# [curves] ECC introduction: point addition on simple Weierstrass form

Dominik Pantůček dominik.pantucek at trustica.cz
Tue Apr 3 06:51:12 PDT 2018

(This is actually a re-post, I accidentally addressed Brad only, sorry
for that).

On 03/24/2018 11:45 PM, Brad Klee wrote:
> Hi Dominik,
>
> Some of these graphs are looking cool, but I'm not sure why your
> worried about integers (m,n). If you include (m,n) then every point
> will actually intersect infinitely many curves. The slopes of these
> curves at the point of intersection are different per curve, but
> reduce to one particular line over a finite field. So (m,n) could
> really be considered extraneous, and parameter k quite sufficient.
you have to include at least m\in{-1,0}, n\in{-1,0} - you can see this
in[1] (I didn't want to draw my own, so I am reusing yours here).
>
> Another thing to think about is intersection geometries combining
> elliptic curves with a non-linear addition curve. For example, Edwards
> curves have a hyperbolic addition rule. In this case the quadratic
> hyperbolas are more similar to cubic elliptic curves than to lines.
> They don't wrap the torus nicely, again require a big set to account
> for all intersections.
>
> Drawings of linear intersections over finite fields are now becoming
> standard fare. We have seen continuous interpretation of hyperbolic
> intersection in a number of references. I have yet to see hyperbolic
> intersection over finite field, so perhaps this would be an
> interesting direction to go in terms of depiction?
Now you have unveiled my secret long-term plan. Yes, I want to slowly go
through all the basics using simple Weierstrass form, then continue with
Montgomery form and finally show the hyperbolic intersections of twisted
Edwards over finite field (there are few things to be solved in that
case, but I am looking forward those challenges :) ).

Speaking of slowly going through all the basics, I produced another
article[2] and accompanying video[3] - this time about point at
infinity. Truth is, I am not satisfied with this one and I am thinking
about creating another one showing the actual intersections at point at
infinity on the projective plane.

Cheers,
Dominik

[1] https://ptpb.pw/AfTT.png
[2] https://trustica.cz/en/2018/03/29/elliptic-curves-point-at-infinity/
[3] https://youtu.be/WnBEZ0qNdV0