Josephson Junction Model: FPGA Implementation and Chaos-Based Encryption of sEMG Signal through Image Encryption Technique

Department of Fundamental Sciences, National Advanced School of Mines and Petroleum Industries, University of Maroua, P. O. Box 46, Maroua, Cameroon Center for Systems Design, Chennai Institute of Technology, Chennai 600069, Tamilnadu, India Department of Physics, Faculty of Science, University of Ngaoundere, P.O. Box: 454, Ngaoundere, Cameroon Department of Telecommunication and Network Engineering, IUT-Fotso Victor of Bandjoun, University of Dschang, P. O. Box: 134, Bandjoun, Cameroon Department of Mechanical, Petroleum and Gas Engineering, National Advanced School of Mines and Petroleum Industries, University of Maroua, P. O. Box 46, Maroua, Cameroon Electrical and Engineering, Telecommunications Department, National Advanced School of Engineering, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon


Introduction
Circuits based on Josephson Junction (JJ) devices received particular attention in literature during the past two decades.
is great interest is justified not only by the interesting characteristics of JJ device including high working frequency, low power consumption, and ultralow noise but also by their exploitation for constructing important technological devices such as ultrahigh sensitive detectors, highdensity computer circuits, quantum-computing devices, superconducting electronic devices (e.g., terahertz pulse generator), and ultrahigh-speed chaotic signal generators [1][2][3][4][5][6][7][8]. Concerning the last application, many works have demonstrated the existence of chaos in several systems using different models of JJ [9][10][11][12][13][14][15]. Among them, linear and nonlinear resistor-capacitor-inductor shunted JJ models (LRCISJJ and NRCIJJ) are the most investigated in the relevant literature. is is due to their fascinating properties very suitable for high-frequency applications such as spread spectrum communication systems. Dana et al. characterized the chaotic dynamics in such models [9]. ey reported some interesting results on the modulation of chaotic oscillation in such devices by an external sinusoidal signal as information. Control and synchronization of the NRCIJJ model using the backstepping design method are discussed in [11,12]. e authors demonstrated that the employed control method is capable to eliminate the chaotic behaviour displayed by the NRCIJJ model and assureed the global asymptotic synchronization between drive-response NRCIJJ models with different system parameters. Remarkable numerical computations were carried out to confirm the feasibility of the developed control technique. Sifeu et al. [16] studied the dynamics and synchronization of the NRCLJJ model. ey used the fractional-order form of the model to develop an application to digital cryptography. Implementing chaotic models with electronic devices has some inconveniences due to the limitations of bandwidth of some electronic devices such as operational amplifiers. To overcome these limitations, the implementation of chaotic models is carried out with FPGA and microcontroller devices. In this regard, the authors of [17] analyzed and implemented with FPGA a fractal JJ with unharmonic current-phase relation. ey applied the system under scrutiny to chaos-based random number generator. Kadjie et al. [18] discussed the implementation with microcontroller of a NRCIJJ model and its applications in electromechanical engineering. e real electrical signals obtained from the implementation of the considered JJ model have been exploited to power an electromechanical pendulum. e numerical simulations revealed periodic and chaotic behaviours in the resulting system. Lai et al. proposed letter reports. In this letter, the authors constructed an interesting no-equilibrium chaotic system from the Lu system. e most striking feature of the new system is that it has hidden attractors and coexisting attractors [19,20]. In [21], the authors investigated a 4D extended Lü system which coexists multiple attractors with respect to different initial conditions. Lai et al. [22] reported a new 4D chaotic system with double memristors. e numerical simulation indicated that the system is capable of yielding infinite coexisting attractors.
In recent years, considerable efforts have been devoted to the designing and investigation of image encryption systems. It is well known that image encryption is a useful technique for secure transmission. e objective of every image encryption algorithm is to generate a noisy image's having topquality capable to keep information secret [23,24]. Several image encryption algorithms have been proposed in relevant works. For example, some algorithms used single low-dimensional chaotic systems, such as logistic map, tent map, Baker map, and cat map, to encrypt images [25][26][27][28][29]. Wang and Zhang investigated an image encryption algorithm based on genetic recombination and 4D Lorenz-like hyperchaotic systems [30]. Huang et al. [31] developed a color image encryption algorithm using fractional-order chaotic sequences.
Motivated by the above discussions, this study designs and implements on FPGA a chaotic NRCISJJ model and applies it to secure sEMG signal through image encryption technique. e innovation of this study is to show that it is possible to secure a 1D signal using image encryption techniques. To our knowledge, the literature devoted to securing signals by encryption does not mention work on the encryption of EMG signals. On the other hand, this same literature mentions several works on the encryption of other electrophysiological signals (EEG and ECG). is lack of work on sEMG encryption highlights the originality of this study. e study is articulated around four sections presented as follows. e FPGA implementation of the NRCISJJ model is presented in Section 2. Section 3 focuses on its application to secure surface electromyographic signals through image encryption technique. Finally, the conclusion of the paper is presented in Section 4.

FPGA Implementation of the NRCISJJ Model
e NRCISJJ model is described by the following dimensionless rate of equations [11]: where t, x, y, and z represent the dimensionless time, the voltage in the junction, the inductor current, and the phase difference, respectively. e parameter i is an external direct current source and β C and β L are capacitive and inductive parameters, respectively. e function g(x) is a piecewise function approximation by current voltage characteristic of the intrinsic junction shunt resistor defined as System (1), describing the NRCISJJ model, is designed in Xilinx system generator Simulink integrated in MATLAB. e blocks of Xilinx system generator tool kit used to design system (1) are configured according to IEEE 754 standard as 32 bit (no. of bits) and 16 bit (binary bit) fixed point, where the latency is set to zero. e forward Euler's algorithm is the digital method used to design the integrator of system (1).
By using the Vivado design tool, the register-transfer level (RTL) architecture required for implementation of NRCISJJ model is presented in Figure 1. Figure 1 is implemented in Kintex 7 XC7K325fFFG676-1 chip. e discretized state equations of system (1) are given by where β R , β L , β C , andi are the parameter values of system (1) and the step size h � 0.01.
2 Complexity e results of numerical simulations and FPGA implementation of NRCISJJ model are illustrated in Figure 2.
Four different shapes of chaotic attractors are depicted in Figure 2 for given values of parameters i, β C , and β L . Moreover, in Figure 2, there is a good qualitative agreement between the numerical simulations and FPGA results.

Application of the Chaotic NRCISJJ Model to
Secure sEMG Signal Using Chaotic Logistic Map Image Encryption Technique e proposed encryption technique is based on that used in image encryption by the chaotic logistics map. Innovation takes place in the chaotic sequence. Indeed, the chaotic sequence of the logistic map is replaced in the algorithm by the chaotic sequence of the NRCISJJ model. e NRCISJJ model is used as a chaotic sequence to apply the sEMG signal encryption technique rather than the chaotic logistic map sequence due to the sensitivity of its key. During the simulation, we found that the key of the NRCISJJ was more sensitive than that of the logistics map. e proposed sEMG signal encryption scheme is shown in Figure 3. e method presented in Figure 3 consists in transforming the sEMG signal (sEMG_1D) in the form of an image (sEMG_2D). Several works have shown that the manipulation of sEMG in 2D gives better results than sEMG in 1D [32,33]. e 2D sEMG signal is easily manipulated for pixel swapping. e process of transforming sEMG_1D into sEMG_2D used is described in [34,35]. Once the signal is transformed into two dimensions as a grayscale image, the encryption algorithm is applied. e following lines summarize step by step the encrypted proposed scheme of Figure 3.
Step 1 : transform the sEMG_1D signal into sEMG_2D as a grayscale image Step 2 : generate the chaotic sequence by the chaotic NRCISJJ with the parameters: i � 1.15, β L � 3, and β C � 0.707, and the initial conditions are (x(0), y(0), z(0)) � (0, 0, 0) Step 3 : generate n-iteration with the NRCISJJ model of system (1) to have h(i) for different orbits Step 4 : match the chaotic sequences with the sEMG_2D image pixels Step 5 : make a permutation in the lines in N-iteration Step 6 : make a permutation in the columns in M-iteration and obtain the encrypted sEMG_2D signal Step 7 : display the encrypted image Step 8 : transform the sEMG_2D into sEMG_1D and display.
e sEMG signals used are the clinical data acquired from a database. PhysioBank ATM database is used in [36]. In the following, different resolutions of surface EMG_2D are used in Table 1. KHEIR1, KHEIR2, and JOUVE3 S_EMG 1D signals belong to holy patients, while EMG_MYOPATHY and EMG_HEALTHY S_EMG 1D signals are for not holy patients.

Correlation Analysis.
e formula for the correlation between two neighboring pixels (w and c, for example) is given by the following relation: Complexity where the integer T refers to the total number of adjoining pixels and D(x) and E(x) are the variance and expectation of x, respectively. Table 2 shows the correlation coefficients of the original sEMG_2D and sEMG_2D encrypted using the proposed approach.
It can be seen from Table 2 that the correlation coefficients of the input images are close to 1, while the correlation coefficients of the cipher images are close to 0, indicating that the pixels of the cipher images are not correlated. ese where M is the total number of samples n i εn, p(n i ) denotes the probability of occurrence of the sample n i , and log with base 2 signifies the entropy expressed in bits. When the signal sEMG_1D is transformed into sEMG_2D in the form of an image, the image obtained has the characteristics of a grayscale image. us, to have a high security encrypted image, the entropy of the encrypted image must be as high as possible, i.e., very close to 8 [37,38]. As can be seen in Table 3, the information entropy of the various encrypted sEMG_2D signals is close to 8, as shown in Table 3.
ese results of Table 3 once again confirm that the data of the sEMG signals are well encrypted and with a high level of security.

Resistance to One Bit Changing Attack.
To ensure the security of an image encryption scheme against differential scanning, two quantitative measures are used: the NPCR   300 100 X i,j Figure 5: Correlations of KHEIR2_2D. e first column is the input signal with its correlation, the second column is the corresponding cipher signal with its correlation, and the third column is the decrypted signal with its correlation.  400 0 X i,j Figure 8: Correlations of HEATHY_2D. e first column is the input signal with its correlation, the second column is the corresponding cipher signal with its correlation, and the third column is the decrypted signal with its correlation.
8 Complexity (number of pixels' change rate) and the UACI (unified average changing intensity). e NPCR represents the rate of different pixels between the two encrypted images, while the UACI represents the difference in average intensity [39]. e formula used to calculate these two percentages is defined as follows [37]: where C 2 refers to the crypted image that is encrypted from the original image by changing only one pixel, while C 1 refers to the cipher image encrypted from the same plain image.
When an NPCR > 99.6094%, a UACI > 33.4635% ensures that an image encryption scheme is secure against this attack [40,41]. Table 4 depicts the values of NPCR and UACI obtained on the sEMG_2D signal.
From the results obtained and presented in Table 4, it appears that all the NPCR values are greater than the optimum value of 99.6094%. As for the values of the UACI, we record two values lower than the optimal value of 33.4635%. Curiously, these two UACI values are obtained on the sEMG signals of patients who are not holy. is observation thus opens another axis of reflection that will furnish our perspective. Apart from these two values, we can conclude that an image encryption scheme is secure against the differential attack. Table 5 shows the analysis of signal quality. After encrypting the signal in 2D, we transform it into 1D to analyze the effect of the encryption on it. e medical personnel who receive the encrypted signal decrypt it for a diagnosis. To avoid misdiagnosis, medical personnel should not receive a corrupt signal. is is why in Table 5 we study the distortion between the original signal and the decrypted signal. is distortion is quantified by the following metrics: the percent root mean square difference (PRD) (%) and the signal to noise ratio (SNR).

Mean Square Error (MSE).
e most commonly used quality measure is mean square error (MSE) and is defined by where y O [n] is the original surface EMG signal, y r [n] is the decrypted surface EMG signal, and N is the number of samples of the signal.
where Log is decimal logarithm. With σ 2 x as the spectral power of the original SEMG signal and σ 2 e as the spectral power of the decrypted error.

Percent Root Mean Square Difference (PRD).
e PRD (percent root mean square difference) is defined by where N is the number of samples of the original sEMG signal, μ is the reference value of the DAC (digital analog converter) used for data acquisition s(n) (μ � 0 for EMG signals), y r [n] is the decrypted sEMG signal, and y O [n] is the original sEMG signal [34]. Analysis of the values in Table 5 shows that the decrypted sEMG signal resembles the original signal. is can be seen through the value of the PRD which must be as small as possible and that of the SNR must be high. is statement can be verified by observing Figures 9-13. Figures 9 to 13 present the original, encrypted, and decrypted sEMG signals of the signals KHEIR1, KHEIR2, JOUVE3, MYOPATHY, and HEALTHY, respectively. For each figure, the first line is the input signal, the second line is the corresponding cipher signal, and the third line is the decrypted signal.

Encryption Time.
In telemedicine, the interaction between two health specialists can be in real time. So, the communications must be as fast as possible. erefore, the encryption and decryption times should be as small as possible. Table 6 shows the encryption and decryption time of the proposed algorithm. e proposed algorithm is implemented in MATLAB R2015a and the "run and time" function is used. e characteristics of the machine are as follows: (i) Name: DESKTOP-FO74VUD (ii) RAM installed: 8,00 Go (7,85 Go useable) (iii) Device ID: DD9D7612-CE5F-4069-85ED-6F57D40 3F31C             ese times for all signals show that the proposed algorithm is fast. e comparison analyses are presented in Table 7.

Conclusion
e field programmable gate array implementation of the nonlinear resistor-capacitor-inductor shunted Josephson junction model and its application to secure surface electromyographic signal were investigated in this study. Different shapes of chaotic attractors were revealed by using numerical simulations and field programmable gate array implementation of the nonlinear resistive-capacitive-inductive shunted Josephson junction model. e aim of this paper was to show that it is possible to encrypt the surface electromyographic signal through image encryption techniques which was achieved. For a first experiment on the encryption of sEMG signals, the results obtained are encouraging. However, it appears that the results are not as powerful as when the encryption technique is applied to surface electromyographic signal of patients with pathology. is can be due to the fact that the signals of holy patients concentrated more information towards the lower frequencies while the signals of pathological patients concentrated the most information towards the higher frequencies.
is will be the subject of a study in future works.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
On behalf of all authors, the corresponding author states that there are no conflicts of interest.