What is an elliptic curve?
Well for a start, it is not the same as an ellipse! But to be more positive: from school mathematics, you probably know the equation for a circle centred on the (a,b) of radius r, which is (x-a)^2 + (y-b)^2 = r^2 where x, y, a, b and r are real numbers. An elliptic curve is also defined by an equation, but it has the slightly more complicated form: y^2 [ + x·y ] = x^3 + a·x^2 + b Notation: · means multiplication, and ^ means raising to a power, so that y^2 means y·y and x^3 means x·x·x. The square brackets mean that the term is optional – sometimes it is there, sometimes it isn’t! Again x and y are variables, a and b are constants. However, these quantities are not necessarily real numbers, instead they may be values from any field. For cryptographic purposes we always use a “finite” field – that is x, y, a and b are chosen from a finite set of distinct values. [In fact the equation given here is not the most general possible, but it will serve for the purposes of this FAQ, and as far
An elliptic curve is not an ellipse! The reason for the name is a little more indirect. It has to do, as we shall explain shortly, with “elliptic integrals”, which arise in computing the arc length of an ellipse. But this happenstance of nomenclature isn’t too significant, since an elliptic curve has different, and much more interesting, properties as compared to an ellipse. Instead, an elliptic curve is simply the locus of points in the x-y plane that satisfy an algebraic equation of the form (with some additional minor technical conditions). This is deliberately vague as to what sort of values x and y represent. In the most elementary case, they are real numbers, in which case the elliptic curve is easily graphed in the usual Cartesian plane. But the theory is much richer when x and y may be any complex numbers (in C). And for arithmetic purposes, x and y may lie in some other field, such as the rational numbers Q or a finite field F. So an elliptic curve is an object that is easily
