Talk:Elliptic curve

By adding a point "at infinity", we obtain the projective version of this curve; every straight line intersects this curve in three points (if the line is tangent to the curve at a point, then that point is counted twice). It is then possible to introduce a group operation on the curve with the following property: if a straight line intersects the curve at the points P, Q and R, then P + Q + R = 0 in the group.

But, for example, the x-axis intersects the curve y² = x³ - x + 1 at only 1 point (call it P); if we consider the other two points as being infinity, this seems to require that P + inf + inf = 0. However, the x-axis is not alone in this property; several lines parallel to the x-axis also intersect at only 1 point, if we select one of these and call the point of intersection Q, then Q + inf + inf = P + inf + inf = 0 implies P = Q. Bzzzt!!!! Is the group operation restricted to those lines which actually intersect at at least two points (where tangency counts as 2 points)? Chas zzz brown 02:00 Jan 22, 2003 (UTC)

My statement above was wrong: it only that way only for algebraically closed base fields. You're right: if the fields isn't algebraically closed, we only consider lines that are tangent to the curve or intersect it in two points. AxelBoldt 17:23 Jan 22, 2003 (UTC)

I'd like to see a bit more about the group aspect of elliptic curves over finite fields (with an eye towards ECC); especially more describing an algebraic (as opposed to geometric) approach to calculating P + Q (see [1] (http://www.certicom.com/research/online) for a nice explanation). Should that be done at this article, at the article for elliptic curve cryptography, or at a new article? Chas zzz brown 20:04 Feb 7, 2003 (UTC)

I think it can still fit here. Can't we give the general group law formula which works for all base fields? AxelBoldt 07:37 Feb 9, 2003 (UTC)

Added the restriction that K not have characteristic 3 - observe that in the diagrams in the article, there is a point P with the property P + P + P = 3P = 0 (in the y² = x³ - x - 1 example, it's the intersection of the y-axis and the curve). Chas zzz brown 19:57 Feb 12, 2003 (UTC)