Talk:Montgomery curve

I suggest replacing the "Daniel J. Bernstein, Peter Birkner, Marc Joye, Tanja Lange and Christiane Peters (2008). Twisted Edwards Curves. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-68159-5." link by http://cr.yp.to/newelliptic/twisted-20080313.pdf to avoid the paywall. The suggested link points to Bernstein's website. — Preceding unsigned comment added by 2001:4CA0:0:F230:9CBB:29C3:9C99:DCA5 (talk) 14:28, 17 July 2012 (UTC)

Equivalence with Weierstrass curves

I feel like the mapping from Montgomery- to Weierstrass-form should be

$\psi :M_{A,B}\rightarrow E(x,y)\mapsto (u,v)=\left({Bx}+{\frac {A}{3}},{By}\right),a={\frac {3-A^{2}}{3B^{2}}},b={\frac {2A^{3}-9A}{27B^{3}}}$

instead of

$\psi :M_{A,B}\rightarrow E(x,y)\mapsto (u,v)=\left({\frac {x}{B}}+{\frac {A}{3B}},{\frac {y}{B}}\right),a={\frac {3-A^{2}}{3B^{2}}},b={\frac {2A^{3}-9A}{27B^{3}}}$

See Okeya, K.; Kurumatani, H.; Sakurai, K. (2000). "Elliptic Curves with the Montgomery-Form and Their Cryptographic Applications". Public Key Cryptography: Third International Workshop on Practice and Theory in Public Key Cryptosystems. Melbourne, Victoria, Australia: Springer Berlin Heidelberg. pp. 238–257. doi:10.1007/978-3-540-46588-1_17.

Jocodes (talk) 13:04, 9 September 2016 (UTC)

Choice of parameters for diagram

The chosen parameters for the curve in the diagram result in a curve shape over the reals that requires very close attention to see the actual shape near the origin. On an initial impression it looks as though there are lines crossing at the origin, but that's not correct.

(For independent reasons I'm skeptical how much useful intuition the shape of the curve over the reals provides, since that is not what is actually used in applications. But if we're going to show that shape, it should not be misleading or require zooming in to see what is really going on.)

For example, the curve y² = x³ + 2.5x² + x shows the general shape much better in my opinion.

Daira Hopwood ⚥ (talk) 13:24, 26 November 2017 (UTC)

I fully agree with that notion. In fact I completely fell for that and needed an expert that today told me that my understanding was completely wrong and that I would have had to look much closer at the image. Can somebody please replace it soon? --Florian Weber (talk) 20:00, 15 April 2019 (UTC)

I've now replaced the image with what I hope is both correct and makes it clearer that there should not be any singularities. --Florian Weber (talk) 09:58, 28 January 2020 (UTC)

Unexplained Term In Example

This might just be my unfamiliarity with the domain, but in case it's an actual error: in "Algorithm and example", $Z_{2}$ is defined with a mysterious $a$ term that isn't defined anywhere else in the page that I can see:

$Z_{2}=4X_{1}(XX_{1}+aX_{1}+1)\,$

I think that $a$ should be $A$ instead, is that right? Given the full algorithm (without the $Z_{1}=1$ simplification) which is:

$Z_{2n}=(4X_{n}Z_{n})((X_{n}-Z_{n})^{2}+((A+2)/4)(4X_{n}Z_{n}))$

(It's also shown again as $A$ a few lines further in the example) — Preceding unsigned comment added by 2601:282:4700:C73:A049:BF47:6664:4651 (talk) 21:37, 28 April 2022 (UTC)

Incorrect sign?

In "Montgomery arithmetic" it states:

«It is possible to do some "operations" between the points of an elliptic curve: "adding" two points $P,Q$ consists of finding a third one $R$ such that $R=P+Q$;»

however it's more typical to state that $P+Q$ gives the inverse of $R$, in other words either $P+Q+R=0$ or $P+Q=-R$. With this in mind, should the sign be flipped on R in this statement? Yielding:

«It is possible to do some "operations" between the points of an elliptic curve: "adding" two points $P,Q$ consists of finding a third one $R$ such that $-R=P+Q$;» — Preceding unsigned comment added by 2601:282:4700:C73:A049:BF47:6664:4651 (talk) 02:52, 29 April 2022 (UTC)