The modular curve X1(N) parametrizes elliptic curves with a point of order N . For N ≤ 50 we obtain plane models for X1(N) that have been optimized for fast computation, and provide explicit birational maps to transform a point on our model of X1(N) to an elliptic curve. Over a finite field Fq, these allow us to quickly construct elliptic curves containing a point of order N , and can accelerat...

Elliptic curves are some specific type of curves known as hyper elliptic curves. Compared to the integer factorization problem(IFP) based systems, using elliptic curve based cryptography will significantly decrease key size of the encryption. Therefore, application of this type of cryptography in systems that need high security and smaller key size has found great attention. Hyperelliptic curve...

In order to find elliptic curves over Q with large rank, J.-F. Mestre [1991] constructed an infinite family of elliptic curves with rank at least 12. Then K. Nagao [1994] and S. Kihara [1997b] found infinite subfamilies of rank 13 and 14, respectively. By specialization, elliptic curves of rank at least 21 [Nagao and Kouya 1994], 22 [Fermigier 1997], and 23 [Martin and McMillen 1997] have also ...

An elliptic curve over a field K is given by an equation of the form y2 = x3 + Ax+B. There is a natural way to add any two points on the curve to get a third point, and therefore the set of points of the curve with coordinates in K form a group, denoted E(K). Elliptic curves have long fascinated mathematicians, as they can be approached from many angles, including complex analysis, number theor...

Constructing non-supersingular elliptic curves for pairing-based cryptosystems have attracted much attention in recent years. The best previous technique builds curves with ρ = lg(q)/lg(r) ≈ 1 (k = 12) and ρ = lg(q)/lg(r) ≈ 1.25 (k = 24). When k > 12, most of the previous works address the question by representing r(x) as a cyclotomic polynomial. In this paper, we propose a new method to find m...

Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...

We look at arithmetic progressions on elliptic curves known as Huff curves. By an arithmetic progression on an elliptic curve, we mean that either the x or y-coordinates of a sequence of rational points on the curve form an arithmetic progression. Previous work has found arithmetic progressions on Weierstrass curves, quartic curves, Edwards curves, and genus 2 curves. We find an infinite number...

2 Elliptic Curves and Maps Between Them 2 2.1 The Group Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Weierstrass Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Maps Between Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Dual Isogenies . . . . . . . . . . . . . . ....

Using a multidimensional large sieve inequality, we obtain a bound for the mean square error in the Chebotarev theorem for division fields of elliptic curves that is as strong as what is implied by the Generalized Riemann Hypothesis. As an application we prove a theorem to the effect that, according to height, almost all elliptic curves are Serre curves, where a Serre curve is an elliptic curve...