esa-sceva/satcom-chunk-collection huggingface dataset:
# A Linear Interpolation and Curvature-Controlled Gradient Optimization Strategy Based on [PERSON]
## 2 LCMAdam Algorithm Design  
### Adam Optimization Algorithm  
The Adam algorithm is an optimization algorithm that adapts the learning rate using first-order and second-order moment estimations of the gradient. The first-order moment estimates the gradient direction, while the second-order moment estimates the gradient magnitude. By combining these two estimators, the Adam algorithm can dynamically adjust the learning rate's size to better accommodate changes in different parameters. The algorithm introduces a bias correction mechanism to solve the problem caused by a too-high learning rate at the initial time. Also, the parameters used for both are defined as first-order moments, \(m_{t}\), and second-order moments, \(v_{t}\), respectively. The moment estimation vectors for the gradient in step \(t\) are shown in Equations (1) and (2):  
\[m_{t}=\beta_{1}m_{t-1}+(1-\beta_{1})g_{t} \tag{1}\]  
\[v_{t}=\beta_{2}v_{t-1}+(1-\beta_{2})g_{t}^{2} \tag{2}\]  
where \(\beta_{1}\) and \(\beta_{2}\) are the exponential decay rates of the first-order and second-order moments, respectively, and \(g_{t}\) is the gradient of the step, \(t\). The formula is shown in (3):  
\[g_{t}=\
abla_{\theta}f_{t}(\theta_{t}) \tag{3}\]  
The parameter values updated by the neural network model at the \(t\)-th iteration are denoted by \(\theta_{t}\), the usage loss function computed by the neural network model at the \(t\)-th iteration is denoted by \(f_{t}\), and the rate of change in the neural network model to the loss function relative to the parameters at the \(t\)-th iteration is denoted by \(\
abla_{\theta}f_{t}(\theta_{t})\).  
To address bias in estimating first- and second-order moments when the decay rate is very small in the initial stage, the Adam algorithm corrects these moments to optimize parameter updates more efficiently. The formulas are shown in (4) and (5):  
\[\hat{m}_{t}=m/(1-\beta_{1}^{t}) \tag{4}\]  
\[\hat{v}_{t}=v/(1-\beta_{2}^{t}) \tag{5}\]  
After each iteration, Equation (6) is used to calculate the update of parameter \(\theta\).  
\[\theta_{t+1}=\theta_{t}-\frac{\alpha}{\sqrt{\vartheta}+\epsilon}\hat{m}_{t} \tag{6}\]  
where \(\theta\) represents the value of the learning rate and \(\epsilon\) represents a sufficiently small but greater-than-zero value that prevents a denominator of zero.  
Although the Adam algorithm performs well in many cases, this algorithm is very sensitive to the learning rate setting, which requires fine-tuning its hyperparameters. In some cases, these hyperparameter settings may cause the algorithm to fail to reach the globally optimal solution, especially when dealing with certain non-convex optimization problems.
###### Abstract
[17] in their paper entitled \"Comparison of Machine Learning Regression Algorithms for Cotton Leaf Area Index Retrieval Using Sentinel-2 Spectral Bands\" compared the algorithm performance of five advanced machine learning regression algorithms, including ANN, support vector regression (SVR), Gaussian process regression (GPR), RF, and gradient boosting regression tree (GBRT), to retrieve cotton leaf area index (LAI) in a relatively comprehensive manner. Although the five models showed different performance, all of the models showed a potential for cotton LAI retrieval. [PERSON] et al. [18] in their paper entitled \"Spatial Data Reconstruction via ADMM and Spatial Spline Regression\" proposed a novel constrained spatial smoothing (CSS) algorithm to reconstruct a spatial field of densities. They evaluated the proposed method from the problem of reconstructing the spatial distribution of cellphone traffic volumes based on aggregate volumes recorded at sparsely scattered base stations. [PERSON] et al. [19] in their paper entitled \"Landslide Prediction with Model Switching\" provided a total solution in the form of an early warning system. The system is called the Model Switch-based Landslide Prediction System (MoSLAPS). To address the data imbalance problem, they also adapted the popular adaptive synthetic sampling (ADASYN) method to landslide prediction. Moreover, to address the low true-positive rate (TPR) problem, they proposed a novel event-class model switch predictor design that significantly improves TPR. [PERSON] [20] in the paper entitled \"A Critical Review of Spatial Predictive Modeling Process in Environmental Sciences with Reproducible Examples in R\" assisted spatial modelers and scientists by critically reviewing the spatial predictive modeling process, developing guidelines for selecting the most appropriate spatial predictive methods, and identifying and developing the most accurate predictive model to generate spatial predictions. [PERSON] et al. [21] in their paper entitled \"Mapping Areal Precipitation with Fusion Data by ANN Machine Learning in Sparse Gauged Region\" showed an efficient method to map areal precipitation with the data fused from the remote-sensing precipitation acquired from Tropical Precipitation Measurement Satellite (TRMM) product and ground gauge precipitation using the ANN method. This work was supported by the National Research Foundation of Korea funded by the Korea government under Grant NRF-2018M1A3A3A02066008 for H.J. and it was also supported by the Basic Research Project of the Korea Institute of Geoscience and Mineral Resources (KIGAM) funded by the Ministry of Science and ICT for S.L. This special issue would not be possible without the contributions of professional authors and reviewers, and the excellent editorial team of Applied Sciences.
# A Method for Building a Mixed-Reality Digital Twin of a Roadheader Monitoring System
## 2 Architecture of Digital Twin Monitoring System
### Overall Architecture of Digital Twin Monitoring System
#### 2.1.2 Digital Twin Virtual Entity  
The digital twin virtual entity is mainly a digital model constructed through bidirectional mapping of the characteristic attributes of the concerned entity. The digital twin monitoring system of the roadheader in this paper focuses on the geometric and behavioral attributes of the roadheader during the cutting process. The geometric attribute describes the main geometric parameters of the roadheader, such as the shape, size, and position of the parts, which can rationally reflect the contour appearance of the physical entity. The behavioral attribute portrays a series of behavioral responses produced by the roadheader under the interference of factors such as environmental disturbances, internal operating mechanisms, and human operations, encompassing the pose transformations of the main moving parts of the roadheader, the alterations in various physical parameters inside the roadheader, and the changes in the physical properties of the working environment.
## 4 Feature Extraction Techniques
### Statistical Features: MRE
#### 4.2.3 Entropy, \(E\)  
The theory of entropy plays a key part in the description of many intricate processes such as communications, statistics, thermodynamics, and so on. Entropy is scientifically associated with the terms disorder, uncertainty, or randomness [28]. Different types of entropy have been used for feature extraction such as approximate entropy (ApEn), sample entropy (SampEn), permutation entropy (PeEn), and so on [29; 30]. However, the Shannon entropy E is used in this article because it can easily measure the mean or average amount of information conveyed in each data point. Therefore, given a single card transaction with \(1\times k\) principal components, \(E\) is estimated as:  
\[E=\sum_{1}^{k}P_{k}log_{2}\left(\frac{1}{P_{k}}\right), \tag{15}\]  
where \(P_{k}\) is the principal component at point \(k\). Accordingly, for \(\alpha\) credit card transactions, E will be formulated as:  
\[E=\begin{bmatrix}E_{1}\\ E_{2}\\ \vdots\\ E_{\alpha}\end{bmatrix}=\begin{bmatrix}\sum_{1}^{k}P_{k}log_{2}\left(\frac{1}{ P_{k}}\right)_{1}\\ \sum_{1}^{k}P_{k}log_{2}\left(\frac{1}{P_{k}}\right)_{2}\\ \vdots\\ \sum_{1}^{k}P_{k}log_{2}\left(\frac{1}{P_{k}}\right)_{\alpha}\\ \end{bmatrix} \tag{16}\]  
Thus, the mean, relative amplitude, and entropy are combined in no particular order to form a feature vector that can be adapted with HMM as shown:  
\[f_{MRE}=[M,R,E]=\begin{bmatrix}M_{1}&R_{1}&E_{1}\\ M_{2}&R_{2}&E_{2}\\ \vdots&\vdots&\vdots\\ M_{\alpha}&R_{\alpha}&E_{\alpha}\\ \end{bmatrix} \tag{17}\]  
It should be noted that the \(f_{MRE}\) is \(h\)-dimensional (where \(h=5\)), which is a significant reduction from the d-dimension (\(d=28\)) or \(k\)-dimension ( \(k\) is determined in Section 7.1). However, with HMM, the computational complexity increases with an increase in the dimension of the feature vector. This implies that MRE-HMM simplifies the HMM pro cess in comparison to PCA-HMM. This proposed MRE feature extraction technique is summarised in Algorithm 2.  
```
Input:\(f\) Output:\(f_{MRE}\)
1: Compute \(\mathcal{M}\) as in (10) and (11)
2: Compute \(\mathcal{R}\) as in (12)-(14)
3: Compute \(E\) as in (15) and (16)
4: Derive \(f_{MRE}\) as in (17)
```  
**Algorithm 2** MRE Feature Extraction
## 3 Results and Discussion
### Geochemical Baseline
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**(a)**} \\ \hline \hline \multicolumn{2}{c}{Elements} & \multicolumn{3}{c}{Main} \\ \multicolumn{2}{c}{Components} \\ \hline \hline \multicolumn{2}{c}{} & PC1 & PC2 \\ As & 0.12 & 0.85 \\ Cu & 0.57 & \(-\)0.06 \\ Pb & 0.47 & \(-\)0.48 \\ Zn & 0.66 & 0.23 \\ Variance (\%) & 42.7 & 29.7 \\ Cumulative Variance (\%) & 42.7 & 72.4 \\ \hline \hline \multicolumn{2}{c}{} & \multicolumn{2}{c}{Main} \\ \hline \multicolumn{2}{c}{Elements} & \multicolumn{2}{c}{Components} \\ \hline \hline \multicolumn{2}{c}{} & PC1 & PC2 \\ As & 0.026 & 0.740 \\ Cu & \(-\)0.263 & \(-\)0.642 \\ Pb & 0.677 & \(-\)0.113 \\ Zn & 0.687 & \(-\)0.162 \\ Variance (\%) & 41.6 & 30.9 \\ Cumulative Variance (\%) & 41.5 & 72.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: **(a)** Principal components for the Casablanca River basin. (b) Principal components for the Aconcagua River basin. Figure 3: Biplots for two main components of the Aconcagua (**a**) and Casablanca (**b**) Rivers, respectively. Cluster analysis was performed using the Ward method with Euclidean distance squared. Figure 4 shows the dendrogram generated for the Aconcagua River basin (a) and Casablanca River (b). The As-Pb-Zn group can be attributed to more than one source of origin, while Cu is at the second level in the Aconcagua River basin. However, in the Casablanca River basin, two groups are observed: one formed by As-Pb-Cu, which indicates that they come from more than one source, and the second by Zn, which suggests that they come from a single source.
## 4 Parametric Analysis
### Support Pressure  
Figure 13 shows the ground deformation at the pipeline's elevation and the pipeline's deformation with different support pressures at the tunnel face. The divergence of the support pressure \(\Delta P\) ranges from 10 kPa to 50 kPa. The ground deformation is larger than the pipeline's deformation. The pipeline's deformation near the tunnel face increases with the increase of \(\Delta P\). To prevent the pipeline's uplift, the support pressure's divergence should be limited to a minor value.
## 2 Network Model
### Underwater Acoustic Channel Model  
Path loss caused by an ever-changing feature of acoustic channels is signal frequency dependent. Underwater ambient noise is the main factor that affects underwater acoustic transmission. In the underwater environment, signal attenuation is related to noise interferences, frequency, turbulence, and distance. In a propagation path without obstacles, a signal's attenuation factor at frequency \(f\) is [24]:  
\[A(d,f)=d^{S}a(f)^{d} \tag{2}\]  
where \(d\) and \(S\) represent the distance and spreading factor, respectively. \(S\) is set to one for shallow water cylindrical propagation; 1.5 for practical propagation; and two for deep water spherical propagation. The absorption coefficient happens as a result of a signal's frequency, and the absorption factor a(\(f\)) shown by the Thorp equation is:  
\[10 loga(f)=\frac{0.11f^{2}}{1+f^{2}}+\frac{44f^{2}}{4100+f}+2.75\times 10^{-4}f^{ 2}+0.003 \tag{3}\]  
Turbulence, ships, wind, and thermal noise are the key factors of underwater ambient noise \(N(f)\), represented as \(N_{t}(f)\), \(N_{s}(f)\), \(N_{m}(f)\) and \(N_{th}(f)\)[21]. Considering the application scenarios of UASNs, shipping noise and sea surface noise are the main factors affecting  
Figure 1: Network model.  
transmission frequency. Therefore, the influence of uncertain noise must be taken into account in the prediction of underwater acoustic transmission quality. Since most ambient noise sources can be described by Gaussian statistics, the following empirical formula gives the estimations of the four noise components [25]:  
\[\begin{array}{c}10\log N_{t}(f)=17-30\log f\\ 10\log N_{s}(f)=40+20*(s-0.5)+26\log f-60\log(f+0.3)\\ 10\log N_{w}(f)=50+7.5\sqrt{w}+20\log f-40\log(f+0.4)\\ 10\log N_{th}(f)=-15+20\log f\end{array} \tag{4}\]  
where \(s\) is the shipping activity factor, and the value of \(s\) is between zero and one; \(w\) is wind speed in m/s. The effective noise level at frequency \(f\) is the sum of the contributions of the above factors:  
\[N(f)=N_{t}(f)+N_{s}(f)+N_{w}(f)+N_{th}(f) \tag{5}\]  
Based on the noise model of the underwater environment, the average signal-to-noise ratio (SNR) is expressed as follows:  
\[\Gamma(d,f)=\frac{\mathrm{P}_{tp}\Big{/}A(d,f)}{N(f)B} \tag{6}\]  
where \(B\) denotes bandwidth and \(\mathrm{P}_{tp}\) is the power for transmission. The bit error rate between nodes for distance \(d\) is [25]:  
\[p_{e}(d)\approx\frac{1}{4 SNR} \tag{7}\]  
Therefore, for the successful packet transmission, \(p(d,m)\) represents the probability that \(m\) bits are transmitted between two nodes across the distance \(d\):  
\[p(d,m)=(1-p_{e}(d))^{m} \tag{8}\]
# Feasibility Investigation of SIW Cavity-Backed Patch Antenna Array for Ku Band Applications
## 3 Antenna Array Design  
The most widely-used techniques for feeding microstrip patch arrays can be categorized into two main kinds: the series and the corporate (parallel) ones. Corporate feeding has a single input port and multiple feed lines arranged in parallel. Each line is terminated with an individual radiating element, e.g., a patch antenna. The basic corporate feed is composed of two-way power dividers, which can be either 3 dB splitters or dividers with different power at the output ports.  
In this paper, the corporate feeding with 3 dB splitters was used, since it allows a feed with the same amplitude and phase of each radiating element. One drawback is that for long feeding lines, insertion losses cannot be negligible, this reducing the array efficiency. Arrays of uniform antennas consist of a repetition of radiating elements in a regular fashion. Each element is also located at a specific distance from the other. Figure 6 shows the base cell antenna employed in this paper to obtain the array and made by 2 \(\times\) 2 patch antennas (four elements) spaced next to each other at 0.66 \(\times\)\(\lambda_{0}\) in both directions of the array plane, where \(\lambda_{0}\) is the wavelength in vacuum at the central frequency of the Ku band f = 11.7 GHz, so that D\({}_{1}\) = D\({}_{2}\) = 25.2 mm. The input impedance of the antenna Z\({}_{a}\) = 94.90 \(\Omega\) was matched to the line of the first two-way power divider. A quarter-wave transformer  
\begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Size** \\ \hline v & 2.750 mm \\ m & 7.523 mm \\ r & 6.888 mm \\ \(\alpha\) & 15\({}^{\circ}\) \\ d & 0.620 mm \\ \hline \end{tabular}
\end{table}
Table 1: _Cont._  
Figure 5: Simulated \(|\)S\({}_{11}\)\(|\) parameter versus the frequency of the patch antenna with lateral feeding and cuts, and with and without the SIW backing cavity.  
with \(Z_{\lambda 1}\) = 67.10 \(\Omega\) was used to match the lines with characteristic impedance \(Z_{\mathrm{a}}\) (matched to the single radiating element) with the line of characteristic impedance \(Z_{\mathrm{d}}\) = 47.45 \(\Omega\). The quarter-wave transformers with \(Z_{\lambda 2}\) = 48.70 \(\Omega\) were designed to match the line with characteristic impedance \(Z_{\mathrm{d}}\) with the input line with \(Z_{\mathrm{i}}\) = 50 \(\Omega\).  
A larger planar array can be obtained as a regular repetition in the horizontal and vertical directions of the base cell layout of Figure 6. Following this approach, we designed the 4 \(\times\) 4, 8 \(\times\) 8, and 16 \(\times\) 16 antenna arrays.
# CFD Analysis and Validation of a Foreign Material Winnowing Machine for Pepper Harvester
## 1 Introduction
Those in the loss rate and separation efficiency were 9.8% and 2.6%, respectively, through the analysis result and validation test. [PERSON] et al. (2018) have designed and fabricated a winnower test device for a chili pepper harvester and conducted a performance test. The conditions that achieved the least loss of chili pepper and the highest separation rate of foreign materials were a winnower inclination angle of 34\({}^{\circ}\) and a rotating speed of 2000 RPM. Winnowing machines presented in the existing literature showed the following trend: the higher the rotating speed of the winnowing machine, the larger the loss of chili pepper fruit and the higher the foreign material separation rate. It is required to find a way to minimize the loss of chili pepper fruits and secure adequate foreign material separation at the same time. The optimization and application of CFD solution were studied to evaluate fluid flow around a target object at a low Reynolds number and numerical study on the drag coefficient. Additionally, the CFD simulation was used to evaluate the flow patterns and indicate the important numerical solutions. To optimize parameters of winnowing equipment for machine-harvested _Lycium barbarum_ L., a discrete element method (DEM) simulation of ripe fruit in the airflow was performed using EDEM software ([PERSON] et al., 2018). Aerodynamic analysis and CFD simulation were conducted to evaluate the model on temperature and relative humidity or predict the terminal velocity of agricultural granular materials or perform the wind tunnel in the agriculture field ([PERSON] et al., 2018; [PERSON] et al., 2018; [PERSON] et al., 2018). Thus, the objective of the present study was to design a foreign material separation winnowing machine for specializing work in chili pepper harvesters through an aerodynamic analysis and validate the same using CFD (Computational Fluid Dynamics, Ansys Fluent 2020 R1) software. Detailed objectives are below. 1. The first objective is to measure the terminal velocities of chili pepper species and conduct an aerodynamic analysis. The terminal velocities of chili pepper fruit, chili pepper stem, and chili pepper leaf move distance by wind are each derived through aerodynamic analysis. 2. The second objective is to validate the winnowing machine through a CFD analysis and test device. The wind speed of CFD flow analysis is verified as a flow test of separation equipment. In this result, the optimal rotation speed and discharge out guides for foreign material separation are decided, which will be applied to the chili pepper harvester.
## 3 Empirical Gramians
### State-Space Empirical Gramians
#### 3.1.4. Empirical Cross Gramian  
Analytically, a cross Gramian for (control-affine) nonlinear gradient systems was developed in [62; 63], yet the computation of this nonlinear cross Gramian (This is also called **cross operator** or **cross map** in this context) is infeasible for large systems. For (nonlinear) Single-Input-Single-Output (SISO) systems, the empirical variant of the cross Gramian is developed in [12], for (nonlinear) Multiple-Input-Multiple-Output (MIMO) systems in [11].  
**Definition 4** (Empirical Cross Gramian).: _Given non-empty sets \(E_{u}\), \(E_{x}\), \(S_{u}\) and \(S_{x}\), the **empirical cross Gramian**\(\widehat{W}_{X}\in\mathbb{R}^{N\times N}\) is defined as:_  
\[\widehat{W}_{X} =\frac{1}{|S_{u}||S_{x}|M}\sum_{k=1}^{|S_{u}|}\sum_{l=1}^{|S_{x}| }\sum_{m=1}^{M}\frac{1}{c_{k}d_{l}}\int_{0}^{\infty}\Psi^{klm}(t)\,\mathrm{d}t,\] \[\Psi^{klm}_{ij} =(x^{km}_{i}(t)-\bar{x}^{km}_{i})(y^{lj}_{m}(t)-\bar{y}^{lj}_{m}) \in\mathbb{R},\]  
_with the state trajectories \(x^{km}(t)\) for the input configurations \(\dot{u}^{km}(t)=c_{k}e^{m}\circ u(t)+\bar{u}\), the output trajectories \(y^{lj}(t)\) for the initial state configurations \(x^{lj}_{0}=d_{l}e^{j}+\bar{x}\), and the offsets \(\bar{u}\), \(\bar{x}^{km}\), \(\bar{x}\), \(\bar{y}^{lj}\)._For an asymptotically stable linear system, delta impulse input \(u_{i}(t)=\delta(t)\) and an arithmetic averages over time as offsets \(\tilde{x}=\frac{1}{T}\int_{0}^{T}x(t)\,\mathrm{d}t\), \(\tilde{y}=\frac{1}{T}\int_{0}^{T}y(t)\,\mathrm{d}t\), the empirical cross Gramian is equal to the cross Gramian \(\widehat{W}_{X}=W_{X}\)[11; 12]. The numerical computation of this empirical Gramian requires \((|S_{x}|\cdot(N+|S_{u}|\cdot M))\) trajectories.
## 3 MMPP/D/1 Simulation for Rare Event
### IS Simulation for Queue  
The goal of the IS method is to generate as many rare events as possible. In queue simulation, to increase the number of packets in the queue, one increases the arrival rate. This way, if the processing speed stays the same, the queue will quickly fill up and rare events will become more numerous. In [1], the author proposed the DIS (Dynamic Importance Sampling) method in which time slots are divided into Regenerative Cycle (RC) which described in Figure 2. Each RC is the time interval from when the queue length is greater than 0 to when its length returns to 0 (Figure 3). DIS is an important concept in IS simulation.  
Now, we will discuss the process of estimating the tail probability using importance sampling (IS) simulation. Let \(Q^{\prime}_{t_{1}},Q^{\prime}_{t_{2}},Q^{\prime}_{t_{3}},\dots\) represent the queue length associated with the time slot sequence \(t_{1},t_{2},t_{3},\dots\) according to the IS simulation distribution. Next, we will present the IS estimate of the tail probability as follows:  
\[P_{IS}(Q>q)=\frac{\frac{1}{N}\sum_{k=1}^{N}\sum_{t_{k}=1}^{T_{k}}\delta_{q}(Q^{ \prime}_{t_{k}})W_{t_{k}}}{\frac{1}{M}\sum_{k=1}^{M}\sum_{t_{k}=1}^{T_{k}}W_{t _{k}}} \tag{3}\]  
\[W_{t_{k}}=\prod_{m_{k}=1}^{t_{k}}\frac{P(Q^{\prime}_{m_{k}-1},Q^{\prime}_{m_{ k}})}{P^{\prime}(Q^{\prime}_{m_{k}-1},Q^{\prime}_{m_{k}})} \tag{4}\]  
where  
* \(M\) and \(N\) represent the quantities of RCs;
* \(t_{k}\) represents a specific time interval within the \(k\)th RC;
* \(T_{k}\) and \(T^{\prime}_{k}\) represent the length of the \(k\)th RC;
* \(W_{t_{k}}\) represents the weighting function;
* The parts \(P(Q^{\prime}_{m_{k}-1},Q^{\prime}_{m_{k}})\) and \(P^{\prime}(Q^{\prime}_{m_{k}-1},Q^{\prime}_{m_{k}})\) are the state transition probability from \((m_{k}-1)\) to \(m_{k}\), respectively.  
In the next section, we will present how to determine the optimal IS distribution.  
Figure 3: The Dynamic Importance Sampling method.  
Figure 2: The Regenerative Cycle.
# Authentication and Billing for Dynamic Wireless EV Charging in an Internet of Electric Vehicles
## 2 Preliminaries
### Digital Signatures  
A digital signature is a cryptographic protocol that verifies the authenticity of a digital document or a message. This protocol is used in asymmetric cryptography to send information over an insecure channel, and the receiver can verify that the message is transmitted by the claimed sender [34]. The widely used digital signature techniques are Rivest, Shamir, and Adleman (RSA), Digital Signature Algorithm (DSA), and Elliptic Curve Digital Signature Algorithm (ECDSA) [35; 36; 37]. ECDSA is the recommended digital signature scheme in the IEEE 1609.2 standard, as it offers a lower computational overhead compared to RSA-based schemes [38; 39]. Three steps are involved in digital signature schemes, namely: key generation, algorithm signing, and signature verification.