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

Authors: Colince Welba; Dhanagopal Ramachandran; Alexendre Noura; Victor Kamdoum Tamba; Sifeu Takougang Kingni; Pascal Eloundou Ntsama; Pierre Ele
Journal: Complexity
 (2022)

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## Abstract

The field programmable gate array (FPGA) implementation of the nonlinear resistor-capacitor-inductor shunted Josephson junction (NRCISJJ) model and its application to sEMG (Surface ElectroMyoGraphic) signal encryption through image encrypted technique are reported in this study. Thanks to the numerical simulations and FPGA implementation of the NRCISJJ model, different shapes of chaotic attractors are revealed by varying the parameters. The chaotic behaviour found in the NRCISJJ model is used to encrypt the sEMG signal through image encryption technique. The results obtained are interesting and open up many perspectives.


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## Body

## 1. Introduction

Circuits based on Josephson Junction (JJ) devices received particular attention in literature during the past two decades. This 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, high-density computer circuits, quantum-computing devices, superconducting electronic devices (e.g., terahertz pulse generator), and ultrahigh-speed chaotic signal generators [1–8]. Concerning the last application, many works have demonstrated the existence of chaos in several systems using different models of JJ [9–15]. Among them, linear and nonlinear resistor-capacitor-inductor shunted JJ models (LRCISJJ and NRCIJJ) are the most investigated in the relevant literature. This 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]. They 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]. The 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. They 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. They 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. The real electrical signals obtained from the implementation of the considered JJ model have been exploited to power an electromechanical pendulum. The 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. The 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. The 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. The objective of every image encryption algorithm is to generate a noisy image’s having top-quality 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 [25–29]. 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. The 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). This lack of work on sEMG encryption highlights the originality of this study.The study is articulated around four sections presented as follows. The FPGA implementation of the NRCISJJ model is presented in Section2. 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.


## 2. FPGA Implementation of the NRCISJJ Model

The NRCISJJ model is described by the following dimensionless rate of equations [11]:(1)dxdt=1βci−y−gxx−sinz,dydt=1βLx−y,dzdt=x,where t, x, y, and z represent the dimensionless time, the voltage in the junction, the inductor current, and the phase difference, respectively. The parameter i is an external direct current source and βC and βL are capacitive and inductive parameters, respectively. The function gx is a piecewise function approximation by current voltage characteristic of the intrinsic junction shunt resistor defined as(2)gx=0.366,ifx>2.9,0.061,ifx≤2.9.System (1), describing the NRCISJJ model, is designed in Xilinx system generator Simulink integrated in MATLAB. The 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. The 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 Figure1.Figure 1
RTL architecture of the NRCISJJ model.Figure1 is implemented in Kintex 7 XC7K325fFFG676-1 chip. The discretized state equations of system (1) are given by(3)xm+1=xm+hi−ym−1−gxm−1xm−1−sinzm−1βC,ym+1=ym+hxm−1−ym−1βL,zm+1=zm+hxm−1,where βR,βL,βC,andi are the parameter values of system (1) and the step size h=0.01.The results of numerical simulations and FPGA implementation of NRCISJJ model are illustrated in Figure2.Figure 2
2D phase portraits of numerical simulations and FPGA implementation of the NRCISJJ model for given values of parameteri and βL: (a) i=1.36 and βL=2.07, (b) i=1.15 and βL=2.52, (c) i=1.15 and βL=2.6, and (d) i=1.15 and βL=3. The other parameter is βC=0.707 and the initial conditions are (x(0), y(0), z(0))=(0, 0, 0).
(a)(b)(c)(d)Four different shapes of chaotic attractors are depicted in Figure2 for given values of parameters i, βC, and βL. Moreover, in Figure 2, there is a good qualitative agreement between the numerical simulations and FPGA results.


## 3. Application of the Chaotic NRCISJJ Model to Secure sEMG Signal Using Chaotic Logistic Map Image Encryption Technique

The 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.The 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. The proposed sEMG signal encryption scheme is shown in Figure3.Figure 3
Schematic diagram of the proposed encryption scheme.The method presented in Figure3 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]. The 2D sEMG signal is easily manipulated for pixel swapping. The 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.The following lines summarize step by step the encrypted proposed scheme of Figure3.Step 1
: transform the sEMG_1D signal into sEMG_2D as a grayscale imageStep 2
: generate the chaotic sequence by the chaotic NRCISJJ with the parameters:i=1.15, βL=3, and βC=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 orbitsStep 4
: match the chaotic sequences with the sEMG_2D image pixelsStep 5
: make a permutation in the lines in N-iterationStep 6
: make a permutation in the columns in M-iteration and obtain the encrypted sEMG_2D signalStep 7
: display the encrypted imageStep 8
: transform the sEMG_2D into sEMG_1D and display.The 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.Table 1
S_EMG 2D and their resolutions.
S_EMG 1DS_EMG 2DRESOLUTIONKHEIR1KHEIR1_2D180∗180KHEIR2KHEIR2_2D180∗180JOUVE3JOUVE3_2D180∗180EMG_MYOPATHYEMG_MYOPATHY_2D148∗148EMG_HEALTHYEMG_HEALTHY_2D53∗53KHEIR1, 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.


### 3.1. Key Sensitivity Analysis

#### 3.1.1. Correlation Analysis

The formula for the correlation between two neighboring pixels (w and c, for example) is given by the following relation:(4)γx,y=Ey−Eyx−ExDyDx,Ex=1T∑i=1Txi,Dx=1T∑i=1Txi−Ex2,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.Table2 shows the correlation coefficients of the original sEMG_2D and sEMG_2D encrypted using the proposed approach.Table 2
The correlation coefficientγ of signals (S_EMG 2D).
S_EMG 2Dγγ of input signalProposed chaotic NRCISJJKHEIR1_2Dγh0.9510−0.0005γv−0.00540.0028γd−0.0050−0.0009KHEIR2_2Dγh0.91700.0095γv0.0017−0.0012γd−0.0009−0.0078EMG_MYOPATHY_2Dγh0.4520−0.0073γv0.0158−0.0069γd0.0067−0.0001EMG_HEALTHY_2Dγh0.7484−0.0021γv0.0984−0.0032γd0.0852−0.0113JOUVE3_2Dγh0.9526−0.0032γv−0.0368−0.0032γd0.0379−0.0076It can be seen from Table2 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. These results confirm that the proposed algorithm can remove the correlation between adjacent pixels in the encrypted sEMG signals. These results sufficiently show that the encryption algorithm is well suited to EMG signals. Figures 4–8 show a visual representation of the correlation coefficients of the original, encrypted, and decrypted sEMG_2D signal.Figure 4
Correlations of KHEIR1_2D. The 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.Figure 5
Correlations of KHEIR2_2D. The 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.Figure 6
Correlations of JOUVE3_2D. The 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.Figure 7
Correlations of MYOPATHY_2D. The first column is the input signal with its correlation, the second column is the corresponding cipher signal with his correlation, and the third column is the decrypted signal with its correlation.Figure 8
Correlations of HEATHY_2D. The 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.From Figures4 to 8, 200, 300, or even 400 pairs of adjacent pixels are randomly selected from the images to show their adjacent pixel distribution maps. This implies the strong correlation effect in the input and decrypted sEMG_2D signals, while there is a weak correlation effect in the encrypted sEMG_2D signals. These figures verify well that there is no significant correlation between pixels of the encrypted sEMG_2D signals. In addition to the encouraging encryption result offered by the proposed algorithm, Figures 4 to 8 also show that the proposed algorithm gives a decrypted sEMG_2D image very close to the original sEMG_2D image.


### 3.2. Information Entropy

For a grayscale image, the intensity has28 possible types of values, so its ideal IFE is 8. The IFE can be defined by(5)En=∑i=0M−1pnilog21pni,where M is the total number of samples niεn,pni denotes the probability of occurrence of the sample ni, 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. Thus, 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.Table 3
Information entropy performance of cipher S_EMG 2D.
S_EMG 2DChaotic NRCISJJKHEIR1_2D7.9941KHEIR2_2D7.9936EMG_MYOPATHY_2D7.9885EMG_HEALTHY _2D7.9388JOUVE3_2D7.9923These results of Table3 once again confirm that the data of the sEMG signals are well encrypted and with a high level of security.


### 3.3. 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 (number of pixels’ change rate) and the UACI (unified average changing intensity). The NPCR represents the rate of different pixels between the two encrypted images, while the UACI represents the difference in average intensity [39].The formula used to calculate these two percentages is defined as follows [37]:(6)NPCR=1M×N∑i+1M∑j=1NDIFi,j×100,UACI=1M×N∑i+1M∑j=1NC2i,j−C1i,j255×100%,with DIFi,j=0,C2i,j=C1i,j,1,C2i,j≠C1i,j,where C2 refers to the crypted image that is encrypted from the original image by changing only one pixel, while C1 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.Table 4
Average performance of NPCR (%) and UACI (%).
S_EMG 2DNPCR (%)UACI (%)KHEIR1_2D99.966035.5353KHEIR2_2D99.959934.9161EMG_MYOPATHY_2D10024.5620EMG_HEALTHY _2D10025.3572JOUVE3_2D99.956835.5353From the results obtained and presented in Table4, 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. This 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.


### 3.4. Quality Metrics Analysis between Plain and Encrypted

Table5 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. The medical personnel who receive the encrypted signal decrypt it for a diagnosis. To avoid misdiagnosis, medical personnel should not receive a corrupt signal. This is why in Table 5 we study the distortion between the original signal and the decrypted signal. This distortion is quantified by the following metrics: the percent root mean square difference (PRD) (%) and the signal to noise ratio (SNR).Table 5
Quality metrics analysis.
S_EMG_1DPRD (%)SNR (dB)MSEKHEIR1_1D1.6635.621165.37KHEIR2_1D0.8741.24317.92EMG_MYOPATHY_1D3.0030.46859.00EMG_HEALTHY _1D1.9334.28183.29JOUVE3_1D0.1456.908.49


#### 3.4.1. Mean Square Error (MSE)

The most commonly used quality measure is mean square error (MSE) and is defined by(7)MSE=1N∑n=1NyOn−yrn2,where yOn is the original surface EMG signal, yrn is the decrypted surface EMG signal, and N is the number of samples of the signal.


#### 3.4.2. Signal-to-Noise Ratio (SNR)

(8)SNR=10Log10σx2σe2.where Log is decimal logarithm. With σx2 as the spectral power of the original SEMG signal and σe2 as the spectral power of the decrypted error.


#### 3.4.3. Percent Root Mean Square Difference (PRD)

The PRD (percent root mean square difference) is defined by(9)PRD=∑n=0N−1yOn−yrn2∑n=0N−1yOn−μ2⋅100%,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), yrn is the decrypted sEMG signal, and yOn is the original sEMG signal [34].Analysis of the values in Table5 shows that the decrypted sEMG signal resembles the original signal. This can be seen through the value of the PRD which must be as small as possible and that of the SNR must be high. This statement can be verified by observing Figures 9–13.Figure 9
Input signal, cipher signal, and decrypted signal of KHEIR1_1D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figure 10
Input signal, cipher signal, and decrypted signal of KHEIR2_1D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figure 11
Input signal, cipher signal, and decrypted signal of JOUVE3_1D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figure 12
Input signal, cipher signal, and decrypted signal ofMYOPATHY_1D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figure 13
Input signal, cipher signal, and decrypted signal of HEALTY_2D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figures9 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.


### 3.5. Encryption Time

In telemedicine, the interaction between two health specialists can be in real time. So, the communications must be as fast as possible. Therefore, the encryption and decryption times should be as small as possible. Table6 shows the encryption and decryption time of the proposed algorithm. The proposed algorithm is implemented in MATLAB R2015a and the “run and time” function is used. The 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-6F57D403F31C(iv)
Product ID: 00330-80000-00000-AA798(v)
System type: 64 bits, processor x64Table 6
Encrypted and decrypted time.
S_EMG 2DResolutionProposed chaotic NRCISJJEncrypted time (s)Decrypted time (s)KHEIR1_2D180∗1800.3361.701KHEIR2_2D180∗1800.3511.676EMG_MYOPATHY_2D148∗1480.2231.241EMG_HEALTHY_2D53∗530.0280.169JOUVE3_2D180∗1800.3611.757These times for all signals show that the proposed algorithm is fast. The comparison analyses are presented in Table7.Table 7
Comparison of the proposed encryption scheme with recent schemes reported in literature, where √ means “achieved” and — means “not achieved.”
[42][43][44][45]Proposed chaotic NRCISJJClinical signalsECG√√———BP√————EEG√—√√—EMG————√Acquisition methodLive sensing—————Database techniquePhysioBank ATMMIT−BIHUCI KDDNTOUPhysioBank ATMEncryption√√√√√Compression—√———Chaos specificationsChaotic mapLogistic mapNot specifiedLogistic mapLogistic mapChaotic NRCISJJLyapunov exponent√————Security analysisSecret key space√————Correlation√—√—√Autocorrelation√———√Secret key sensitivity√——√√Plain signal sensitivity√————Floating frequency√————Information entropy√———√Mean square error√—√—√Pick signal-to-noise ratio√————Structural similarity index√————Encryption time√——√√Distortion criteriaSignal-to-noise ratio————√Percentage residual deviation—√—√√


## 3.1. Key Sensitivity Analysis

### 3.1.1. Correlation Analysis

The formula for the correlation between two neighboring pixels (w and c, for example) is given by the following relation:(4)γx,y=Ey−Eyx−ExDyDx,Ex=1T∑i=1Txi,Dx=1T∑i=1Txi−Ex2,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.Table2 shows the correlation coefficients of the original sEMG_2D and sEMG_2D encrypted using the proposed approach.Table 2
The correlation coefficientγ of signals (S_EMG 2D).
S_EMG 2Dγγ of input signalProposed chaotic NRCISJJKHEIR1_2Dγh0.9510−0.0005γv−0.00540.0028γd−0.0050−0.0009KHEIR2_2Dγh0.91700.0095γv0.0017−0.0012γd−0.0009−0.0078EMG_MYOPATHY_2Dγh0.4520−0.0073γv0.0158−0.0069γd0.0067−0.0001EMG_HEALTHY_2Dγh0.7484−0.0021γv0.0984−0.0032γd0.0852−0.0113JOUVE3_2Dγh0.9526−0.0032γv−0.0368−0.0032γd0.0379−0.0076It can be seen from Table2 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. These results confirm that the proposed algorithm can remove the correlation between adjacent pixels in the encrypted sEMG signals. These results sufficiently show that the encryption algorithm is well suited to EMG signals. Figures 4–8 show a visual representation of the correlation coefficients of the original, encrypted, and decrypted sEMG_2D signal.Figure 4
Correlations of KHEIR1_2D. The 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.Figure 5
Correlations of KHEIR2_2D. The 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.Figure 6
Correlations of JOUVE3_2D. The 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.Figure 7
Correlations of MYOPATHY_2D. The first column is the input signal with its correlation, the second column is the corresponding cipher signal with his correlation, and the third column is the decrypted signal with its correlation.Figure 8
Correlations of HEATHY_2D. The 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.From Figures4 to 8, 200, 300, or even 400 pairs of adjacent pixels are randomly selected from the images to show their adjacent pixel distribution maps. This implies the strong correlation effect in the input and decrypted sEMG_2D signals, while there is a weak correlation effect in the encrypted sEMG_2D signals. These figures verify well that there is no significant correlation between pixels of the encrypted sEMG_2D signals. In addition to the encouraging encryption result offered by the proposed algorithm, Figures 4 to 8 also show that the proposed algorithm gives a decrypted sEMG_2D image very close to the original sEMG_2D image.


## 3.1.1. Correlation Analysis

The formula for the correlation between two neighboring pixels (w and c, for example) is given by the following relation:(4)γx,y=Ey−Eyx−ExDyDx,Ex=1T∑i=1Txi,Dx=1T∑i=1Txi−Ex2,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.Table2 shows the correlation coefficients of the original sEMG_2D and sEMG_2D encrypted using the proposed approach.Table 2
The correlation coefficientγ of signals (S_EMG 2D).
S_EMG 2Dγγ of input signalProposed chaotic NRCISJJKHEIR1_2Dγh0.9510−0.0005γv−0.00540.0028γd−0.0050−0.0009KHEIR2_2Dγh0.91700.0095γv0.0017−0.0012γd−0.0009−0.0078EMG_MYOPATHY_2Dγh0.4520−0.0073γv0.0158−0.0069γd0.0067−0.0001EMG_HEALTHY_2Dγh0.7484−0.0021γv0.0984−0.0032γd0.0852−0.0113JOUVE3_2Dγh0.9526−0.0032γv−0.0368−0.0032γd0.0379−0.0076It can be seen from Table2 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. These results confirm that the proposed algorithm can remove the correlation between adjacent pixels in the encrypted sEMG signals. These results sufficiently show that the encryption algorithm is well suited to EMG signals. Figures 4–8 show a visual representation of the correlation coefficients of the original, encrypted, and decrypted sEMG_2D signal.Figure 4
Correlations of KHEIR1_2D. The 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.Figure 5
Correlations of KHEIR2_2D. The 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.Figure 6
Correlations of JOUVE3_2D. The 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.Figure 7
Correlations of MYOPATHY_2D. The first column is the input signal with its correlation, the second column is the corresponding cipher signal with his correlation, and the third column is the decrypted signal with its correlation.Figure 8
Correlations of HEATHY_2D. The 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.From Figures4 to 8, 200, 300, or even 400 pairs of adjacent pixels are randomly selected from the images to show their adjacent pixel distribution maps. This implies the strong correlation effect in the input and decrypted sEMG_2D signals, while there is a weak correlation effect in the encrypted sEMG_2D signals. These figures verify well that there is no significant correlation between pixels of the encrypted sEMG_2D signals. In addition to the encouraging encryption result offered by the proposed algorithm, Figures 4 to 8 also show that the proposed algorithm gives a decrypted sEMG_2D image very close to the original sEMG_2D image.


## 3.2. Information Entropy

For a grayscale image, the intensity has28 possible types of values, so its ideal IFE is 8. The IFE can be defined by(5)En=∑i=0M−1pnilog21pni,where M is the total number of samples niεn,pni denotes the probability of occurrence of the sample ni, 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. Thus, 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.Table 3
Information entropy performance of cipher S_EMG 2D.
S_EMG 2DChaotic NRCISJJKHEIR1_2D7.9941KHEIR2_2D7.9936EMG_MYOPATHY_2D7.9885EMG_HEALTHY _2D7.9388JOUVE3_2D7.9923These results of Table3 once again confirm that the data of the sEMG signals are well encrypted and with a high level of security.


## 3.3. 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 (number of pixels’ change rate) and the UACI (unified average changing intensity). The NPCR represents the rate of different pixels between the two encrypted images, while the UACI represents the difference in average intensity [39].The formula used to calculate these two percentages is defined as follows [37]:(6)NPCR=1M×N∑i+1M∑j=1NDIFi,j×100,UACI=1M×N∑i+1M∑j=1NC2i,j−C1i,j255×100%,with DIFi,j=0,C2i,j=C1i,j,1,C2i,j≠C1i,j,where C2 refers to the crypted image that is encrypted from the original image by changing only one pixel, while C1 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.Table 4
Average performance of NPCR (%) and UACI (%).
S_EMG 2DNPCR (%)UACI (%)KHEIR1_2D99.966035.5353KHEIR2_2D99.959934.9161EMG_MYOPATHY_2D10024.5620EMG_HEALTHY _2D10025.3572JOUVE3_2D99.956835.5353From the results obtained and presented in Table4, 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. This 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.


## 3.4. Quality Metrics Analysis between Plain and Encrypted

Table5 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. The medical personnel who receive the encrypted signal decrypt it for a diagnosis. To avoid misdiagnosis, medical personnel should not receive a corrupt signal. This is why in Table 5 we study the distortion between the original signal and the decrypted signal. This distortion is quantified by the following metrics: the percent root mean square difference (PRD) (%) and the signal to noise ratio (SNR).Table 5
Quality metrics analysis.
S_EMG_1DPRD (%)SNR (dB)MSEKHEIR1_1D1.6635.621165.37KHEIR2_1D0.8741.24317.92EMG_MYOPATHY_1D3.0030.46859.00EMG_HEALTHY _1D1.9334.28183.29JOUVE3_1D0.1456.908.49


### 3.4.1. Mean Square Error (MSE)

The most commonly used quality measure is mean square error (MSE) and is defined by(7)MSE=1N∑n=1NyOn−yrn2,where yOn is the original surface EMG signal, yrn is the decrypted surface EMG signal, and N is the number of samples of the signal.


### 3.4.2. Signal-to-Noise Ratio (SNR)

(8)SNR=10Log10σx2σe2.where Log is decimal logarithm. With σx2 as the spectral power of the original SEMG signal and σe2 as the spectral power of the decrypted error.


### 3.4.3. Percent Root Mean Square Difference (PRD)

The PRD (percent root mean square difference) is defined by(9)PRD=∑n=0N−1yOn−yrn2∑n=0N−1yOn−μ2⋅100%,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), yrn is the decrypted sEMG signal, and yOn is the original sEMG signal [34].Analysis of the values in Table5 shows that the decrypted sEMG signal resembles the original signal. This can be seen through the value of the PRD which must be as small as possible and that of the SNR must be high. This statement can be verified by observing Figures 9–13.Figure 9
Input signal, cipher signal, and decrypted signal of KHEIR1_1D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figure 10
Input signal, cipher signal, and decrypted signal of KHEIR2_1D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figure 11
Input signal, cipher signal, and decrypted signal of JOUVE3_1D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figure 12
Input signal, cipher signal, and decrypted signal ofMYOPATHY_1D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figure 13
Input signal, cipher signal, and decrypted signal of HEALTY_2D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figures9 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.


## 3.4.1. Mean Square Error (MSE)

The most commonly used quality measure is mean square error (MSE) and is defined by(7)MSE=1N∑n=1NyOn−yrn2,where yOn is the original surface EMG signal, yrn is the decrypted surface EMG signal, and N is the number of samples of the signal.


## 3.4.2. Signal-to-Noise Ratio (SNR)

(8)SNR=10Log10σx2σe2.where Log is decimal logarithm. With σx2 as the spectral power of the original SEMG signal and σe2 as the spectral power of the decrypted error.


## 3.4.3. Percent Root Mean Square Difference (PRD)

The PRD (percent root mean square difference) is defined by(9)PRD=∑n=0N−1yOn−yrn2∑n=0N−1yOn−μ2⋅100%,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), yrn is the decrypted sEMG signal, and yOn is the original sEMG signal [34].Analysis of the values in Table5 shows that the decrypted sEMG signal resembles the original signal. This can be seen through the value of the PRD which must be as small as possible and that of the SNR must be high. This statement can be verified by observing Figures 9–13.Figure 9
Input signal, cipher signal, and decrypted signal of KHEIR1_1D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figure 10
Input signal, cipher signal, and decrypted signal of KHEIR2_1D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figure 11
Input signal, cipher signal, and decrypted signal of JOUVE3_1D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figure 12
Input signal, cipher signal, and decrypted signal ofMYOPATHY_1D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figure 13
Input signal, cipher signal, and decrypted signal of HEALTY_2D. The first is the input signal, the second is the corresponding cipher signal, and the third is the decrypted signal.Figures9 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.


## 3.5. Encryption Time

In telemedicine, the interaction between two health specialists can be in real time. So, the communications must be as fast as possible. Therefore, the encryption and decryption times should be as small as possible. Table6 shows the encryption and decryption time of the proposed algorithm. The proposed algorithm is implemented in MATLAB R2015a and the “run and time” function is used. The 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-6F57D403F31C(iv)
Product ID: 00330-80000-00000-AA798(v)
System type: 64 bits, processor x64Table 6
Encrypted and decrypted time.
S_EMG 2DResolutionProposed chaotic NRCISJJEncrypted time (s)Decrypted time (s)KHEIR1_2D180∗1800.3361.701KHEIR2_2D180∗1800.3511.676EMG_MYOPATHY_2D148∗1480.2231.241EMG_HEALTHY_2D53∗530.0280.169JOUVE3_2D180∗1800.3611.757These times for all signals show that the proposed algorithm is fast. The comparison analyses are presented in Table7.Table 7
Comparison of the proposed encryption scheme with recent schemes reported in literature, where √ means “achieved” and — means “not achieved.”
[42][43][44][45]Proposed chaotic NRCISJJClinical signalsECG√√———BP√————EEG√—√√—EMG————√Acquisition methodLive sensing—————Database techniquePhysioBank ATMMIT−BIHUCI KDDNTOUPhysioBank ATMEncryption√√√√√Compression—√———Chaos specificationsChaotic mapLogistic mapNot specifiedLogistic mapLogistic mapChaotic NRCISJJLyapunov exponent√————Security analysisSecret key space√————Correlation√—√—√Autocorrelation√———√Secret key sensitivity√——√√Plain signal sensitivity√————Floating frequency√————Information entropy√———√Mean square error√—√—√Pick signal-to-noise ratio√————Structural similarity index√————Encryption time√——√√Distortion criteriaSignal-to-noise ratio————√Percentage residual deviation—√—√√


## 4. Conclusion

The 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. The 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. This 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. This will be the subject of a study in future works.



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*Source: 4510236-2022-02-28.xml*