB2C3289D12C9BB8B8D8896D972D14B93857F95F96D9EB517999EED6D6D7” using the MECHMS.Namely, (a) shows the plaintext time signal m(t), (b) the integral of the plaintext m(t), (c) the ciphertext of the embed-ded message m(t) and the reduced value of η1(t), (d) the reconstructed plaintext m(t) and (e) the error em between theoriginal plaintext m(t) and the reconstructed plaintext m(t). “Intruder” can see only the third graph from the top.

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above mentioned problems and it can be used forthe encryption of the digital data using the full car-rier of the generated signal. On the other hand, thedetailed analysis provided in the next subsectionshows that to enhance the security of the messagetransmission one can change the part of the secretkey in the transmitter and the receiver during somesuitable selected time intervals.

5.4. Estimation of the initialcondition and the bifurcationparameter by the adaptiveobserver

Kerckhoffs’s principle was stated by Auguste Kerck-hoffs. It proclaims that everything about the cryp-tosystem is public knowledge, except the secret key.Later on, Claude Shannon reformulated this prin-ciple as “the enemy knows the system” [Shannon,1949]. Therefore, we suppose that the “intruder”knows everything about the proposed secure cryp-tosystem, except the secret key, which consists ofthe initial condition and the bifurcation parameterτ of the chaotic system in the transmitter.

Figure 5 illustrates the extraction of the trans-mitted message generated by the transmitter shown

in Fig. 2 without knowledge of the initial condi-tion and the constant bifurcation parameter τ esti-mation after 3900 of iterations by the adaptiveobserver (18). Here, Fig. 5(a) shows the originalmessage m(t). Figure 5(b) illustrates the ciphertextm(t) + η1 openly transmitted through the commu-nication channel to the receiver. Figure 5(c) showsthe estimation of the constant bifurcation parame-ter τ , one can see that the value of the parameterτ ≈ 0.4696 estimated by the adaptive observer (18)is still quite far from the correct value τ = 0.5 ofthe bifurcation parameter in the transmitter. Never-theless, the adaptive observer (18) can reconstructthe original message m(t) with a small error as well[see Fig. 5(d)]. Figure 6 demonstrates the full timehistories of the errors between the transmitter ofimproved chaotic masking scheme and the adap-tive observer (18) and the progress of the estima-tion of the constant bifurcation parameter τ . Onecan see that the states of the adaptive observer andthe estimated value of the bifurcation parameter τconverge to the states of the transmitter and thecorrect value of the constant bifurcation parame-ter τ during a short time of ≈ 3.5 sec. Therefore,the secret key component τ of the improved mask-ing scheme (see Fig. 2) must be changed at least

Iterations3900 3920 3940 3960 3980 4000

m(t

)

×10 -6

0

5

10

Iterations3900 3920 3940 3960 3980 4000

m(t

)+η 1

-1.5

-1.45

-1.4

-1.35

(a) (b)

Iterations3900 3920 3940 3960 3980 4000

τ

0.4696

0.4696

0.4696

0.4696

0.4696

Iterations3900 3920 3940 3960 3980 4000

m(t

)+e 1

×10 -4

-1.5

-1

-0.5

0

(c) (d)

Fig. 5. Time histories related to the encryption and the decryption of the binary plaintext “00000101010110100001000110001110100000000010111101100101010100101010001001110010000000111110000111110” using the adaptive observer (18).Namely, (a) shows the binary plaintext time signal m(t), (b) the ciphertext m(t) + η1(t), (c) the estimated bifurcationparameter τ and (d) the reconstructed plaintext with the error m(t) + e1. “Intruder” can see the graph (b) which gives apossibility of the estimation of the bifurcation parameter τ (c) and the reconstruction of the plaintext (a) during quite shorttime interval. Here, the improved chaotic masking scheme described in Fig. 2 is analyzed.

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Iterations0 2000 4000 6000 8000 10000

Err

ors

-100

0

100e

1

e2

e3

Iterations0 2000 4000 6000 8000 10000

τ

-1

-0.5

0

0.5

(a) (b)

Fig. 6. Time histories of (a) the errors e1, e2, e3 between (17) and (18) and (b) the value of the estimated bifurcationparameter τ .

Iterations0 2000 4000 6000 8000 10000

Err

ors

-50

0

50

100

Iterations0 2000 4000 6000 8000 10000

τ

-1.5

-1

-0.5

0

0.5

(a) (b)

Iterations0 2000 4000 6000 8000 10000

τ

0

0.2

0.4

0.6

(c)

Fig. 7. Time histories related to estimation of the parameter τ by the adaptive observer (18). Here, (a) shows the errorse1, e2, e3 between (17) and (18), (b) the value of estimated bifurcation parameter τ and (c) the value of the parameter τ inthe transmitter (17) which switches every 1024 iterations between values 0.1 and 0.5.

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every three seconds, to exclude the possibility ofthe decryption of the transmitted information bythe adaptive observer attack.

Figure 7 illustrates the error dynamics andthe estimate of the varying bifurcation parameterτ in the improved chaotic masking scheme. Here,the bifurcation parameter τ is changed every 1024of iterations (≈ 1 sec) taking two different values.Switching between two bifurcation parameter val-ues during the encryption of the message improvesthe chaotic masking transmitter shown in Fig. 2 andit prevents to estimate the state and the bifurcationparameter by the adaptive observer (18).

Using the adaptive observer (18) for the esti-mation of the varying bifurcation parameter τ andtransmitter states of the MECHMS are illustratedin Fig. 8. During all the time the synchroniza-tion error between the transmitter (14) and the

1 2 3 4 5Iterations 10 4

-10

0

10

20

Err

ors

e1

e2

e3

(a)


0

0.2

τ

(b)


0

0.1

0.2

τ

(c)

Fig. 8. Time histories related to the estimation of theparameter τ by the adaptive observer (18) for the MECHMS.Here, (a) shows the errors e1, e2, e3 between (14) and (18),(b) the value of the estimated bifurcation parameter τ and(c) the value of the parameter τ in the transmitter (14) whichswitches every 1024 iterations between the values 0.1 and 0.2.

adaptive observer (18) is large. Furthermore, theadaptive observer cannot estimate a varying bifur-cation parameter τ . Therefore, decryption of theoriginal message with the adaptive observer (18)is not possible. That actually shows a goodstrength against very specific adaptive observerattack and demonstrates the possibility of usingthe proposed MECHMS in the chaos-based securitycommunication.

6. Conclusions and Outlooks

The GLS family has been analyzed and used forthe chaotic masking of the digital information. Moreprecisely, the MECHMS has been used to avoid thecorruption of the synchronization by the transmit-ted message. This enabled to use a digital modula-tion with a very small amplitude thereby enhancingthe security of the chaotic masking. It was shownthat the proposed digital communication methodhas a potential of introducing the high degree ofsecurity at the low receiver complexity. At the sametime, it requires a reasonable amount of data toencrypt a single bit, thereby making the continuous-time chaotic system for the digital data encryptionrealistic. The proposed cryptosystem has a goodresistance against the adaptive observer attack. Fur-ther research will be devoted to the application ofthe proposed algorithm to encrypt the informationtransmitted between nodes in the complex networks[Celikovsky et al., 2013]. Another goal of futureresearch is to study the message embedded synchro-nization in analogue chaotic masking as well.

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Message Embedded Chaotic Masking Synchronization Scheme