A New Hyperchaotic Map for a Secure Communication Scheme with an Experimental Realization

: Using different chaotic systems in secure communication, nonlinear control, and many other applications has revealed that these systems have several drawbacks in different aspects. This can cause unfavorable effects to chaos-based applications. Therefore, presenting a chaotic map with complex behaviors is considered important. In this paper, we introduce a new 2D chaotic map, namely, the 2D inﬁnite-collapse-Sine model (2D-ICSM). Various metrics including Lyapunov exponents and bifurcation diagrams are used to demonstrate the complex dynamics and robust hyperchaotic behavior of the 2D-ICSM. Furthermore, the cross-correlation coefﬁcient, phase space diagram, and Sample Entropy algorithm prove that the 2D-ICSM has a high sensitivity to initial values and parameters, extreme complexity performance, and a much larger hyperchaotic range than existing maps. To empirically verify the efﬁciency and simplicity of the 2D-ICSM in practical applications, we propose a symmetric secure communication system using the 2D-ICSM. Experimental results are presented to demonstrate the validity of the proposed system.


Introduction
Numerous phenomena have been comprehended by studying the complex behaviors in many natural and non-natural dynamical systems. Understanding the chaotic behavior, which is a kind of nonlinear complex dynamical behavior, has provided a significant description of these systems. Although dynamical systems with chaotic behaviors are deterministic, long-term prediction of their behaviors is impossible [1]. Moreover, the sensitivity, topological mixing, and orbits density are the main characteristics of the chaotic systems [2][3][4]. Therefore, chaotic systems have valuable applications in various fields including computer science, telecommunication, physics, engineering, etc. [5][6][7][8][9][10]. In particular, due to the similarity between the characteristics of chaotic systems and the diffusion and confusion properties of cryptography [11], a wide body of chaos-based cryptographic applications has been presented in the last few years [12][13][14][15][16][17].
For the time being, discrete-time systems and continuous-time systems are the major types of chaotic systems. The former type is described by a difference equation, and it can be implemented 1. We introduce an analytical framework to understand the dynamical behavior of the 2D-ICSM including stability of its fixed points, bifurcation diagram, and Lyapunov Exponents. 2. We experimentally evaluate the complexity, sensitivity, and randomness of the 2D-ICSM using Sample Entropy, cross-correlation coefficient, and NIST-800-22 statistical test, respectively. 3. To demonstrate the efficiency and simplicity of the 2D-ICSM in practical applications, we design a secure communication system, and then experimented tested it on an optical channel with Arduino microcontrollers.
This paper is organized as follows. Section 2 introduces the 2D-ICSM and studies the stability of its equilibria. In Section 3, we analyze the dynamics of the 2D-ICSM. Section 4 demonstrates the high sensitivity and randomness of the 2D-ICSM. Section 5 demonstrates the detailed complexity performance of the 2D-ICSM. In Section 6, we introduce the proposed secure communication system. Section 7 illustrates the implementation of the communication system. Conclusions are presented in Section 8.

The 2D Infinite-Collapse-Sine Model
In this section, we introduce a new 2D chaotic map, called the 2D infinite-collapse-Sine model (2D-ICSM), and then discusses its stability analysis.

Definition of 2D-ICSM
Among existing 1D discrete-time dynamical systems, the infinite collapse model is considered as one of the best maps that show robust chaotic performance [38]. Mathematically, its dynamical equation is given by where β is the control parameter and x is the state variable. The dynamical behavior of this map can be illustrated by depicting its bifurcation diagram and trajectory in the phase plane, as shown in Figure 1. It can be seen that the numerical solution of this map is in the range of [−1, 1]. Besides that, the bifurcation diagram of this map shows that the chaotic attractor appears in limited regions of its parameter. Meanwhile, several non-chaotic regions can be observed as the parameter α increasing. Furthermore, Figure 1b demonstrates that its trajectory only occupies a small space in the phase plane. Such behaviors are widely observed in the existing 1D chaotic maps such as Logistic, Tent, and Sine maps. To tackle the aforementioned issues, we propose a new 2D chaotic map, which consists of four terms with two control parameters. Mathematically, it is defined as follows, where α is the amplitude parameter and β is the internal frequency parameter. It can be seen that the proposed 2D infinite-collapse-Sine model (2D-ICSM) is mainly designed by using three components including a linear variable, 1D Sine map, and 1D infinite collapse map. The linear state variable x n is used to modulate the output of a 1D infinite collapse map. Therefore, it can enhance the chaotic behavior of the state variable y n+1 . Meanwhile, the 1D Sine map is employed to boost randomness to the state variable x n+1 .

Stability Analysis
For discrete-time systems, the fixed point of a function form a graphical point of view is an element in the domain that maps to itself by the function. For instance, P is a fixed point of the function F(x) only when F n (P) = P. To simplify the calculation of the fixed points of 2D-ICSM, we reduce its dimension to become 1D as follows, The fixed points of the 2D-ICSM are calculated for two different sets of system parameters. For each set of parameters, we obtain the fixed points of the variable y by Equation (3), and subsequently, the corresponding points of the variable x can be easily obtained by the first equation of the system (2). Figure 2 illustrates how the fixed points of the variable y can be obtained using the graphical method. From this figure, one can notice that the number of fixed points is increased as the values of the amplitude and internal frequency parameters increase. Now, let us collect some fixed points from each set of system parameters to investigate their stability. First, we have extracted the following fixed point from the first set of the system parameters, Second, we have extracted three different fixed points from the second set as follows, (1) , y (1) = (0.9168, 4.1324), (2) , y (2) = (0.7607, 3.5738), The stability of the above-fixed points can be determined by obtaining the Jacobian matrix, which is given by Using the above matrix, the 2D-ICSM is Linearized at any arbitrary fixed point P i = (x * , y * ) as follows, Thus, the eigenvalues are obtained by solving the following equation, It is well-known that the stability of fixed points is dependent on the eigenvalues. When an eigenvalue is within the interval [−1, 1], then the fixed point exhibits a stable state. Otherwise, it shows an unstable state. Moreover, the stability of the obtained fixed points is as illustrated in Table 1. All the selected fixed points of the 2D-ICSM are unstable.

Dynamical Behaviors
This section investigates the dynamical behaviors of 2D-ICSM through the bifurcation diagram, Lyapunov Exponents, and phase space.

Bifurcation Diagram and Lyapunov Exponents
Typically, the bifurcation diagram and Lyapunov Exponents are used to determined the non-chaotic and chaotic regions of a dynamical system when one of its parameters varies. Furthermore, the Lyapunov exponent is used to evaluate the chaotic properties of a dynamical system. In other words, it could recognize the chaotic and hyperchaotic behaviors of the system. A system is recognized as chaotic when there is one positive Lyapunov Exponent value for each parameter value, whereas the hyperchaotic system has more than one positive Lyapunov Exponent value. The hyperchaotic system exhibits a higher level of randomness, and the generated sequences by the hyperchaotic system show extreme unpredictability.
To investigate the dynamics of 2D-ICSM, we depict its bifurcation diagram and Lyapunov Exponents with the initial values (0.5, 0.5) and for the parameters 0 ≤ α ≤ 8 and β = 12, as shown in Figure 3. It can be seen that 2D-ICSM is hyperchaotic among the whole parameter range, which indicates that its sequences are extremely hard to be predicted.

Hyperchaotic Attractor
The set of numerical values, which is generated by a chaotic/hyperchaotic map with specific sets of initial values and control parameters, is called chaotic/hyperchaotic attractor. For a 2D map, its attractor can be described by a group of points that occupies a particular region in the phase space. A chaotic/hyperchaotic model has better performance when its attractor is geometrically complicated and occupies a larger range in the phase space. To illustrate the hyperchaotic range of the 2D-ICSM, Figure 4f depicts its attractor in the 2D phase space with the parameters α = 6 and β = 12. Besides that, this figure plots the attractors of several existing chaotic and hyperchaotic models to demonstrate the complicated behavior of the 2D-ICSM. It can be observed that the hyperchaotic attractor of 2D-ICSM fully occupies a 2D phase space ranging x ∈ [−1, 1] and y ∈ [− 9,9]. This means that 2D-ICSM can generate more unpredictable hyperchaotic sequences and it has a better competitive ergodicity property than existing models.

Performance Evaluations
In this section, the sensitivity of the initial conditions and the control parameters is measured by the cross-correlation coefficient. Furthermore, the quantitative values of the randomness of sequences generated by the 2D-ICSM are determined using NIST-800-22 randomness tests.

Cross-Correlation Coefficient
To estimate the sensitivity of the initial conditions and the control parameters of the 2D-ICSM, we use the cross-correlation coefficient (CCF); its equation is given by where A(α) represents the mean value of the time series α t , meanwhile A(β) represents the mean value of the time series β t . When CCF(α t , β t ) is close to 0, then it can be indicated that these two-time series are diverging. Figure 5 presents the sensitivity of the 2D-ICSM with the parameters α ∈ [0, 8] and β = 12. In this figure, the sensitivity is estimated by calculating the CCF between the original time series and the modified time series. It is important to mention here that the modified time series was generated by the 2D-ICSM using a very small error, e = 5 × 10 −5 , which was added to the initial value x 0 and the parameter α, as shown in Figure 5a,b, respectively. It can be observed that the 2D-ICSM has a high level of sensitivity to its initial values and control parameters.

Chaos-Based Pseudorandom Number Generator
A chaotic map could be a suitable source to generate pseudorandom numbers when it has high sensitivity, good ergodicity, and extreme unpredictability. The existence of these features in a chaotic map can be determined by the NIST-800-22 randomness tests.
It is, therefore, crucial to determine the existence of such features in the 2D-ICSM to examine its ability to be a PRNG. In this regard, we propose a simple strategy, which directly employs the chaotic sequences as pseudorandom numbers by converting each of their values to a 32-bit binary stream using the IEEE 754 float standard. Figure 6 displays the NIST SP800-22 test results of pseudorandom numbers generated by the 2D-ICSM. In this figure, the generated sequence by 2D-ICSM has a length of 100, 000, 000 binary bits. It is important to state here that a chaotic map can pass the statistical tests of NIST-800-22 only when the corresponding p-values are greater than the experimental significance level [40]. Consequently, the results in Figure 6 demonstrates the high randomness of the generated pseudo random numbers by the 2D-ICSM.

Complexity-Based Sample Entropy
In this section, the complexity of 2D-ICSM is investigated through a fundamental algorithm, namely, Sample Entropy (SamEn). The authors of [36] presented SamEn to calculate how much extra information is required to predict the (t + 1)th output of a trajectory using its previous (t) outputs. SamEn with larger values indicate a lower degree of regularity of a chaotic map. In other words, the chaotic map exhibits a high level of complexity and unpredictability.
The SamEn algorithm for a given time series {x(i), i = 0, 1, 2, . . . , N − 1} is outlined as follows: 1. Reconstruction: the time series can be reconstructed as follows, where m is embedding dimension and τ is time delay.
2. Counting the vector pairs: For a given tolerance parameter r, let B i be the number of vectors X j such that here, d[X i , X j ] is the distance between X i and X j , which is defined as 3. Calculating θ m (r): According to the obtained number of vector pairs, we can get then calculate θ m (r) by 4. Calculating SamEn: Repeating the above steps we can get θ m+1 (r), then SamEn is given by Moreover, Figure 7b depicts the SamEn results of the 2D-ICSM and different chaotic and hyperchaotic maps. It is quite clear that the 2D-ICSM has the largest SamEn values, which indicates that one needs more information to predict the generated sequences by this map.  [14], 2D-SIMM [13], and 2D-SLMM [12], respectively.

Chaos Based Cryptography
This section investigates the performance of 2D-ICSM in cryptography applications by designing a symmetric secure communication system. Figure 8 displays the schematic diagram of the proposed symmetric secure communication scheme. As can be observed in this figure, the proposed communication system is designed to transmit a message M(s) between two points in which the 2D-ICSM is employed to encrypt the information.

Arduino Transmitter
In the proposed communication system, the Arduino is considered as the core of transmission. Here, we employ the Arduino Uno R3 microcontroller boards, which are simple and implemented at a low cost. It has 14 digital input/output pins (6 of them can be used as PWM outputs), 6 analog inputs, a 16 MHz crystal oscillator, a USB connection, and a reset button [41].
First, we convert the signal from the bipolar form to the unipolar form, and this is due to the fact that the Arduino analog inputs only accept unipolar signals in the range from 0 V to 5 V. The communication starts with a message M(s), which is sent to the analog input A0 of the Arduino transmitter, as shown in Figure 9. Second, the input signal is converted from analog to digital using an embedded 8-bit ADC at a maximum rate of 8000 samples per second. Finally, this signal is encrypted by the 2D-ICSM and Delta modulator.

Delta Modulation
The Delta modulation is a simple and robust A / D conversion method [42]. It has a comparator in the forward path and an integrator in the feedback path of a simple control loop, as shown in Figure 10. The signal M(k) is the input of the comparator. Meanwhile, U(n) is the integrated output, which has a binary form. The value of Delta modulation depends on the current sample, if it is less than the previous sample, then zero is transmitted as a signal, whereas if the current sample is greater than the previous one, then number one is sent as a transmitted signal. Algorithm 1 illustrates the pseudocode of the Delta modulation process.

Encryption Process
The encryption process begins after obtaining the signal M b (k) from the Delta modulator. First, the secret key of the proposed encryption scheme is mainly generated from the initial values and a control parameter of the 2D-ICSM, as illustrated in Figure 11. As can be seen from this figure, the secrete key consists of 5 parts with 232 bits in which (x 0 , y 0 ) and α are the initial values control parameter of the 2D-ICSM. Meanwhile, s and z are added to increase the security by increasing the key space and changing the initial values and control parameters. It is crucial to state here that the proposed key is symmetric, which means that it can be used for both encryption and decryption processes. Now, the obtained signal M b (k) can be encrypted using the hyperchaotic sequence generated by the 2D-ICSM. Algorithm 2 illustrates the pseudo-code of generating the secrete key and the encryption process.

Algorithm 2:
The encryption process.

Input:
The secret key of size 232 bits, M b (k), and the length of input data Q. Output: The serial data M e (k) and s i .
Generate the hyperchaotic sequence c i using the 2D-ICSM; 10 for i=1 to Q do 11 Obtain N-bit key vectors s i from the chaotic sequence c i ;

end
It is crucial to mention here that the encrypted signal will be the pulses (1's on and 0's off) that is sent from pin 1 in Arduino (board 1) to the electronic laser circuit as shown in Figure 9. The laser diode output depends on the diode injected current instead of voltage. The Arduino pin 1 is used to power the diode directly. Its processor draws 30 mA with 40 mA outputs. When the current enters into the laser diode circuit (LDC), it should be controlled by a modulated data stream. This current has high speed, where this circuit inverts the signal phase before the Laser diode (LD) is injected. Figure 12 shows a simulation example of the LDC. The inputs of the data stream are modulated through the LDC directly. The output of LD as an emitted light represents the reaction of "one" or "zero" logic. The direct modulation is considered as the most commonly used. It is utilized to modulate the light intensity for the transmission of information through free space. Besides that, the laser beam transmitted by the photodiode is received in the form of light pulses. The photodiode acts as a semiconductor device used to convert light into a current that converts the received beam into an electrical signal within a voltage range between two volts (0-0.5). The encoded signal received by the photodiode passes various stages to amplify the signal voltage and return it to 5 volts, as can be seen in Figure 13. Furthermore, the binary signal is inverted and then sent it to pin 0 of Arduino (Board 2), as shown in Figure 12.

Decryption Process
The decryption signal is referred to by M d (k). This output signal is sent to Pin 5, which in turn sends it immediately to delta demodulation, which is designed by an analog electronics via the operating amplifier. Figure 13 shows the delta demodulation composite of a Delta demodulation integrator. An op-amplifier as a low pass filter is used to send the signals and obtain the final M r (s), which is equal to the original signal M(s).

Experimental Implementation
This section investigates the simplicity of the proposed secure communication system in simulation and hardware implementation.

Simulation Implementation
The simulation results of the proposed secure communication system are presented here by Matlab 2017b programs and implemented in a computer with specification Core i3-2.00 GHz, Intel CPU, and 4 GB RAM. The decimal values of the secret key are selected as x 0 = 0.9382, y 0 = 0.3171, α = 4.6516, s = 0.5782, and z = 149936. Figure 14a-f depicts the simulation results of the proposed communication system. Empirical correctness of the system can be observed through the retrieved signal M r (s), which is completely identical to the original signal M(s) that has been encrypted, and then sent through a free-space optical channel.

Key Sensitivity Analysis
The employed key for the encryption scheme is considered highly sensitive when the encrypted message cannot be recovered, as a slight difference in one of the key components would result in an incorrect decrypted ciphertext. Therefore, we hereby investigate the key sensitivity of the proposed secure communication scheme, as shown in Figure 15. In this figure, a signal has been encrypted by k 1 , and using the same key, we could recover the signal, as shown in Figure 15a-c. However, when we change the 14th decimal place in the parameters, or initial conditions, or both to obtain three other keys, namely, k 2 , k 3 , and k 4 , respectively, Figure 15d-f demonstrates that these keys fail to recover the original signal, which means that the proposed communication system has a robust and sensitive key.

Hardware Implementation
The encryption algorithm is applied to a signal between (0 Hz-200 Hz) based on two microcontroller boards of Arduino Uno R3. The properties of these boards are non-expensive, have a simple design, and powerful microcontrollers that depend on the ATmega328 chip. The input and output pins are digital which consists of 14 digits, six of them are used as pulse width modulation (PWM) outputs, the other six are analog inputs. Besides that, the other 2 digits are used as 16 MHz crystal oscillator, the USB connection, and a reset button. These materials can be implemented by utilizing the C++ language. The computer with specification Core i3-2.00 GHz, Intel CPU, and RAM 4 GB is used to run the software that is designed by the C++/C programing language, which is used to implement the two Arduino boards. These experiments have been implemented in a lab using a 200 MHz digital oscilloscope and a digital function generator. It is shown in Figure 16. The experiment was conducted in the laboratory. The results presented in Figure 17a refers to the sent message that appears in blue, while the retrieved message appears in yellow, and Figure 17a refers to the serial binary data.

Conclusions
In summary, this paper introduces the 2D-ICSM, which is a new hyperchaotic map designed using the 1D infinite collapse model as seed. The fixed points of certain parameters of the 2D-ICSM have been calculated, and then the stability of these points was analyzed by the graphical method. Performance evaluations including Lyapunov exponents, bifurcation diagram, cross-correlation coefficient, phase space diagram, NIST-800-22 randomness test, and Sample Entropy algorithm showed that the 2D-ICSM has a wide hyperchaotic range, high sensitivity, good ergodicity, sufficient level of randomness, and extreme complexity performance. Therefore, the 2D-ICSM could be an ideal source for many chaos-based practical applications. To demonstrate the efficiency of 2D-ICSM, we proposed a secure communication system, which is designed to transmit a message between two points. The input message is modulated using a simple Delta modulator and then encrypted using the 2D-ICSM. In the receiver side, the 2D-ICSM along with Delta demodulation are employed to retrieve the original message. It is crucial to state that the transmitted message by the proposed communication system could be an image, a text, or a sound. Simulation and empirical results have verified the efficiency and simplicity of the proposed secure communication system.