The Regularity of Solutions to Mixed Boundary Value Problems of Second-Order Elliptic Equations with Small Angles

This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.

ELLIPTIC EQUATIONS IN DIVERGENCE FORM WITH DISCONTINUOUS COEFFICIENTS IN DOMAINS WITH CORNERS

We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C1,αestimates across the discontinuity surfaces and provide an example to illustrate the issue regarding the regularity at the corners.

THE WEIGHTED KATO SQUARE ROOT PROBLEMOF ELLIPTIC OPERATORS HAVING A BMOANTI-SYMMETRICPART

Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.

Message Verification Protocol Based on Bilinear Pairings and Elliptic Curves for Enhanced Security in Vehicular Ad Hoc Networks

Vehicular ad hoc networks(VANETs)provide intelligent navigation and efficient route management,resulting in time savings and cost reductions in the transportation sector.However,the exchange of beacons and messages over public channels among vehicles and roadside units renders these networks vulnerable to numerous attacks and privacy violations.To address these challenges,several privacy and security preservation protocols based on blockchain and public key cryptography have been proposed recently.However,most of these schemes are limited by a long execution time and massive communication costs,which make them inefficient for on-board units(OBUs).Additionally,some of them are still susceptible to many attacks.As such,this study presents a novel protocol based on the fusion of elliptic curve cryptography(ECC)and bilinear pairing(BP)operations.The formal security analysis is accomplished using the Burrows–Abadi–Needham(BAN)logic,demonstrating that our scheme is verifiably secure.The proposed scheme’s informal security assessment also shows that it provides salient security features,such as non-repudiation,anonymity,and unlinkability.Moreover,the scheme is shown to be resilient against attacks,such as packet replays,forgeries,message falsifications,and impersonations.From the performance perspective,this protocol yields a 37.88%reduction in communication overheads and a 44.44%improvement in the supported security features.Therefore,the proposed scheme can be deployed in VANETs to provide robust security at low overheads.

Remote sensing image encryption algorithm based on novel hyperchaos and an elliptic curve cryptosystem

Remote sensing images carry crucial ground information,often involving the spatial distribution and spatiotemporal changes of surface elements.To safeguard this sensitive data,image encryption technology is essential.In this paper,a novel Fibonacci sine exponential map is designed,the hyperchaotic performance of which is particularly suitable for image encryption algorithms.An encryption algorithm tailored for handling the multi-band attributes of remote sensing images is proposed.The algorithm combines a three-dimensional synchronized scrambled diffusion operation with chaos to efficiently encrypt multiple images.Moreover,the keys are processed using an elliptic curve cryptosystem,eliminating the need for an additional channel to transmit the keys,thus enhancing security.Experimental results and algorithm analysis demonstrate that the algorithm offers strong security and high efficiency,making it suitable for remote sensing image encryption tasks.

Elliptical encirclement control capable of reinforcing performances for UAVs around a dynamic target

Most researches associated with target encircling control are focused on moving along a circular orbit under an ideal environment free from external disturbances.However,elliptical encirclement with a time-varying observation radius,may permit a more flexible and high-efficacy enclosing solution,whilst the non-orthogonal property between axial and tangential speed components,non-ignorable environmental perturbations,and strict assignment requirements empower elliptical encircling control to be more challenging,and the relevant investigations are still open.Following this line,an appointed-time elliptical encircling control rule capable of reinforcing circumnavigation performances is developed to enable Unmanned Aerial Vehicles(UAVs)to move along a specified elliptical path within a predetermined reaching time.The remarkable merits of the designed strategy are that the relative distance controlling error can be guaranteed to evolve within specified regions with a designer-specified convergence behavior.Meanwhile,wind perturbations can be online counteracted based on an unknown system dynamics estimator(USDE)with only one regulating parameter and high computational efficiency.Lyapunov tool demonstrates that all involved error variables are ultimately limited,and simulations are implemented to confirm the usability of the suggested control algorithm.

Elliptically polarized high-order harmonic generation of Ar atom in an intense laser field

High-order harmonic generation(HHG) of Ar atom in an elliptically polarized intense laser field is experimentally investigated in this work.Interestingly,the anomalous ellipticity dependence on the laser ellipticity(ε) in the lower-order harmonics is observed,specifically in the 13rd-order,which displays a maximal harmonic intensity at ε ≈ 0.1,rather than at ε = 0 as expected.This contradicts the general trend of harmonic yield,which typically decreases with the increase of laser ellipticity.In this study,we attribute this phenomenon to the disruption of the symmetry of the wave function by the Coulomb effect,leading to the generation of a harmonic with high ellipticity.This finding provides valuable insights into the behavior of elliptically polarized harmonics and opens up a potential way for exploring new applications in ultrafast spectroscopy and light–matter interactions.

Legendre-Jacobi’s Elliptic Integrals Shed Light on the Luminosity Distance in Cosmology

This article concerns the integral related to the transverse comoving distance and, in turn, to the luminosity distance both in the standard non-flat and flat cosmology. The purpose is to determine a straightforward mathematical formulation for the luminosity distance as function of the transverse comoving distance for all cosmology cases with a non-zero cosmological constant by adopting a different mindset. The applied method deals with incomplete elliptical integrals of the first kind associated with the polynomial roots admitted in the comoving distance integral according to the scientific literature. The outcome shows that the luminosity distance can be obtained by the combination of an analytical solution followed by a numerical integration in order to account for the redshift. This solution is solely compared to the current Gaussian quadrature method used as basic recognized algorithm in standard cosmology.

Sensitive Information Security Based on Elliptic Curves

The elliptic curve cryptography algorithm represents a major advancement in the field of computer security. This innovative algorithm uses elliptic curves to encrypt and secure data, providing an exceptional level of security while optimizing the efficiency of computer resources. This study focuses on how elliptic curves cryptography helps to protect sensitive data. Text is encrypted using the elliptic curve technique because it provides great security with a smaller key on devices with limited resources, such as mobile phones. The elliptic curves cryptography of this study is better than using a 256-bit RSA key. To achieve equivalent protection by using the elliptic curves cryptography, several Python libraries such as cryptography, pycryptodome, pyQt5, secp256k1, etc. were used. These technologies are used to develop a software based on elliptic curves. If built, the software helps to encrypt and decrypt data such as a text messages and it offers the authentication for the communication.

Estimation of fracture size and azimuth in the universal elliptical disc model based on trace information

The geometric characteristics of fractures within a rock mass can be inferred by the data sampling from boreholes or exposed surfaces.Recently,the universal elliptical disc(UED)model was developed to represent natural fractures,where the fracture is assumed to be an elliptical disc and the fracture orientation,rotation angle,length of the long axis and ratio of short-long axis lengths are considered as variables.This paper aims to estimate the fracture size-and azimuth-related parameters in the UED model based on the trace information from sampling windows.The stereological relationship between the trace length,size-and azimuth-related parameters of the UED model was established,and the formulae of the mean value and standard deviation of trace length were proposed.The proposed formulae were validated via the Monte Carlo simulations with less than 5%of error rate between the calculated and true values.With respect to the estimation of the size-and azimuth-related parameters using the trace length,an optimization method was developed based on the pre-assumed size and azimuth distribution forms.A hypothetical case study was designed to illustrate and verify the parameter estimation method,where three combinations of the sampling windows were used to estimate the parameters,and the results showed that the estimated values could agree well with the true values.Furthermore,a hypothetical three-dimensional(3D)elliptical fracture network was constructed,and the circular disc,non-UED and UED models were used to represent it.The simulated trace information from different models was compared,and the results clearly illustrated the superiority of the proposed UED model over the existing circular disc and non-UED models。

Asymmetric Key Cryptosystem for Image Encryption by Elliptic Curve over Galois Field GF(2^(n))

Protecting the integrity and secrecy of digital data transmitted through the internet is a growing problem.In this paper,we introduce an asymmetric key algorithm for specifically processing images with larger bit values.To overcome the separate flaws of elliptic curve cryptography(ECC)and the Hill cipher(HC),we present an approach to picture encryption by combining these two encryption approaches.In addition,to strengthen our scheme,the group laws are defined over the rational points of a given elliptic curve(EC)over a Galois field(GF).The exclusive-or(XOR)function is used instead of matrix multiplication to encrypt and decrypt the data which also refutes the need for the inverse of the key matrix.By integrating the inverse function on the pixels of the image,we have improved system security and have a wider key space.Furthermore,through comprehensive analysis of the proposed scheme with different available analyses and standard attacks,it is confirmed that our proposed scheme provides improved speed,security,and efficiency.

Sharp power-type Heronian and Lehmer means inequalities for the complete elliptic integrals

In the article,we prove that the inequalities H_(p)(K(r);E(r))>π/2;L_(q)(K(r);E(r))>π/2 hold for all r 2(0;1)if and only if p≥3=4 and q≥3=4,where Hp(a;b)and Lq(a;b)are respectively the p-th power-type Heronian mean and q-th Lehmer mean of a and b,and K(r)and E(r)are respectively the complete elliptic integrals of the first and second kinds.

Data Mining with Privacy Protection Using Precise Elliptical Curve Cryptography

Protecting the privacy of data in the multi-cloud is a crucial task.Data mining is a technique that protects the privacy of individual data while mining those data.The most significant task entails obtaining data from numerous remote databases.Mining algorithms can obtain sensitive information once the data is in the data warehouse.Many traditional algorithms/techniques promise to provide safe data transfer,storing,and retrieving over the cloud platform.These strategies are primarily concerned with protecting the privacy of user data.This study aims to present data mining with privacy protection(DMPP)using precise elliptic curve cryptography(PECC),which builds upon that algebraic elliptic curve infinitefields.This approach enables safe data exchange by utilizing a reliable data consolidation approach entirely reliant on rewritable data concealing techniques.Also,it outperforms data mining in terms of solid privacy procedures while maintaining the quality of the data.Average approximation error,computational cost,anonymizing time,and data loss are considered performance measures.The suggested approach is practical and applicable in real-world situations according to the experimentalfindings.

A Low-Cost and High-Performance Cryptosystem Using Tripling-Oriented Elliptic Curve

Developing a high-performance public key cryptosystem is crucial for numerous modern security applications.The Elliptic Curve Cryptosystem(ECC)has performance and resource-saving advantages compared to other types of asymmetric ciphers.However,the sequential design implementation for ECC does not satisfy the current applications’performance requirements.Therefore,several factors should be considered to boost the cryptosystem performance,including the coordinate system,the scalar multiplication algo-rithm,and the elliptic curve form.The tripling-oriented(3DIK)form is imple-mented in this work due to its minimal computational complexity compared to other elliptic curves forms.This experimental study explores the factors playing an important role in ECC performance to determine the best combi-nation that leads to developing high-speed ECC.The proposed cryptosystem uses parallel software implementation to speed up ECC performance.To our knowledge,previous studies have no similar software implementation for 3DIK ECC.Supported by using parallel design,projective coordinates,and a fast scalar multiplication algorithm,the proposed 3DIK ECC improved the speed of the encryption process compared with other counterparts and the usual sequential implementation.The highest performance level for 3DIK ECC was achieved when it was implemented using the Non-Adjacent Form algorithm and homogenous projection.Compared to the costly hardware implementations,the proposed software implementation is cost effective and can be easily adapted to other environments.In addition,the power con-sumption of the proposed ECC is analyzed and compared with other known cryptosystems.thus,the current study presents a detailed overview of the design and implementation of 3DIK ECC.

Resonant interactions among two-dimensional nonlinear localized waves and lump molecules for the(2+1)-dimensional elliptic Toda equation

The(2+1)-dimensional elliptic Toda equation is a high-dimensional generalization of the Toda lattice and a semidiscrete Kadomtsev–Petviashvili I equation.This paper focuses on investigating the resonant interactions between two breathers,a breather/lump and line solitons as well as lump molecules for the(2+1)-dimensional elliptic Toda equation.Based on the N-soliton solution,we obtain the hybrid solutions consisting of line solitons,breathers and lumps.Through the asymptotic analysis of these hybrid solutions,we derive the phase shifts of the breather,lump and line solitons before and after the interaction between a breather/lump and line solitons.By making the phase shifts infinite,we obtain the resonant solution of two breathers and the resonant solutions of a breather/lump and line solitons.Through the asymptotic analysis of these resonant solutions,we demonstrate that the resonant interactions exhibit the fusion,fission,time-localized breather and rogue lump phenomena.Utilizing the velocity resonance method,we obtain lump–soliton,lump–breather,lump–soliton–breather and lump–breather–breather molecules.The above works have not been reported in the(2+1)-dimensional discrete nonlinear wave equations.

Experimental Investigation of Material Removal in Elliptical Vibration Cutting of Cortical Bone

To benefit tissue removal and postoperative rehabilitation,increased efficiency and accuracy and reduced operating force are strongly required in the osteotomy.A novel elliptical vibration cutting(EVC)has been introduced for bone cutting compared with conventional cutting(CC)in this paper.With the assistance of high-speed microscope imaging and the dynamometer,the material removals of cortical bone and their cutting forces from two cutting regimes were recorded and analysed comprehensively,which clearly demonstrated the chip morphology improvement and the average cutting force reduction in the EVC process.It also revealed that the elliptical vibration of the cutting tool could promote fracture propagation along the shear direction.These new findings will be of important theoretical and practical values to apply the innovative EVC process to the surgical procedures of the osteotomy.

A Secure Hardware Implementation for Elliptic Curve Digital Signature Algorithm

Since the end of the 1990s,cryptosystems implemented on smart cards have had to deal with two main categories of attacks:side-channel attacks and fault injection attacks.Countermeasures have been developed and validated against these two types of attacks,taking into account a well-defined attacker model.This work focuses on small vulnerabilities and countermeasures related to the Elliptic Curve Digital Signature Algorithm(ECDSA)algorithm.The work done in this paper focuses on protecting the ECDSA algorithm against fault-injection attacks.More precisely,we are interested in the countermeasures of scalar multiplication in the body of the elliptic curves to protect against attacks concerning only a few bits of secret may be sufficient to recover the private key.ECDSA can be implemented in different ways,in software or via dedicated hardware or a mix of both.Many different architectures are therefore possible to implement an ECDSA-based system.For this reason,this work focuses mainly on the hardware implementation of the digital signature ECDSA.In addition,the proposed ECDSA architecture with and without fault detection for the scalar multiplication have been implemented on Xilinxfield programmable gate arrays(FPGA)platform(Virtex-5).Our implementation results have been compared and discussed.Our area,frequency,area overhead and frequency degradation have been compared and it is shown that the proposed architecture of ECDSA with fault detection for the scalar multiplication allows a trade-off between the hardware overhead and the security of the ECDSA.

Smart Grid Communication Under Elliptic Curve Cryptography

Smart Grids(SGs)are introduced as a solution for standard power dis-tribution.The significant capabilities of smart grids help to monitor consumer behaviors and power systems.However,the delay-sensitive network faces numer-ous challenges in which security and privacy gain more attention.Threats to trans-mitted messages,control over smart grid information and user privacy are the major concerns in smart grid security.Providing secure communication between the service provider and the user is the only possible solution for these security issues.So,this research work presents an efficient mutual authentication and key agreement protocol for smart grid communication using elliptic curve crypto-graphy which is robust against security threats.A trust authority module is intro-duced in the security model apart from the user and service provider for authentication.The proposed approach performance is verified based on different security features,communication costs,and computation costs.The comparative analysis of experimental results demonstrates that the proposed authentication model attains better performance than existing state of art of techniques.

Elliptically polarized high-order harmonic generation in nitrogen molecules with cross-linearly polarized two-color laser fields

Circularly and elliptically polarized high-order harmonics have unique advantages when used in studying the chiral and magnetic features of matter.Here,we studied the polarization properties of high-order harmonics generated from alignment nitrogen molecules driven by cross-linearly polarized two-color laser fields.Through adjusting various laser parameters and targets,such as the relative phase,the crossing angle,the intensity ratio of the driving fields,and the molecular alignment angle,we obtained highly elliptically polarized high-order harmonics with the same helicity in a wide spectral range.This provides a possible effective way to generate elliptically polarized attosecond pulses.Finally,we showed the probability of controlling the spectral range of elliptically polarized harmonics.

Generation of elliptical airy vortex beams based on all-dielectric metasurface

Elliptical airy vortex beams(EAVBs) can spontaneously form easily identifiable topological charge focal spots. They are used for topological charge detection of vortex beams because they have the abruptly autofocusing properties of circular airy vortex beams and exhibit unique propagation characteristics. We study the use of the dynamic phase and Pancharatnam–Berry phase principles for generation and modulation of EAVBs by designing complex-amplitude metasurface and phase-only metasurface, at an operating wavelength of 1500 nm. It is found that the focusing pattern of EAVBs in the autofocusing plane splits into |m| + 1 tilted bright spots from the original ring, and the tilted direction is related to the sign of the topological charge number m. Due to the advantages of ultra-thin, ultra-light, and small size of the metasurface, our designed metasurface device has potential applications in improving the channel capacity based on orbital angular momentum communication, information coding, and particle capture compared to spatial light modulation systems that generate EAVBs.