Quote from: remotemass on March 23, 2013, 11:17:14 AM

Quote from: markm on March 23, 2013, 11:04:17 AM

I didn't know there were any that cannot be used. Hmm now you have me wondering why some cannot be used.

I don't know enough about elliptic curves to make you more clear on that.
But note that the public key is made of 65 bytes. The first byte is always 0x04, I think, and the next pair of 32 bytes correspond to the (x,y) coordinates of a point in the curve.
So the reason must be that because not all points (x,y) can be used, so that not all public keys can be used and therefore not all private keys.
Just my thoughts about it. I actually have not much clues about Elliptic Curves stuff. This is as much as I get of it.

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.

Having said that, even though every number *can* be used as a private key, the set of numbers {n, n+R, n+2R, ...} will have the same corresponding public keys, where R is the order of the base element G (and usually also the order of the group itself).