Cayley table for group of points on elliptic curve

[ephedyn]

Homework Statement

Construct the Cayley table for the group of points on the elliptic curve $$y^2=x^3+x+4$$ over the field F7 of integers modulo seven.

The Attempt at a Solution

So I'm familiar with the constructing a Cayley table, but I've not encountered the group of points on the elliptic curve before.

1. Am I right that over the field of integers modulo seven, there are only 4 group elements: the point at infinity, (0,2), (2,0) and (5,1)? (I took x=0,1,2,... found only these integer values of y.)

2. If I understand this correctly, the composition law is to construct a line intersecting any of these two points, and find a third point of intersection between this line and the curve. The problem is, if I take any of the 2 points above, I don't seem to obtain other group elements and maintain closure. Did I misunderstand something in step 1?

 

[micromass]

You're missing a few values in (1). For example, (0,-2) (or (0,5) equivalently). Remember that most numbers have two square roots! So I would recheck your solutions in (1).

Also, I don't think your group law is correct. To my knowledge, to add points P and Q, we draw the line between P and Q and take the third intersection point. Call this R. Then draw the line between R and the point on infinity, then the third intersection point is P+Q.