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In many domains of science and technology, as the need for secured transmission of information has grown over the years, a variety of methods have been studied and devised to achieve this goal. In this paper, we present an information securing method using chaos encryption. Our proposal uses only one chaotic oscillator both for signal encryption and decryption for avoiding the delicate synchronisation step. We carried out numerical and electronic simulations of the proposed circuit using electrocardiographic signals as input. Results obtained from both simulations were compared and exhibited a good agreement proving the suitability of our system for signal encryption and decryption.

During the last decades, the demand for cryptographic techniques to secure transmitted information has increased. For multiple reasons, the need to protect information arises. A number of attempts have been carried-out in research. Basically, the literature proposes two main approaches of information encryption. The first is done through algorithms of encoding and decoding implemented in software. Several techniques were used to encrypt data streams, but interception was possible in case of the encryption key hacking. Authors therefore massively turned to the circuit-based approach. A particular class that has received much attention in research these last years is chaos-based techniques. The idea of using chaotic signals to transmit secured information appeared at the beginning of 90’s after it had been proved by Peccora and Carroll that the chaotic system can be synchronized [1-3]. The robustness in multipath environments, resistance to jamming, and low probability of interception are essential when dealing in communication systems. Properties like sensitivity to initial conditions, random-like behaviour, nonlinear dynamics found in chaotic oscillators are an advantage to fight against interception. The principle of these techniques is to use oscillators that generate chaos to modulate the information signal that has to be transmitted. After reception, the signal is demodulated and the secret information recovered.

Some authors concentrated on designing different oscillators for chaos generation. For example, chaotic behaviours of Duffing, Chua, Colpitts and Van der Pol oscillators [4-9], just to name a few, have been studied.

A second group of authors was preoccupied by synchronization of chaotic oscillators involved in the emission and reception parts, using a variety of techniques [1-4,10-16]. Some of these had secured communications as one of the applications.

Mindful of the fact that synchronisation is very sensitive to noise, some authors have tried a number of techniques excluding any need for synchronisation. The first of this type is chaos shift keying (CSK) [17,18]. CSK is a method of digital modulation. Depending on the current value of the N-ary message symbol, the signal x_{i}(t) (i = 1, ∙∙∙N) from one of N chaos generators with different characteristics is transmitted. The main drawback of the CSK is that the threshold level required by the decision circuit depends on the signal to noise ratio (SNR). A special case of CSK is the chaotic on-off keying (COOK) [

However, the threshold value can be kept constant and the distance can be doubled by applying the differential CSK (DCSK) [20,21]. In DCSK, the two channels are formed by time division. For every message symbol, the reference signal is first transmitted, followed by the modulated reference carrying the message symbol. The principal drawback of DCSK arises from the fact that every information bit is transmitted by two sample functions because the bit rate is halved.

This problem can be avoided using frequency modulation DCSK (FM-DCSK) [22,23], where the transmitted energy per bit belonging to one symbol, is kept constant. Here, as in the DCSK technique, every information bit is transmitted by two sample functions, where the first part serves as a reference, while the second part carries the information. The operation of the modulator is the same as in DCSK, the only difference is that not the chaotic, but the FM modulated signal is the input of the DCSK modulator. The limitation of standard FM-DCSK system is the fact that only one information-bearing is transmitted after the reference signal.

Several different methods have been proposed in the literature to increase the data rate of DCSK, of which one of the most efficient is the quadratic chaos shift keying (QCSK) [24,25] scheme. The basic idea underlying the QCSK scheme is the generation of chaotic signals which are orthogonal over a specified time interval. This allows the creation of a basis of chaotic functions from which arbitrary constellations of chaotic signals can be constructed. For instance, in QCSK, a linear combination of two chaotic basis functions is used to encode four symbols. The key point for exploiting this idea in a communication system is that one must be able to generate the chaotic basis functions starting from a single chaotic signal. The same concept holds for conventional digital communication schemes such as QPSK, where the quadrature component can be obtained from the phase one by means of a simple phase shifter. The main drawback of this method is its high complexity.

Among several systems proposed, one of the best performances has been achieved by the differential chaos shift keying (DCSK) scheme and its variation utilizing frequency modulation, which is FM-DCSK. This is the reason why, our method draws its inspiration from this technique.

In this paper, we suggest a very simple encryptionand-decryption system organized around a multiplexer and demultiplexer and based on the DCSK philosophy. Apart from its great simplicity, our system provides, as will be seen later a good signal to noise ratio. Finally, unlike in many of the aforementioned systems where the decrypted signal is obtained by estimation, in our proposal the final signal is actually deducted from the sent signal.

In the next section, we describe and model the circuits used in our system. This is followed in Section 3 by results obtained during our simulations. These results are discussed in Section 3. The conclusion of the paper is the object of Section 4.

The general diagram of the secured transmission system is given in

We shall now briefly describe the different elements of the system and their functioning.

There are basically two classes of signal encryption with chaos. In the first class of systems, which is generally more complicated, the information to be hidden is injected in the system producing the chaotic signal. This approach has the disadvantage of imposing the modification of the decoding system and is more suitable when the signal to be coded is of high amplitude. The second class of systems would allow generation of the chaos which is then added to or multiply by the signal to be hidden. This technique, which is simpler in its design is appropriate for low amplitude signals like ECG and would not need the modification of the receiving system.

The central element of the encrypting bloc is the chaos generator which, in our case, is a colpitts oscillator. Let’s describe the model of the chaotic oscillator used.

The Colpitts oscillator is one of the most researched and easiest oscillators. _{1} and C_{2}. The non linear component of the circuit is the Bipolar Junction Transistor (BJT) Q2N2222.

The current source, of conductance polarizes the BJT. Applying Barkhausen criterium to this oscillator, the resonance frequency can be computed

Applying Kirchhoff current and voltage laws to the circuit, we have:

where and are the BJT parameters:,

Let’s introduce some dimensionless variables for convenient numerical analysis:

If serves as control variable for the system, and posing the set of Equations (2) becomes:

where the dot denotes the differentiation.

Changing origins, (4) becomes:

In order to study stability around the equilibrium point, let’s rewrite the system above using the formalism where X is a three-dimensional vector and F, a function of X and of time. Performing a first order development in equilibrium point’s neighbourhood by means of the Jacobian matrix M_{F}, we can have an approximation of the system dynamics when subjected to a perturbation where

with (6)

Therefore, (7)

The characteristic equation of the Jacobian matrix about the equilibrium is:

given that (8)

In the triplet of eigenvaluesthe first two lead to oscillations while the last pulls the system towards equilibrium. The bifurcation diagram is given in

(Equation (3)) as a function of μ. It can be noticed that for:

- the system tends to stabilize around a single frequency value;

- the system oscillates between two frequency values;

- from many bifurcations points appear and the system exhibits a chaotic behaviour.

A usual test for chaos is calculation of Lyapunov exponents. It is common to refer to the largest one as the Maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. The Lyapunov exponents give the average exponential rates of divergence or convergence of nearby orbits in the phasespace.

In systems exhibiting exponential orbital divergence, small differences in initial conditions which we may not be able to resolve get magnified rapidly leading to loss of predictability. Such systems are chaotic. In ^{th} order Runge Kutta technique, with Step 0.01, three values of Lyapunov exponents (Lamda 1, Lamda 2, Lamda 3) are obtained: (positive value), Lamda 2 = 0.0, Lamda 3 = −0.70 (negative value). These results validate the bifurcation diagram of

The chaotic signal studied above is used to encrypt the ECG signal. Encryption and decryption techniques which are described below are depicted in

The informative signal (ECG) is added to the chaotic signal. The output of the adder, which is the encrypted signal, is multiplexed with the same chaotic signal and yields Signal C. The latter is transmitted to the receiver through a physical link (copper wire in our case). At the receiver end, decryption is obtained by first of all demultiplexing Signal C and then subtracting the two output of the demultiplexer. A low pass filter is added at the output of the subtractor to discard noise from the decrypted signal.

The different functional units of our system are indicated in

The overall circuit was built according to the scheme shown in

C_{1} and C_{2} are the two capacitors found on the colpitts oscillator (_{1} and C_{2} were plotted as a function of time. One can easily notice that both from Matlab (a, c) and Multisim (b, d) simulations, the waveforms obtained are chaotic.

Using the value the system (5) was solved numerically using Matlab by means of fourth-order Runge-Kutta algorithm and yielded the phase portrait of

across C_{1} as a function of the voltage across C_{2} under Multisim, the phase portrait of