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

Dominik Pantůček dominik.pantucek at trustica.cz
Sat Mar 24 13:00:26 PDT 2018

Hello everyone again,

I hope this does not bother anyone, but I'd like to present you next
doubling. For those following my earlier discussion with Brad, you
should find the explanation how the rational points over given finite
field form - more specifically, from which complete set of curves are
they actually produced. The general idea is for any EC in simple
Weierstrass form satisfying y^2=x^3+ax+b we define a function
f(x,y)=y^2-x^3-ax-b and the original curve is then specified as
f(x,y)=0. The infinite set of all curves from which some cross the
rational points of given finite field GF(p) is then given by the
equation f(x+mp,y+np)+kp=0 \forall m,n,k\in Z

The visualization[2] may look messy or you may see it immediately - the
torus rotation timing was set in sync with showing the basic f(x,y)=0
variant when the origin [0,0] is facing the viewer.

As always, I'd love to hear any feedback regarding the visualizations or
the writing style. My longer-term goal is to finish simple Weierstrass
form and continue with isogeny to Montgomery form and showing the same
operations on Montgomery form - ideally both over R and GF(p) and also
in X:Z coordinates. But that is still some way to go...

Cheers,
Dominik

[1] https://trustica.cz/en/2018/03/22/elliptic-curves-point-doubling/
[2] https://youtu.be/jfQEOHiwE0k