# Counting Points On Elliptic Curves

Counting Points On Elliptic Curves

An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve. There have been several approaches to do so, and the algorithms devised have proved to be useful tools in the study of various fields such as number theory, and more recently in cryptography and Digital Signature Authentication (See elliptic curve cryptography and elliptic curve DSA). While in number theory they have important consequences in the solving of Diophantine equations, with respect to cryptography, they enable us to make effective use of the difficulty of the discrete logarithm problem (DLP) for the group, of elliptic curves over a finite field, where q = pk and p is a prime. The DLP, as it has come to be known, is a widely used approach to Public key cryptography, and the difficulty in solving this problem determines the level of security of the cryptosystem. This article covers algorithms to count points on elliptic curves over fields of large characteristic, in particular p > 3. For curves over fields of small characteristic more efficient algorithms based on p-adic methods exist.

Read more about Counting Points On Elliptic CurvesApproaches To Counting Points On Elliptic Curves, Naive Approach, Baby-step Giant-step, Schoof's Algorithm, Schoof–Elkies–Atkin Algorithm, See Also, Bibliography

### Other articles related to "counting points on elliptic curves, elliptic curves, counting points on":

Counting Points On Elliptic Curves - Bibliography
... Smart Elliptic Curves in Cryptography, Cambridge University Press, 1999 ... Enge Elliptic Curves and their Applications to Cryptography An Introduction ... Musiker Schoof's Algorithm for Counting Points on ...

### Famous quotes containing the words curves, points and/or counting:

One way to do it might be by making the scenery penetrate the automobile. A polished black sedan was a good subject, especially if parked at the intersection of a tree-bordered street and one of those heavyish spring skies whose bloated gray clouds and amoeba-shaped blotches of blue seem more physical than the reticent elms and effusive pavement. Now break the body of the car into separate curves and panels; then put it together in terms of reflections.