In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K 2, the Cartesian product of K with itself.

† Elliptic curves with points in Fp are flnite groups. † Elliptic Curve Discrete Logarithm Prob-lem (ECDLP) is the discrete logarithm problem for the group of points on an elliptic curve over a flnite fleld. † The best known algorithm to solve the ECDLP is exponential, which is why elliptic curve groups are used for cryptography.

4 天之前 · Elliptic curves are curves defined by a certain type of cubic equation in two variables. The set of rational solutions to this equation has an extremely interesting structure, including a group law. The theory of elliptic curves was essential in Andrew Wiles' proof …

2025年1月31日 · Informally, an elliptic curve is a type of cubic curve whose solutions are confined to a region of space that is topologically equivalent to a torus. The Weierstrass elliptic function P(z;g_2,g_3) describes how to get from this torus to the algebraic form of an elliptic curve.

What is an elliptic curve? The equation x 2 a2 + y b2 = 1 de nes an ellipse. An ellipse, like all conic sections, is a curve of genus 0. It is not an elliptic curve. Elliptic curves have genus 1. The area of this ellipse is ˇab. What is its circumference?

Elliptic curves as plane cubics Weierstrass determined the field of meromorphic functions that are doubly periodic with respect to a given lattice. His work led to a description of the corresponding elliptic curve as a cubic curve in the complex projective plane CP2. Recall that CP2 is the space of complex lines through

De nition 4. For Ka eld, an elliptic curve is a nonsingular cubic curve of genus 1, or, equivalently, is the set of solutions over Kto the following y2 = ax3 + bx2 + cx+ d where a6= 0 and the polynomial in xdoes not have a multiple root. Below are two examples of cubic curves that are not elliptic curves, the rst being y 2= x3 + x and the

We will return to the geometric development of elliptic curves. However, we will begin with the complex analyic picture in which an elliptic curve will, somewhat suprisingly, naturally arise.

2017年9月19日 · Elliptic curves over the field of complex numbers. An elliptic curve over ${\mathbb C}$ is a compact Riemann surface of genus 1, and vice versa. The group structure turns $X$ into a complex Lie group, which is a one-dimensional complex torus ${\mathbb C}/\Lambda$, where $\Lambda$ is a lattice in the complex plane ${\mathbb C}$.

One can use elliptic curves to factor integers, although probably not RSA moduli. There is a probabilistic algorithm for proving primality that uses elliptic curves. An ellipticcurve group maybe used directlyin cryptographic algorithms in many of the same waysthemultiplicativegroupofintegersmod-ulo p can be used. In these applications, the