A Novel Chaotic System with a Line Equilibrium: Analysis and Its Applications to Secure Communication and Random Bit Generation ^†

Abstract

In this study, a novel two-parameter, three-dimensional chaotic system is constructed. The system has no linear terms and its equilibrium is a line, so it is a system with hidden attractors. The system is first studied by computation of its bifurcation diagrams and diagram of Lyapunov exponents. Then, the system is applied to two encryption related problems. First, the problem of secure communications is considered, using the symmetric chaos shift keying modulation method. Here, the states of the chaotic system are combined with a binary information signal in order to mask it, safely transmit it through a communication channel, and successfully reconstruct the information at the receiver end. In the second problem, the states of the system are utilized to design a simple rule to generate a bit sequence that possesses random properties, and is thus suitable for encryption related applications. For both applications, simulations are performed through Matlab to verify the soundness of the designs.

1. Introduction

Since its first emergence in the 1960s [1], chaos theory has emerged in multiple scientific fields, ranging from physics to biology, medicine, engineering, communications and economics. Chaotic systems are deterministic dynamical systems, modelled by a set of differential or difference equations. Their unique feature is that their solution can be extremely prone to changes in the initial conditions of the system, or to parameter changes. This means that solution trajectories starting from almost identical initial key values will diverge over time, giving rise to completely different behaviors.

The above-mentioned feature makes chaotic systems hard to predict. This property, combined with their deterministic nature, makes chaotic systems suitable in security applications that require the use of complex dynamics, like encryption [2,3,4], communications [5,6,7,8,9], random number generators [10] and more [11].

Thus, there is an ongoing interest in creating novel chaotic systems, since their demand in applications is continuous. To design a chaotic system, researchers developed several different approaches. One way is to consider an existing chaotic system, and modify it by adding more terms to the differential/difference equations describing the system, or modifying an existing term, or even adding a new state to the system and changing its order [12,13,14]. In such cases, the new system should preferably have a more complex behavior compared to the original one, which is usually indicated by the highest value that the Lyapunov exponent can achieve.

Another approach, which is the method used here, is to design a new system by utilizing nonlinear functions that are documented to yield chaotic behavior, like the hyperbolic sine function, the exponential function, the cubic power, or the absolute value; see, for example [15,16,17,18,19,20] and the references found therein.

Interestingly, when considering chaotic systems, it is preferable if the resulting system has hidden attractors [21]. An attractor is called hidden if its basin of attraction does not intersect with any open neighbourhood of an equilibrium. Systems with hidden attractors have gained attention due to their complex behavior, which makes them suitable for applications. Examples of recent works on such systems are [7,18,21,22,23,24,25,26].

Based on the above, this work considers a chaotic system with a hidden attractor, having a line equilibrium. The system has two parameters, and is purely nonlinear, having no linear terms. A dynamical analysis on the system is performed by studying its bifurcation diagrams, the Poincaré map, and the Lyapunov exponents diagram.

Moreover, to showcase the applicability of the system to chaos based encryption applications, the system is applied to two problems. First, the problem of secure communications is considered. Here, the aim is to find a method to combine an information signal, with the values of the states of the chaotic system, resulting in a scrambled signal that carries the masked information in it. Then, this signal can be safely transmitted through a communication channel, and after appropriate signal processing, the original information can be unmasked and safely retrieved at the receiver end [6,27]. The method used here is the symmetric chaos shift keying (SCSK) modulation, applied in [7,28]. The present work extends the work of [29] by including an extensive bit-error-rate (BER) performance analysis of the proposed communication scheme. The simulation results show that a combination of the states of the chaotic system with a sine function can yield better performance under a noisy transmission channel.

In addition to the above application, the design of a pseudo-random bit generator (PRBG) is considered. In this application, the states of the chaotic system are used to generate a sequence of bits, that possesses properties similar to those of a random sequence. Such bit generators are highly useful, since they can be used to encrypt binary information signals. The proposed PRBG generates three bits per iteration, and the generated sequence passes all statistical tests of the NIST test suite.

The rest of the paper is structured as follows: In Section 2, the proposed chaotic system is presented and studied. In Section 3, the problem of secure communications is considered, followed by the problem of random bit generation in Section 4. Section 5 concludes the paper with a discussion on future topics of interest.

2. The Proposed Chaotic System

In this section, the 3-D chaotic system is proposed and studied. The system is given by the following differential equations:

$$\left\{\begin{array}{c}\dot{x}=yz\hfill \\ \dot{y}=x^3-y^3\hfill \\ \dot{z}=a\left|x\right|-b{y}^3-xy\hfill \end{array}\right.$$

(1)

where x, y, z are the state variables and a, b are positive parameters. System (1) is chaotic for a wide range of parameter values. For example, when parameters a and b take the values $a=0.65$, $b=0.1$, the chaotic behavior of the system is presented in the phase portraits of Figure 1 and Figure 2. Moreover, the state trajectories are shown in Figure 3, and in Figure 4—the phase portraits of each state with respect to its derivative are depicted.

Moreover, although the system (1) is structurally simple, it is purely nonlinear, having no linear terms, and two independent parameters $(a,b)$. In addition, the equilibria of system (1) lie in the line $(0,0,z)$. So, it belongs to the recently discovered dynamical systems with hidden attractors [7,21,23,30,31].

For a detailed analysis of system’s (1) dynamical behavior, the system is investigated numerically by using the fourth order Runge–Kutta algorithm. The bifurcation diagram is first computed. This diagram depicts the points of intersection where the trajectory cuts the plane $y=0$ with $dy/dt<0$. In this way, the bifurcation diagram of variable x versus the parameter a, for $b=0.1$ reveals the richness of system’s dynamical behavior (Figure 5). It can be seen that the system enters chaos following a period doubling route, as the parameter a increases. The system remains chaotic for a wide range of parameter values, which is interrupted by small periodic windows. As an additional simulation, the bifurcation diagram of variable z versus the parameter a, for $b=0.1$ is shown in Figure 6, from which the same dynamical behavior can be observed. Also, Figure 7 depicts the Poincaré map of the system, as the trajectory cuts the plane $y=0$ with $dy/dt<0$, for $a=0.65$, $b=0.1$. The subfigure also depicts clearly the plane $y=0$ that cuts the attractor, with respect to which the Poincaré map is taken.

Figure 8 depicts the diagram of the two largest Lyapunov exponents with respect to bifurcation parameter a, which are computed using the well-known Wolf et al. [32] algorithm. The third exponent is discarded from the graph, since its values are negative and too low to depict. As can be seen, when the system is periodic, the maximal Lyapunov exponent (MLE) is equal to zero, while when it is chaotic, the system has a positive MLE, as it is expected according to the theory. So, from Figure 8, the system’s (1) chaotic behavior is found for a wide range of parameter values, confirming the bifurcation diagram of Figure 5. Moreover, as parameter a increases, the value of the MLE also increases, and the behaviour of the system becomes more complex.

3. Application to Secure Communications

3.1. Symmetric Chaos Shift Keying Modulation

In this section, the proposed chaotic system is applied to the problem of secure communications. The methodology considered is the symmetric chaos shift keying (SCSK) modulation, which was applied in [7,28]; see also [27,33,34,35,36,37,38,39] for similar approaches. The system design is shown in Figure 9. The system is comprised of two units, a transmitter and a receiver, connected through a noisy communication channel.

In the transmitter unit, the information signal is multiplied (modulated) with a chaotic signal generated by the chaotic system, resulting in a masked signal. The generated masked signal is then transmitted to the receiver. In the simulations, the information signal is taken as a binary signal with a period of 1 s. For the chaotic signal used, different options were considered, as will be seen in the following subsection, generated from (1), with $a=0.65$, $b=0.1$. During transmission, it is also assumed that the channel is disturbed by an additive white Gaussian noise (AWGN). In the receiver end, the received masked signal is multiplied by the chaotic signal generated by the same chaotic system as in the receiver, and the resulting signal is integrated in the correlator. This way, the bit energy of the signal is calculated and the binary information signal is reconstructed by passing the resulting signal through a threshold detector, with threshold set to 0. So, this threshold detector generates a ”1” if its input is positive, and a ”−1” if its input is negative.

Note that in this design, the same chaotic signal is used in the transmitter and receiver ends; hence, the term symmetric is used in its description. So in the receiver end, in order to replicate the chaotic signal, knowledge of the systems initial conditions $x_0,y_0,z_o$ and parameter values $a,b$ is required. These five parameters are called the key space of the design. So assuming a 16-digit accuracy, an upper bound of the key space is ${10}^{5·16}={10}^{80}={\left({10}^3\right)}^{26.6}\approx {\left(2^{10}\right)}^{26.6}=2^{266}$. This is higher than the upper bound of $2^{100}$ that is generally required to resist brute force attacks [40].

The SCSK communication system was studied under the presence of AWGN in the communications channel. The retrieved signal is delayed by $0.5$ s, due to the sample/hold block on the receiving unit. Therefore, there is a $0.5$ s delay between the transmitted information signal and the retrieved information signal on the receiving unit.

3.2. Bit Error Rate Performance

To study the bit-error-rate (BER) performance of the system under different noise intensities, we considered several cases of chaotic signals S used in the transmission. The signals considered were $S=100·x$, $S=100·sin(x+y+z)$, $S=100·sin3(x+y+z)$, $S=100·sin5(x+y+z)$. For each case, the simulation results for 100 s are shown in Figure 10, Figure 11, Figure 12 and Figure 13 for $Eb/No=-10$ dB. From the simulations, it can be seen that the combination of all the states with a sine function yields a more complex signal, especially when the gain term inside the sine function increases. In addition, it is seen that, in all cases, the information signal is successfully retrieved at the receiver end.

The BER performance results are shown in Figure 14. The noise intensity is considered in the range of −10 dB to 3 dB, and the simulation time is 40,000 s. It is verified that more complex chaotic signals yield a consistently better BER performance compared to simply using a state of the chaotic system.

4. Application to Random Bit Generation

As an additional application of the proposed system, we will consider the design of a random bit generator (PRBG). The problem consists of designing a simple rule to generate a bit sequence, using the values of the states of a chaotic system [10,41,42,43,44,45,46,47]. The resulting bit sequence, also called bitstream, should have the statistical properties of a random sequence. This can be tested through a series of statistical tests, provided by the National Institute of Standards and Technology (NIST) [48]. The test suite consists of 15 statistical tests. Each test returns a p-value, and if it is higher than a chosen significance level, the test is considered successful and the bitstream is considered random. The significance level here is taken as the default $0.01$.

The PRBG generates bits using the following simple rule. The system (1) is simulated using ode45 in Matlab, with a simulation step of $h=0.01$. Then, in each discrete time instance i, the following rule is applied to generate the bitstream:

$$\begin{array}{ccc}\hfill Z_i& =& [{10}^7·x_i,{10}^8·y_i,{10}^9·z_i]\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill B_i& =& \lfloor mod(Z_i,2)\rfloor \hfill \end{array}$$

(3)

where $x_i,y_i,z_i$ are the values of the states at the discrete time instance i, and $\lfloor ·\rfloor$ denotes the integer part of the argument. The coefficients ${10}^7,{10}^8,{10}^9$ were chosen after some trial and error, until the obtained sequence was indeed random, as will be seen next. The complete bit sequence $\mathcal{B}$ is obtained as

$$\mathcal{B}=[B_0,B_1,B_2,\dots ]$$

(4)

In addition, to reduce cross-correlation in the generated bitstreams, the first 200 s of the system simulation are discarded. It is clear that with the above rule, 3 bits can be generated in each discrete time. So, for example, given the time step $h=0.01$, to generate $3·{10}^6$ bits, the system (1) needs to be simulated for ${10}^4+200$ s. Similar approaches were used in [42,46,49], using different operators to generate the bits.

To test the algorithm, a sequence of $50·{10}^6$ bits is generated and tested. The sequence was generated using parameter values $a=0.65$, $b=0.1$ and initial conditions $(0,0.1,0.1)$. The NIST test results are shown in

Table 1. NIST statistical test results, if $p\ge 0.01$, the test is successful.

If $\mathit{p}\ge \mathit{\alpha }$, the Test Is Successful
No.	Statistical Test	p-Value	Proportion	Result
1	Frequency	0.023545	50/50	Success
2	Block Frequency	0.191687	49/50	Success
3	Cumulative Sums	0.935716	49/50	Success
4	Runs	0.171867	49/50	Success
5	Longest Run	0.350485	49/50	Success
6	Rank	0.935716	49/50	Success
7	FFT	0.779188	50/50	Success
8	Non-Overlapping Template	0.319084	50/50	Success
9	Overlapping Template	0.137282	48/50	Success
10	Universal	0.191687	50/50	Success
11	Approximate Entropy	0.085587	50/50	Success
12	Random Excursions	0.010606	29/29	Success
13	Random Excursions Variant	0.186566	29/29	Success
14	Serial	0.574903	50/50	Success
15	Linear Complexity	0.262249	50/50	Success

. It can be seen that all tests are successful, so the resulting sequence is indeed random.

In addition, Figure 15 depicts the autocorrelation and cross-correlation for a bit sequence of length ${10}^4$. For a random bit sequence, the auto-correlation should have a delta form, while the cross-correlation should be close to zero [41]. This is indeed verified from the diagrams. For the cross-correlation, the initial conditions for the two bitstreams were chosen as $(0,0.1,0.1)$ and $(0,0.1,0.1+{10}^{-16})$.

Finally,

Table 2. Key space of different techniques.

This Work	$2^{266}$
[46]	$2^{70}$
[44]	$2^{160}$
[42]	$2^{259}$
[50]	$2^{279}$
[51]	$2^{425}$
[41]	$2^{448}$

shows a comparison between the key space of different generators.

5. Conclusions

In this work, a 3D chaotic system with a line equilibrium was proposed. The system was studied through the computation of its bifurcation diagrams and Lyapunov exponents diagram. To study the applicability of the system to encryption and security related applications, the system was applied to a secure communications scheme using the SCSK modulation method, yielding satisfactory BER performance. Moreover, the problem of random bit generation was considered. The proposed design can generate three bits per iteration, with statistical randomness for the generated sequence.

It is worth noting here that as far as the numerical evaluation of the chaotic system in the transmitter and receiver ends is concerned, there are instances where errors can arise, that may lead to degradation of the design. Examples can include round-off errors that arise when a different error tolerance is chosen between transmitter and receiver, or errors that arise when a different integration step is chosen or different numerical methods are used to solve the system; see for example, discussions in the recent works [52,53,54]. This is one of the main obstacles that have so far held up the possible commercialization of chaos-based secure communications.

Here, since no implementation of the design is performed, we assumed, as many works do, that the receiver and transmitter systems are simulated in devices with the same configuration settings. Yet, it is within our scope of research interest to physically implement such a design, to see when and how such erros arise and how thay can be tackled. Thus, future aspects of this work would be the experimental implementation of the communications design [55], and the further application of the proposed PRBG to image encryption.