Arithmetic of Elliptic Curves

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.

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Currently, a lot of standardized elliptic curve cryptographic algorithms are widely used. Due to the constantly increasing execution speed demands, implementation optimization problem become actual. Using elliptic curve arithmetic on a different information security devices require to form an mathematical programming problem of elliptic curves calculation. Solution of this problem allows to find optimal way of calculation in the current condition. This approach increases the speed of cryptographic algorithms on information security devices. The paper present one of the possible ways to optimize computations in elliptic curve arithmetic by using mathematical programming. An optimization problem is formed that describes the calculation of scalar multiplication. Operations composition and multiplication methods composition are introduced to form the corresponding optimization problem. Adduction of elliptic curve calculations to the mathematical programming problem allow to research applicability of known optimization methods to computations in elliptic curves.

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.

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.

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.

In this article, we establish a non-triviality statement for Heegner points which was conjectured by B. Mazur [10], and has subsequently been used as a working hypothesis by a few authors in the study of the arithmetic of elliptic curves.

It is known that every positive integer n can be represented as a finite sum of the form $n={\textstyle\sum} a_i 2^{i}$, where $a_i\in \{0,1,-1\}$ for all i, and no two consecutive ai's are nonzero. Such sums are called nonadjacent representations. Nonadjacent representations are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. In this paper, we investigate if other digit sets of the form {0,1,x}, where x is an integer, provide each positive integer with a nonadjacent representation. If a digit set has this property, we call it a nonadjacent digit set (NADS). We present an algorithm to determine if {0,1,x} is an NADS and, if it is, we present an algorithm to efficiently determine the nonadjacent representation of any positive integer. We also present some necessary and sufficient conditions for {0,1,x} to be an NADS. These conditions are used to exhibit infinite families of integers x such that {0,1,x} is an NADS, as well as infinite families of x such that {0,1,x} is not an NADS.