ОПТИМИЗАЦИОННЫЕ ЗАДАЧИ ВЫЧИСЛЕНИЙ В ЭЛЛИПТИЧЕСКИХ КРИВЫХ

This paper investigates the distribution of prime elements over finite fields through the interplay between elliptic curves and sheaf-theoretic methods. We define new algebraic-geometric invariants of primes by interpreting them as Frobenius traces on elliptic curves defined over , and further characterize their behavior using cohomological data of associated étale sheaves. By analyzing the point counts of elliptic curves modulo , we derive a structural correspondence between the statistical distribution of primes and the arithmetic of elliptic curves, including supersingular loci and trace formulas. The use of sheaf cohomology allows us to refine classical density results and propose new congruence-based filtrations on prime sets. This fusion of arithmetic geometry and finite field analysis offers a deeper geometric interpretation of prime phenomena in positive characteristic settings.

Computing l-isogenies between elliptic curves defined over a finite field Fq of small characteristic is of some importance for the computation of the cardinalities of elliptic curves using Schoof-Atkin-Elkies method. In a previous publication [3] we showed that this could be achieved in O(l3+e) multiplications in \(\mathbb{F}_q \)using formal groups. This method has been implemented by Lercier and Morain who obtained spectacular results in this direction [7, 8]. Nevertheless, the use of formal groups seems to be a serious deterent both for man and machine to perform such a work. More recently Lercier proposed an algorithm specific for characteristic 2 that has the same assymptotic complexity but is faster by some significative constant factor. In this paper we propose a general algorithm which does not use formal groups. Instead we take advantage of the elementary Galois properties of the p-torsion. This algorithm has the same complexity as the previous ones if we don't use fast multiplication techniques. But, contrary to the previous methods, it allows the use of fast multiplication for polynomials and then turns out to run in O(l2+e) multiplications in the field \(\mathbb{F}_q \). Our algorithm has also the advantage that it is made exclusively of very classical routines in polynomial and elliptic curve arithmetic. Also one may expect that the implementation of this method should require less work than the previous ones thus bringing new people to this kind of calculation.

Book Review: The arithmetic of elliptic curves

In Mathematics, especially number theory, one often comes across problems which arise naturally and are easy to pose, but whose solutions require very sophisticated methods. What is known as ‘The Congruent Number Problem’ is one such. Its statement is very simple and the problem dates back to antiquity, but it was only recently that a breakthrough was made, thanks to current developments in the Arithmetic of elliptic curves, an area of intense research in number theory.

Arithmetic of Elliptic Curves

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The cryptosystem of elliptic curve has gained more and more researchers' attention from home and abroad in recent years as it owns a short key and a fine security. This paper elaborates the definition of elliptic curve which is used to establish cryptosystem and also makes some explanation to the realization of the Add Point Operation on the elliptic curve together with the fast Add Point Operation of its corresponding coordinate system. In addition, the security of the elliptic curve cryptosystem is also analyzed starting from the theory of the elliptic curve in this paper. And the issue of production of the safe elliptic curve is discussed. Finally, the elliptic curve cryptosystem has aroused great attention of the researchers as the public key system of the next generation. The production arithmetic of the safe elliptic curve, the security analysis of the elliptic curve cryptosystem together with the fast arithmetic on the elliptic curve will be the hotspot and the issues that are remained to be solved in its future development.

A time domain objective function based on the normalized Euclidean distance (NED) matrix of general short pulse network is proposed and is employed to optimize a short pulse power divider (PD). The NED matrix is obtained directly from the time domain port signals, which describes the performance of the short pulse networks comprehensively and compactly. Frobenius norm of the difference between the non-optimal and the optimal NED matrix is used as the objective function in the optimization. A short pulse PD is optimized using the proposed NED matrix based objective function and then fabricated. The feasibility and effectiveness of the NED matrix based objective function are validated by the opti mization procedure and the measured results of the short pulse PD.

The objective of this paper consists in the development, implementation, testing and validation of a new mathematical model based on the eXtended Finite Element Method (XFEM) and Genetic Algorithm (GA) specially designed for modeling the mobile material discontinuities with direct application in optimization of electric structures in 3D space. This new mathematical concept eliminates the conventional use of discrete elements and provides an efficient, stable, accurate and fast calculation scheme, removing the finite elements network re-meshing at each step of optimization.

We discuss the computational application of Heegner points to the study of elliptic curves over Q, concentrating on the curves E d : Dy2 = x3 − x arising in the “congruent number” problem. We begin by briefly reviewing the cyclotomic construction of units in real quadratic number fields, which is analogous in many ways to the Heegner-point approach to the arithmetic of elliptic curves, and allows us to introduce some of the key ideas in a more familiar setting. We then quote the key results and conjectures that we shall need concerning elliptic curves and Heegner points, and show how they yield practical algorithms for finding rational points on E d and other properties of such curves. We conclude with a report on more recent work along similar lines on the elliptic curves x3 + y3 = A.

The integration and application of a new multi-objective tabu search optimization algorithm for Fluid Structure Interaction (FSI) problems are presented. The aim is to enhance the computational design process for real world applications and to achieve higher performance of the whole system for the four considered objectives. The described system combines the optimizer with a well established FSI solver which is based on the fully implicit, monolithic formuFlation of the problem in the Arbitrary Lagrangian-Eulerian FEM approach. The proposed solver resolves the proposed uid-structure interaction benchmark which describes the self-induced elastic deformation of a beam attached to a cylinder in laminar channel ow. The optimized ow characteristics of the aforementioned geometrical arrangement illustrate the performance of the system in two dimensions. Special emphasis is given to the analysis of the simulation package, which is of high accuracy and is the core of application. The design process identifies the best combination of ow features for optimal system behavior and the most important objectives. In addition, the presented methodology has the potential to run in parallel, which will significantly speed-up the elapsed time. Finite Element Method (FEM), Fluid-Structure Interaction (FSI), Multi-Ojective Tabu search (MOTS2). Copyright © 2013 Tech Science Press.

Efficiency is one of the important criteria for the design and evaluation of the construction quality. Power losses in planetary transmissions consist of losses in the gear contact, losses in bearings, losses in sealings, losses due to oil viscosity etc.The calculation of the gear transmission efficiency is generally confined to losses depending on friction on tooth sides, i.e. on the calculation of contact power losses. Nevertheless, it is not suitable for using efficiency as only criterion for optimal solution choice, but as a one of the set of criteria. The objective of this paper, is the application of multicriteria optimization to the planetary gear trains with efficiency as one of the criteria. Apart the determination of the set of the Pareto optimal solutions, the weighted coefficient method for choosing an optimal solution from this set is included in order to prioritize particular criterion.

This article presents an application of the Accelerated Particle Swarm Optimization based approach to minimize the cost of grounding grids in high voltage substations while maintaining the legitimacy of safety necessities. The cost effectiveness and the constraints are adapted to formulate optimized solutions for the grounding grid planning problem. The proposed approach attempts to calculate the optimal values of the grounding grid parameters while respecting a set of constraints within pre-set acceptable limits. Several test cases of grounding grid planning, some of them for real projects, are demonstrated to validate the proposed method. The superiority and efficacy of Accelerated Particle Swarm Optimization have been acknowledged in terms of grounding grid cost minimization, stable performance and short CPU computational time.

We present new algorithms computing 3P and $$2P+Q$$2P+Q by removing the same part of numerators and denominators of their formulas, given two points P and Q on elliptic curves defined over prime fields and binary fields in affine coordinates. Our algorithms save one or two field multiplications compared with ones presented by Ciet, Joye, Lauter, and Montgomery. Since $$2P+Q$$2P+Q takes $$\frac{1}{3}$$13 proportion, 28.5i¾?% proportion, and 25.8i¾?% proportion of all point operations by non-adjacent form, binary/ternary approach and tree approach to compute scalar multiplications respectively, 3P occupies 42.9i¾?% proportion and 33.4i¾?% proportion of all point operations by binary/ternary approach and tree approach to compute scalar multiplications respectively, utilizing our new formulas of $$2P+Q$$2P+Q and 3P, scalar multiplications by using non-adjacent form, binary/ternary approach and tree approach are improved.

Keeping networks security has never been such an imperative task as today. In this paper, by combining particle swarm optimization and clustering algorithm, a new detection method, Intrusion Detection based on Unsupervised Clustering and PSO algorithm (IDCPSO), is proposed. Particle swarm optimization algorithm is used to optimize the clustering results and obtain the optimal detection result. IDCPSO needs unlabeled data for training and automatically establish clusters and detect intruders by labeling normal and abnormal groups. Computer experiment results with KDD cup show that this algorithm is effective for intrusion detection.