Associativity

Associativity on elliptic curves refers to a fundamental property of the addition operation defined on points of the curve (Washington, L. C., 2008, pp20-35, Section 2.4).

Associativity: for any three points P, Q, and R on the elliptic curve, the following holds true:

(P + Q) + R = P + (Q + R)

No matter how you associate these points (i.e. group with parentheses) when adding them, the result is the same. This property is crucial for the Group structure of the elliptic curve, which is foundational for their applications in cryptography and advanced number theory.

To assist with visualising what this means, for an elliptic curve - take the following steps, related to the image above (also view the animation below, with these steps in mind):

1. Identify Points: Select three points P, Q, and R on an elliptic curve.

   1. in the image above, you can see points P (orange), Q (blue), and R (green) marked

2. Add Points: Compute P + Q and Q + R.

   1. in the image above:

      1. find the points P and Q. Follow the line through these points, and find the point at which it crosses the curve at another point, -(P+Q), the small green dot in between P and Q. From that point follow the vertical line, down, to (P+Q).

      2. find the points Q and R. The line through these two points crosses the elliptic curve at -(Q+R), the small black dot in between Q and R. From that point follow the vertical line, up, to (Q+R).

3. Complete Addition: Add R to the result of P + Q and add P to the result of Q + R.

   1. in the image above:

      1. the line that connects (P+Q) to R, crosses the elliptic curve at one more point, -((P+Q) + R). From this point follow the vertical line, up, to ((P+Q)+R).

      2. the line that connects (Q+R) to P, crosses the elliptic curve at one more point, -(P+(Q+R)). From this point, follow the vertical line, up, to (P+(Q+R)).

4. Check Equality: Verify that both methods give the same result on the curve.

   1. in the image above, note that ((P+Q)+R) is the same point as (P+(Q+R)). By definition (if our assertions regarding the additive inverse hold true), this means that the point -((P+Q)+R) is the same as the point -(P+(Q+R)). This indicates that (at least for the values chosen) associativity holds true visually, and apparently geometrically.

This is by no means a proof, of any kind. It simply indicates, visually and loosely, that the rule appears to hold true for the values chosen.

This associative property is algebraically verified through geometric operations of point addition on elliptic curves, which typically involve defining lines through points, using these to find intersections with the curve, and investigating the properties of the resulting cubic functions.

A rigorous proof of which will not be undertaken here (see Washington, L. C., 2008, pp20-35, Section 2.4).

Rather, as a further exploration of subject, we supply an animation of varying values for P, Q and R and the relative results for +/-(P+Q), +/-(Q+R), +/-((P+Q) + R), +/-(P+(Q+R)).

NOTE: if you can't see the video clearly enough, on your screen, try hitting the "full screen" icon (the broken square, in the bottom-left icons on the video player).

Again, not as any kind of proof - this animation explores, illustrates and visualises the concept that: no matter the original choice of P, Q, and R... the resultant values for ((P+Q) + R), and (P+(Q+R)) are the same.