«
»

Definition of Elliptic Curves

By ellipticcurves

I think it is about time we defined what an elliptic curve is. It is a nonsingular cubic curve C with a distinguished point, usually denoted by \mathcal O. We say that the elliptic curve is defined over a field K if the curve C is defined over K and {\mathcal O} \in C(K).

It turns out that for every field L/K, the set C(L) can be endowed with a structure of an abelian group as follows.

The line through any two points P and Q of C(L) intersects C in a third point, which we denote by P*Q. If P=Q, we use the line tangent to C at this point. As we saw above, P*Q \in C(L). Notice that this operation is commutative, though not associative. Also notice the following “cancellation rule”:

(Q * P) * P = Q.

Now we define

P + Q := (P*Q)*{\mathcal O}

and

-P := P*({\mathcal O} * {\mathcal O}).

  1. Observe that this addition operation is commutative.

  2. Also

    P + {\mathcal O} = (P * {\mathcal O}) * {\mathcal O} = P,

    so {\mathcal O} is the identity for the addition operation.

  3. In addition,

    (-P) + P = ((P*({\mathcal O} * {\mathcal O})) * P)*{\mathcal O} = ({\mathcal O} * {\mathcal O}) * {\mathcal O} = {\mathcal O}.

    This means that -P is the additive inverse of P.

  4. The only property we need to check is the associativity of the addition operation. This is not trivial, so we postpone this until later (unspecified) date.
If {\mathcal O} is a flex point, then {\mathcal O}*{\mathcal O}={\mathcal O}, and the negation operation simplifies to -P = P * {\mathcal O}.

This entry was posted on September 19, 2008 at 9:41 pm and is filed under elliptic curves. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

Leave a Reply