Wikipedia: Elliptic curve

Difference (from prior author revision) (major diff, minor diff)

Changed: 5c5

If we add a point at infinity, an elliptic curve forms an abelian group (as long as a certain constraint is met on the values of a and b, which ensures that the polynomial x³ + ax + b does not have a double zero). Each point on the curve is an element of the group, and a geometrical construction allows us to define addition of points in a consistent manner.

If we add a point at infinity, an elliptic curve forms an abelian group (as long as a certain constraint is met on the values of a and b, which ensures that the polynomial x³ + ax + b does not have a multiple zero and the curve does not have a singularity). Each point on the curve is an element of the group, and a geometrical construction allows us to define addition of points in a consistent manner.

Changed: 9c9

Elliptic curves on finite fields are used in some cryptographic applications. (see Elliptic curve cryptography)

Elliptic curves on finite fields are used in some cryptographic applications, see elliptic curve cryptography.

The elliptic curve equation is: y² = x³ + ax + b

An elliptic curve over some field F is the set of pairs of elements (x,y) in F satisfying the above equation, for a given a and b in F.

If we add a point at infinity, an elliptic curve forms an abelian group (as long as a certain constraint is met on the values of a and b, which ensures that the polynomial x³ + ax + b does not have a multiple zero and the curve does not have a singularity). Each point on the curve is an element of the group, and a geometrical construction allows us to define addition of points in a consistent manner.

Specifically, to add distinct points we draw a straight line through them, and find the unique third point which this line meets. The reflection of this line in the x-axis is the sum of the two original points.

Elliptic curves on finite fields are used in some cryptographic applications, see elliptic curve cryptography.