Hello everyone again, I hope this does not bother anyone, but I'd like to present you next article[1] in my introductory series about ECC. This time about point 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 _______________________________________________ Curves mailing list Curves@moderncrypto.org https://moderncrypto.org/mailman/listinfo/curves