Elliptic curve
Elliptic curve
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 K2, the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions (x, y) for:
"For about 1500 years, from the time of Diophantus to Newton, elliptic curves were known only as curves defined by certain cubic equations. This put them just a step beyond the conic sections, and some of their geometric and arithmetic properties can in fact be viewed as generalisations of properties of conics."
"The equations of conic sections involve at most the square of x and y, never the third or higher powers. In contrast, the equation of an elliptic curve has an x^3-term; a typical example is y^2 = x^3 - x."
"We now know that equations of degree 3 are in some kind of border area between equations of lower degree (easy) and equations of higher degree (very hard)."
