Quote from: remotemass on March 23, 2013, 11:42:42 AM

Quote from: prezbo on March 23, 2013, 11:24:45 AM

Elliptic curves form a group under addition. The private key is an integer, while the public key is a point on the group, so if n is a private key, and G is the "base" point, then n*G = G+G+...+G would be the public key corresponding to n. There is no bound on how large n can be, since a group is closed under it's operator.

Can the G that you mention, the "base" point, be expressed with (X,Y) coordinates?

I barely understand Elliptic Curves with real numbers, but I suppose ECDSA uses Elliptic Curves over Finite Fileds (or is it over Prime Field?!).

I'm still trying to figure if we use a curve or a set of curves.

Can you, give me an idea of how secp256k1 curve and its relevant points look like in a cartesian way? Or it does not work like that at all?! Huh

Certainly. secp256k1 is defined over one, predefined elliptic curve. The base point is also predetermined, usually a group generator. The group operator (+) is defined in a purely geometric way as shown in this picture

,
here P1+P2 = P3.

You can find some more information on secp256k1 in this pdf. Note that the order of base point G is exactly the number you specified in your OP.