Nonlinearities in Elliptic Curve Authentication

Elliptic curves are beautiful mathematical objects that again and again appear in the most surprising places. Their history certainly originates at least in ancient Greece, whereas the study of arithmetic properties of elliptic curves as objects in algebra, geometry, and number theory traces back to the nineteenth century. Curiously, the earliest use of the term “elliptic curve” in the literature seems to have been by James Thomson in 1727 in “A Poem sacred to the Memory of Sir Isaac Newton”: “He, first of Men, with awful Wing pursu’d the Comet tro’ the long Elliptic Curve.” In 1985, Koblitz and Miller independently proposed to use elliptic curves in cryptography which can only be described as a magnificent and practical application of elliptic curves. This paper intends to mostly present a low-brow introduction of elliptic curves and their use in real-world applications of public-key cryptography.

This article is concerned with the numerical solution of three-dimensional elliptic partial differential equations (PDEs) using the trivariate spectral collocation approach based on the Kronecker tensor product. By using the quasilinearization method, the nonlinear elliptic PDEs are simplified to a linear system of algebraic equations that can be discretized using the spectral collocation method. The method is based on approximating the solutions using the triple Lagrange interpolating polynomials, which interpolate the unknown functions at selected Chebyshev–Gauss–Lobatto (CGL) grid points. The CGL points are preferred to ensure simplicity in the conversion of continuous derivatives to discrete derivatives at the collocation points. The collocation process is carried out at the interior points to reduce the size of differentiation matrices. This work is aimed at verifying that the algorithm based on the method is simple and easily implemented in any scientific software to produce more accurate and stable results. The effectiveness and spectral accuracy of the numerical algorithm is checked through the determination and analysis of errors, condition numbers and computational time for various classes of single or system of elliptic PDEs including those with singular behavior. The communicated results indicate that the proposed method is more accurate, stable and effective for solving elliptic PDEs. This good accuracy becomes possible with the usage of few grid points and less memory requirements for numerical computation.

We consider the solution of elliptic partial differential equations with random input data. In particular, we are interested in the mean and the correlation of the solution of these partial differential equations. Once the correlation is available, the variance can be computed from its diagonal. If the dependence of the solution on the random input data is linear and the mean of the input data is available, the mean of the solution can be computed whenever the corresponding partial differential equation can be solved. When the correlation of the input data is available, the correlation of the solution is given as the solution of a higher dimensional partial differential equation in the product domain. A Galerkin discretization yields a matrix equation with a typically densely populated right-hand side. This and the higher dimension of the problem makes the equation prohibitively expensive to solve. Since existing approaches to these correlation equations are known to struggle for roughly correlated input data, we use hierarchical matrices and their corresponding arithmetic to represent and solve the matrix equation. The regularity assumptions required for hierarchical matrices can be justified on sufficiently smooth domains. The feasibility of the approach is illustrated for finite element and boundary element discretizations and several specialities of hierarchical matrices for finite element discretizations are discussed. An almost linear scaling with respect to the dimension of the used finite element spaces is verified. The nonlinear dependence of the solution on the random input data is exemplarily discussed for the case of random domains. Extending previous perturbation approaches, we can linearize the problem and can compute the mean and the correlation up to third order accuracy in almost linear time. We discuss that these spaces can become even fourth order accurate under certain circumstances. A full convergence analysis is presented, which shows that higher order boundary elements are required for the solution of the problem. Therefore, we develop a fast multipole method for higher order boundary element methods on parametric surfaces.

Aiming at the heat conduction equation, the heat conduction model is a very typical example of the partial differential equation. This paper focuses on the development and preliminary knowledge of the partial differential equation, the establishment of the heat conduction model and the solution of the heat conduction model. The partial differential equation with fractional Laplace operator is a kind of typical fractional partial differential equation, which has important applications in the fields of science and engineering. Fractional Laplace operator is a kind of nonlocal quasi-differential operator, which is the infinitesimal generator of L é vy steady-state process. It is essentially different from the classical Laplace operator, which leads to the disappearance of some classical properties, which generally brings difficulties to the study of this kind of problems. It is very difficult to solve the explicit solution of partial differential equation with fractional Laplace operator. In order to improve the accuracy and speed of solving partial differential equation, a high precision method for solving partial differential equation based on deep learning network is proposed. A multilayer radial basis function neural network is constructed and its structural layers are determined. By introducing subnetworks into each layer of the multilayer radial basis function neural network, a composite multilayer radial basis function neural network is constructed to improve the real function approximation performance and operation accuracy of the multilayer radial basis function neural network. The high-precision compound multi-layer radial basis function neural network is used to solve the partial differential equation. By giving a specific example of solving the partial differential equation, the solution effect of the method is tested. The results show that the method has very high solution accuracy under the four-layer network structure, which can improve about 1.5 orders of magnitude compared with the first layer. Under different number of training samples, it has higher solution accuracy and speed, and the comprehensive performance is superior. Therefore, it is a practical but challenging work to study the existence of solutions of partial differential equations with fractional Laplace operators. This paper mainly describes the research progress and trends of the existence of solutions of several classes of partial differential equations with fractional Laplace operators, including some of the work done by the author in this field in recent years.

The positive integer points and integral points of elliptic curves are very important in the theory of number and arithmetic algebra, it has a wide range of applications in cryptography and other fields. There are some results of positive integer points of elliptic curve y2 = x3 + ax + b, a, b ∈ Z In 1987, D. Zagier submit the question of the integer points on y2 = x3 − 27x + 62, it count a great deal to the study of the arithmetic properties of elliptic curves.In 2009, Zhu H L and Chen J H solved the problem of the integer points on y2 = x3 − 27x + 62 by using algebraic number theory and P-adic analysis method.In 2010, By using the elementary method, Wu H M obtain all the integral points of elliptic curves y2 = x3 − 27x − 62.In 2015, Li Y Z and Cui B J solved the problem of the integer points on y2 = x3 − 21x − 90 By using the elementary method.In 2016, Guo J solved the problem of the integer points on y2 = x3 + 27x + 62 by using the elementary method.In 2017, Guo J proved that y2 = x3 − 21x + 90 has no integer points by using the elementary method.Up to now, there is no relevant conclusions on the integral points of elliptic curves y2 = x3 + 3x − 14, which is the subject of this paper.By using congruence and Legendre Symbol, it can be proved that elliptic curve y2 = x3 + 3x − 14 has only one integer point: (x, y) = (2, 0).

Assessing the effectiveness of artificial neural networks on problems related to elliptic curve cryptography

In 1967 Roger Penrose introduced what is now called twistor geometry (Penrose 1967). This geometry is an amplification of the geometry of lines in projective space studied extensively by Felix Klein and others in the late 19th century. Penrose made the fundamental observation that this classical geometry could be used in conjunction with a Radon transform style integral geometry to provide rather complete sets of solutions of numerous partial differential equations, both linear and nonlinear, which arise naturally in theoretical physics. The Radon transform, which involves integration over straight lines in R 3, was used by F. John to generate solutions of hyperbolic partial differential equations in 1942. Penrose integrates over complex projective lines, real 2-spheres, and the data of the integration is complex-analytic functions of 3 complex variables. This is analogous to (and indeed related to) Weierstrass' representation of solutions of the minimal surface equation in terms of triples of holomorphic functions of a single complex variable which satisfy an algebraic relation. In this paper we discuss the origins of the differential equations which are amiable to twistor-geometric study from the realms of theoretical physics. Twistor geometry is most suitable for physical theories which are relativistically invariant, including Maxwell's description of electromagnetic phenomena from the 19th century (Maxwell's equations), Einstein's theory of gravitation (the equations of general relativity), and the contemporary descriptions of elementary particles (quantum field theory). Twistor geometry interacts in a nontrivial manner with the mathematical models involved in all of the above physical theories. In particular, it has led to new solutions of some of the nonlinear partial differential equations involved (Einstein's equations, Yang-Mills equations, equations of a magnetic monopole). We will describe some recent progress on twistorgeometric representations of solutions of classical field equations. There are two aspects to this. First, there is the problem of translating a given system of differential equations into twistor-geometric language. This has been done successfully in many different cases, and will be described in Section 3. Once a given problem has been transferred to a twistor-geometric context, it becomes, in general, a specific problem involving algebraic geometry, several complex variables, and algebraic topology involving holomorphic data on complex manifolds. The only partial differential equations remaining in the twistor context are the CauchyRiemann equations characterizing holomorphic functions. In effect, the partial differential equation has been transformed into a geometric problem. This is analogous to the Fourier transform transforming a partial differential equation with constant coefficients into an algebraic problem, where differentiation is replaced by the (algebraic) symbol of the differential operator in question. The second aspect of the recent progress on the twistor-geometric representation of the solutions of the classical field equations concerns itself with solving the geometric problem in question and transforming back to space-time (or Euclidean space), generating (sometimes new) solutions of the classical field equations.

This paper explores the solutions of certain fractional partial differential equations using two methods; the first method involves separation of variables, which is a common technique for solving partial differential equations. However, since many equations cannot be separated in this way, the tensor product of Banach spaces method is applied to find the atomic solutions. To solve the resulting ordinary differential equations, the reproducing Kernel Hilbert space method is used to find numerical solutions, which are then used to find the numerical solution of the partial differential equation. The residual errors indicate that this method is effective and powerful. In summary, this paper presents a study on the solutions of certain fractional partial differential equations using two methods and demonstrates the effectiveness of these methods in finding numerical solutions.

Several research papers experimentally validated the derived theoretical solution of the partial differential equation (PDE) of Transient Land Drainage (TLD) using laboratory sand tanks, Hele-Shaw models, resistance networks, and field data. In this research, the main objective is to use an Arduino microcontroller linked with the LabVIEW software to derive experimentally-by using an RC analog circuit connected to the Arduino microcontroller - a solution to Kirchhoff’s partial differential equation describing the unsteady electrical flow through the RC circuit. The RC circuit electrically simulates a non-dimensional virtual aquifer for which an experimental Fourier sine series solution in terms of the analog components is to be derived and used for comparison with the theoretical Fourier sine series solution of the transient partial differential equation of land drainage. The derived Fourier sine series solution of the partial differential equation governing the unsteady electrical flow through the RC analog is written in a non-dimensional form to enable its general application to the case study of this research. Isochrones of the water table drawdown based on those derived from the analog solution are similar in trend and behavior to those of the theoretical solution, indicating that both Fourier solutions are physically and mathematically identical.

We investigate the existence and Borel summability of formal power series in one variable with holomorphic coefficients, solutions of nonlinear partial differential equations with shrinkings having non-Kowalewski type in the complex domain. Sufficient conditions are given in terms of the shape of the functional partial differential equations and initial conditions. The existence and asymptotic behavior of local sectorial holomorphic solutions of these nonlinear functional partial differential equations are obtained as a by-product. We apply the results to the study of asymptotic the behavior of solutions of some advanced-argument partial differential equations in a neighborhood of infinity.

This article presents a software and laboratory suite for studying the mathematical foundations and practical applications of elliptic curve cryptography (ECC). The suite is implemented in Python using the PyQt6 framework and the sympy library for cryptographic computations. The program provides an interactive interface for entering elliptic curve parameters, visualizing points on the curve, constructing Cayley tables for point addition, and checking group properties. Key features of the suite include the implementation of the Tonelli–Shanks algorithm for finding absolute square roots, the ability to work with curves over finite fields of large order, and a bilingual interface (Russian/English). The developed suite can be used in educational settings to teach the fundamentals of elliptic curve cryptography.

This upper-level undergraduate textbook provides a modern view of algebra with an eye to new applications that have arisen in recent years. A rigorous introduction to basic number theory, rings, fields, polynomial theory, groups, algebraic geometry and elliptic curves prepares students for exploring their practical applications related to storing, securing, retrieving and communicating information in the electronic world. It will serve as a textbook for an undergraduate course in algebra with a strong emphasis on applications. The book offers a brief introduction to elementary number theory as well as a fairly complete discussion of major algebraic systems (such as rings, fields, and groups) with a view of their use in bar coding, public key cryptosystems, error-correcting codes, counting techniques, and elliptic key cryptography. This is the only entry level text for algebraic systems that includes an extensive introduction to elliptic curves, a topic that has leaped to prominence due to its importance in the solution of Fermat’s Last Theorem and its incorporation into the rapidly expanding applications of elliptic curve cryptography in smart cards. Computer science students will appreciate the strong emphasis on the theory of polynomials, algebraic geometry and Groebner bases. The combination of a rigorous introduction to abstract algebra with a thorough coverage of its applications makes this book truly unique.

In this work, a new graph has been defined as a main point to design a new version of an asymmetric encryption scheme. This graph is formed based on the scalar multiplication operation on elliptic curve defined over a prime field, which is called an elliptic scalar multiplication (ESM) graph. Some mathematical facts to show the ESM graph is a complete graph are proved theoretically. On the complete elliptic scalar multiplication (CESM) graph, an asymmetric encryption algorithm has been implemented. Fast computations on the proposed algorithm are Z obtained through the matrices representation and determination of the minimum spanning tree (MST) to compute the ciphertext. More secure computations with the CESM graphic encryption algorithm in comparison with the previous asymmetric encryption algorithms. So, the CESM graphic asymmetric encryption algorithm is considered as a new sight for elliptic curve cryptographic usages.

This paper presents a comprehensive study of Partial Differential Equations (PDEs), beginning with a general definition and classification into three primary types: parabolic, hyperbolic, and elliptic equations. We focus on the development of finite difference schemes for numerically solving specific PDEs and Ordinary Differential Equations (ODEs), demonstrating their effectiveness through selected examples implemented in Matlab. A general study of some finite difference schemes for the numerical resolution of specific PDEs and ODEs has been conducted. Well-selected numerical examples have been proposed, and numerical implementation of different examples has been carried out using MATLAB. Our study concluded that the numerical methods of finite difference schemes for numerical accuracy of definite PDEs and ODEs have had good effectiveness and this has been verified by the numerical examples that have been implemented using MATLAB software.

Interface solutions of partial differential equations with point singularity