A New Robust Additive Hyperchaos Masking Algorithm for Secure Digital Communications

Said SADOUDI*

*Laboratoire Systèmes de Communications Ecole Militaire Polytechnique

Algiers, Algeria sadoudi.said@gmail.com

Camel TANOUGAST** and Mohamad Salah AZZAZ*

**Laboratoire Interfaces, Capteurs et Microélectroniques Université de Lorraine

Metz, France

Abstract— In this paper, a new robust additive hyperchaos masking algorithm for secure digital chaotic communications is proposed and implemented in FPGA technology. Through the hardware implementation, we determine the experimental conditions to ensure hyperchaotic synchronization and robust securing of embedded communications. The obtained real-time results under these conditions confirm the effectiveness of the proposed solutions.

Keywords— Additive chaos masking; hyperchaotic synchronization; FPGA; VHDL.

I. INTRODUCTION Since the discovery of identical synchronization between

two chaotic systems by Pecora and Carroll [1], the use of chaotic synchronization in communication systems has been widely investigated by scientists. The authors have noted that particularly useful classes of chaotic systems are those that possess a self-synchronization property [1]. A chaotic system is self-synchronizing if it can be decomposed into at least two subsystems: a drive system (transmitter) and a stable response subsystem (receiver) that synchronize when coupled with a common signal. Through this idea, the authors in [2] have noted that for some synchronized chaotic systems, the ability to synchronize is robust. For example, in the Lorenz system, the synchronization is highly robust to perturbations in the drive signal. This property leads to the first interesting chaotic communication scheme introduced in [2-3] and known as Additive Chaos Masking (ACM). This technique was rapidly abandoned by the scientists because of its sensitivity to the additional information signal and channel noise, and it is not robust to secure communication [4-5].

Milanovich and Zaghloul, in [6], have proposed a significant improving of the ACM technique performances by introducing a simple modification which consists of the reinjection of the transmitted signal, the combination of information and chaotic signals, into the transmitter chaotic generator. Since this last is identical to that of the receiver, the information signal can be decoded correctly from the received signal. Later, the authors in [7] have confirmed the good performances of this technique [6], named it Chaotic Synchronization via Dynamic Feedback Modulation (CS-DFM), and compared it to the ACM based techniques by a quantitative comparison in the sense of digital communication performances in the presence of noise.

However, the CS-DFM technique suffers from some very considerable disadvantages. The fact that it is based on ACM principle, two problems arise: the security issue and the loss of the chaotic dynamics of the generators. For the first, previous works have proposed approaches to break ACM technique [4-5]. For the second, careful must be done when using ACM to preserve the chaotic behavior of the chaotic generators [2-3]. Consequently, conditions to the information amplitude must be imposed. In this context and for the analog chaotic communication, the authors in [3] have given an experimental condition for the ACM technique. They suggested that the amplitude of the information signal must be lower of 20 dB than the amplitude of the modulated chaotic signal to ensure the preservation of the chaotic dynamics and then robust chaotic masking. But, since chaotic systems are sensitive to the variation of initial conditions, most of the analog chaotic communication schemes will fail due to parameters mismatch.

Recently, a numerical generation of chaos, based on FPGA implementation technology, has been widely investigated since the problem of parameters mismatch does not exist [8-11]. Numerical methods, contrary to the analog ones, provide accuracy and large possibility integration in embedded applications especially for data encryption and secure communications.

In this paper, we propose hardware implementation of new Additive Hyperchaos Masking (AHM) algorithm for Digital Hyperchaotic Communications, through which we determine and give experimental solutions to the insufficiencies of CS-DFM [6] in terms of robustness and chaos secure reliability. The proposed AHM is based on the Hyperchaotic Synchronization via Dynamic Feedback Modulation (HS-DFM) technique, which is the extension of the CS-DFM developed and proved theoretically in [6], to insure synchronization between hyperchaotic systems. To test and validate our solutions, we introduce our AHM in a Digital Hyperchaotic Communication System (DHCS) which is implemented in Xilinx Virtex II-Pro FPGA technology [12] and where experimental conditions are imposed to insure the preservation of the keys generators hyperchaotic behaviors, a robust hyperchaotic masking and then a correct information recovery. We use the hyperchaotic Lorenz system as keys generator, implemented in FPGA by using the method developed in [8-9]. The obtained real-time results validate our proposed solution.

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II. NEW ADDITIVE HYPERCHAOS MASKING ALGORITHM

A. Conditions to Preserve Chaotic Dynamic To find which continuous chaotic generator is suitable for

our communication system with overcoming the problem of the loss of chaotic dynamics, we have tested three autonomous chaotic systems, Genesion-Tesi, Jerk and Linz-Sprott, a nonautonomous Duffing system and a hyperchaotic Lorenz system. All these systems are implemented by using the method developed in [8-9]. This method is optimal since it uses a direct VHDL description of the fourth order Runge-Kutta (RK-4) numerical resolution technique. To determine the conditions, we use the ACM principle (Fig. 1). The test consist of adding a constant value A, as the information signal m(t) and represented on 32 bits (16Q16) fixed-point data format (see Tab.1), to the chaotic signal x(t) and inject the resulting signal s(t) to the chaotic system of the transmitter (see Fig.1). Then, verify the correct chaotic behavior of the generated chaotic signal x(t). Precisely, the goal of this test is to find the maximum value of information signal amplitude for which the chaotic system loses its chaotic behavior.

The results of the tests are presented in Tab.1, where yes

means that the chaotic behavior is preserved and no: not preserved. From the table, we note that the hyperchaotic Lorenz system is highly robust to perturbations in the drive signal. This confirms the same conclusion of [2] about the three dimensions Lorenz chaotic system.

B. Proposed Algorithm After several experimental tests (Tab.1), we conclude that

to preserve the hyperchaotic dynamics of the keys generators and to improve the security level, in the general case, we must realize the AHM presented in Fig. 2, where N is the size coding and all of the signal samples are coded with (nQp) fixed-point data format. The parameters n (bits of high weight) and p (bits of low weight) are the number of bits of the decimal and fraction parts respectively.

From Fig. 2, we formulate the new AHM algorithm as

follows:

1) Determine experimentally the maximum value of Amax for which the key generator do not loses its hyperchaos dynamic (Tab.1), such as A is coded as follows

[ ]1,0 ,221

1

0

∈+= −

=

−

=∑∑ RRRA q

p

q

jn

j

(3)

2) From Amax, determine the upper value of the index j for which R=1. Then, calculate the maximum possible number of bits (Nm=j+1+p) to encode the information samples mi.

3) Encode the information samples mi and those of the hyperchaotic signal yi with Nm bits wordlenght.

4) Extend the encoded samples mi and yi to N bits by concatenating the remaining (N - Nm) high weight bits to zero (Fig.2).

5) Encrypt the information signal with one of the fourth hyperchaotic signals by using the XOR function as follows:

iii ymv ⊗= . (4)

6) Realize the additive hyperchaos masking with another hyperchaotic signal :

iii vxs += . (5)

Then, the condition for preserving the hyperchaotic behavior of the ciphering keys is resumed to determine the number of bits Nm needed to encode the masked information without perturbing the hyperchaotic dynamic. For example, from Tab. 1, we note that we can secure data information encoded in 23 bits for Lorenz, in 17 bits for Genesio-Tesi and Linz-Sprott and in 15 bits for Jerk.

Fig. 2. The new additive hyperchaos masking algorithm principle.

TABLE I. CHAOTIC DYNAMIC PRESERVATION TESTS

A Keys generators

Genesio-Tesi Jerk Linz-

Sprott Duffing Hyperchaotic Lorenz

(00002FFF) yes yes yes yes yes (00005FFF) yes yes yes no yes (00007FFF) yes yes yes no yes (0000EFFF) yes yes yes no yes (0000FFFF) yes no yes no yes (0001FFFF) yes no yes no yes (0002FFFF) no no no no yes (0003FFFF) no no no no yes (0004FFFF) no no no no yes

… … … … … … (006FFFFF) … ... … … yes (007FFFFF) no no no no no

Nm 17 15 17 14 23

Fig. 1. ACM principle.

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III. DIGITAL HYPERCHAOTIC COMMUNIU

In order to validate experimentaly thadditive hyperchaos masking, we introduce a realized digital hyperchaotic communicatthe hyperchaotic Lorenz system as ciphering

A. Hyperchaotic Lorenz system model The hyperchaotic Lorenz system is d

following four-dimensional (4D) nonliequations system [13]:

( )( )

⎪⎪⎩

⎪⎪⎨

⎧

−=

−=−−=

−=

dxczxyz

yzbxyxyax

ω

The system exhibits hyperchaotic bparameter values a = 10, b = 28, c = 8/3 andthe initial conditions x0 =y0=z0=w0=-10. The result, using the RK-4 numerical resolutionhyperchaotic Lorenz signals x, y, z and w is g

B. DHCS scheme The realized DHCS is presented in Fig

increase the security level compared to the pscheme consists on three points. Firstly, systems as ciphering keys generators, whicomplex dynamics than 3D chaotic systems. information signal m with the hyperchaotic the XOR function before the additive maskfact, if an intruder apply the breaking techaccess to the encrypted signal v but not to tKnowing that, the bitwise XOR cipher is moris random and is as long as the message (soFinally, ensure the hyperchaotic behavior p

Fig. 3. Matlab Simulation results: x, y, z and w hyperc

UCATION SYSTEM

he proposed new our algorithme in

tion system using keys generator.

described by the inear differential

(6)

+ wy

behavior for the d d = -5, and with Matlab simulation

n technique, of the given in Fig. 3.

g. 4. Our idea to previous CS-DFM use hyperchaotic

ich generate more Secondly, mix the signal y by using

king operation. In hnique [5], he will the information m. re secure if the key o it never repeats). preservation of the

encryption keys whatever the simposing condition in the encothe resulting transmitted signal

xs =

where, the symbol ⊗ represent

At the receiver, since the will be identical to x and y noise, the information signal received signal by using

[sm =ˆ

Note that, the proof of the kind of chaotic communicationin [6]. We have followed texperimentally hyperchaotic syproposed DHCS.

IV. REAL-TIME IMPLEMEN

The realized DHCS presenimplemented on two seprated FPGA circuit [12].

A. Experimental framework To test and validate the propo

the experimental framework transmitter and the receiver aDevelopment System [12].

Fig. 5. Photo of the

Fig. 4. The impleme

chaotic Lorenz signals.

signal information amplitude by oded information (Sec. II). Thus, s is

[ ]my ⊗+ (7)

ts the XOR function.

reconstructed signals xr and yr respectively in the absence of m can be decoded from the

] ryxs r ⊗− (8)

chaotic synchronization in this n scheme (CS-DFM) is detailed the same manner for proving ynchronization (HS-DFM) in the

NTATION OF FPGA TECHNOLOGY nted by the scheme of Fig. 4 is Xilinx Virtex-II Pro XC2VP30

osed approach, we have realized depicted in Fig. 5. Both the

are mounted on a Virtex-II Pro

e experimental device.

ented DHCS scheme.

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The experimental transmission test consists of transmitting sinusoidal information encoded and masked (secured) according to the proposed algorithm principle (Fig. 2). At the receiver, we verify, after the demodulation operation (unmasking), the correct recovery of the transmitted sinusoidal information on a digital oscilloscope.

B. Real-Time Results The different real-time results of the DHCS

implementation for N=32 bits are shown in Fig. 6 and Fig. 7, where Fig. 6 presents the signals at the transmitter. We have considered a sinusoidal signal as information (Nm=23 bits) (Fig.6-a). After the additive hyperchaos masking, we note that the transmitted signal s(t) and the encrypted signal v(t) have a hyperchaotic form (Fig. 6-b). Figs. (7-a) and (7-b) present the results of the HS-DFM at the receiver of the hyperchaotic signals x(t) and y(t) and Fig. (7-c) shows the correct recovery of the information signal m(t). These experimental results validate our proposed solution to insure a robust additive hyperchaotic masking in a DHCS.

V. CONCLUSION In this paper, we have proposed new robust additive

hyperchaos masking algorithm for secure digital communications. We have tested and validated the proposed algorithm by implementing a digital hyperchaotic communication system in FPGA technology. Experimental conditions are determined to ensure the hyperchaotic dynamics preservation, the hyperchaotic synchronization and the robust securing. The obtained real-time results under these conditions confirm the effectiveness of our solutions. The proposed new additive hyperchaos masking is suitable for the wired digital communications.

REFERENCES [1] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,”

Phys. Rev. Lett., 64(8):821-824, 1990.

[2] K.M. Cuomo, A.V. Oppenheim, and S.H. Strogatz, “Synchronization of

Lorenz-based chaotic circuits with applications to communications,” IEEE Trans. Cir. and Syst.-II, vol. 40, pp. 626-633, 1993.

[3] K.M. L. Kocarev, K.S. Halle, K. Eckert and L.O. Chua, “Experimental Demonstration of Secure Communications via Chaotic Synchronization,” Int. J. Bif. Chaos, vol. 2, pp. 709-713, 1992.

[4] T. Yang, “A survey of chaotic secure communication systems,” Int. J. Computational Cognition, vol. 2, pp. 81-130, 2004.

[5] G. Alvares, F. Montoya, M. Romera and G. Pastor, “Breaking two secure communication systems based on chaotic masking,” IEEE Trans. Cir. Sys.: Express Briefs, vol. 51, pp. 505-506, 2004.

[6] V. Milanovic and M. E. Zaghloul, “Improved Masking Algorithm for Chaotic Communications Systems,” Elec. Lett., vol 32, pp.11-12, 1996.

[7] C. C. Chen and K. Yao, "Stochastic-calculus-based numerical evaluation and performance analysis of chaotic communication systems," IEEE Trans. on Cir. and Sys. I, vol. 47, pp. 1663-1672, 2000.

[8] S. Sadoudi, C. Tanougast and M. S. Azzaz, “First experimental solution for channel noise sensibility in digital chaotic communications,” Progress In Electromagnetics Research C, Vol. 32, 181-196, 2012.

[9] S. Sadoudi, C. Tanougast, M.S. Azzaz, A. Dandache and A. Bouridane, “Real-time FPGA implementation of Lü chaotic generator for cipher embedded systems,” in Proc. IEEE Int. Symp. Signals, Circ. Syst., pp. 234–238, 2009.

[10] L.S. Indrusiak, E.C. Dutra e Silva Junior, & M. Glesner, “Advantages of the Linz-Sprott weak nonlinearity on the FPGA implementation of chaotic systems: a comparative analysis,” Proc. Int. Symp. Signals, Circuits and Sys. 2, 753 – 756. 2005.

[11] G.Y. Wang, X.L. Bao, and Z.L. Wang, “Design and FPGA Implementation of a new hyperchaotic system,” Chinese Phy. B 17, No. 10, 2006.

[12] Xilinx, “Xilinx University Program Virtex-II Pro Development System,” Xilinx, UG069 (v1.1) April 9, 2008

[13] R. Barboza, “Dynamics of a hyperchaotic Lorenz system,” Int. J. of Bifurcation and Chaos, vol. 17, no. 12, pp. 4285–4294, 2007.

(a)

(b)

(c)

Fig. 7. At the receiver: (a) hyperchaotic signals x(t), xr(t) and theirsynchronization, (b) hyperchaotic signals y(t) and yr(t) and theirsynchronization and (c) transmitted and recovered information signal.

(a) (b)

(c)

Fig. 6. At the transmitter: (a) information signal m(t) and hyperchaotic signaly(t) and (b) transmitted signal s(t) and (c) the encrypted signal v(t).

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