[cc’ing the list at Andrew’s suggestion] Thanks! That is exactly the kind of explanation I was looking for. (Thanks also to Robert Ransom who also responded off-list.)
On Nov 3, 2016, at 1:54 PM, Andrew Egbert <backun...@gmail.com> wrote: > Ah- must have unsubscribed or something (feel free to post this to the list). > I can try to explain intuitively whats happening, and why the degree of the > polynomial decreases. > Imagine you have a curve of some sort in 2-dimensions, this will be an > equation with x, y (two variables). Now imagine you look at the curve in > three dimensions. > If it really is still a one-dimensional object, it will need to have 3 > variables (otherwise it will be a surface if ‘z’ is not specified). > > Resolving singularities of curves is often (not always) a similar process. > Imagine you have a curve with a ‘cusp’ which is sort of like a sharp > ‘singular’ point. > (You can google image search plane curve cusp to get an idea) > Now imagine that instead of a sharp point, you are actually looking at a > place where the curve is going ‘downwards’ in a third dimension (so in fact > it is a smooth curve). > This is sort of what’s happening. > Best, > Andrew > >> On Nov 3, 2016, at 1:48 PM, Ron Garret <r...@flownet.com> wrote: >> >> Not sure what “bad response” you’re referring to here because this is the >> only message I’ve received from you. I took a look at page 1, and I do >> understand the change of variables that transforms curve25519 into Ed25519 >> and vice-versa. It’s the more general case that I don’t yet fully >> understand. >> >> I have a working theory though: because the transformation involves a change >> of variables, the letters X and Y have completely different semantics in the >> Edwards formula than in the other forms. >> >> On Nov 3, 2016, at 1:36 PM, Andrew Egbert <backun...@gmail.com> wrote: >> >>> Sorry that was a bad response, since I missed the last sentence of your >>> post- I’ve written out the transformation on page 1 of my thesis here: >>> https://divisibility.files.wordpress.com/2016/02/thesismarch18.pdf (also >>> available at my github) >>>> On Nov 3, 2016, at 12:30 PM, Ron Garret <r...@flownet.com> wrote: >>>> >>>> >>>> On Nov 1, 2016, at 2:40 PM, Trevor Perrin <tr...@trevp.net> wrote: >>>> >>>>> It would be be great if there were better surveys on modern ECC and >>>>> engineering issues. If someone wanted to suggest a reading list / >>>>> bibliography that would be a nice contribution (but also a bunch of >>>>> work). >>>> >>>> I decided it would be a useful exercise for me to undertake to write such >>>> a survey (even if I couldn’t actually finish it), and right away I ran >>>> into a snag. I was trying to reconcile all the different forms of >>>> elliptic curve formulas commonly found in the literature, and found the >>>> following promising-looking lead on mathworld: >>>> >>>> http://mathworld.wolfram.com/EllipticCurve.html >>>> >>>> Ax^3 + Bx^2y + Cxy^2 + Dy^3 + Ex^2 + Fxy + Gy^2 + hHx + Iy + J = 0 >>>> >>>> This is consistent (AFAICT) with the definition given in section 4.4.2.a >>>> of Cohen and Frey. But then there are Edwards curves, which have a x^2y^2 >>>> term in them. How do those fit in? >>>> >>>> In fact, as I started thinking about this I realized that Edwards curves >>>> are really weird because they’re quartic and not cubic (aren’t they?) and >>>> all elliptic curves are supposed to be cubic (aren’t they?) How can a >>>> fourth-order polynomial be birationally equivalent to a third-order >>>> polynomial? >>>> >>>> I tried taking a look at some of the proofs that Edwards curves are >>>> birationally equivalent to Montgomery curves but they went way over my >>>> head. Is there a more elementary way of understanding this? >>>> >>>> Thanks, >>>> rg >>>> >>>> _______________________________________________ >>>> Curves mailing list >>>> Curves@moderncrypto.org >>>> https://moderncrypto.org/mailman/listinfo/curves >>> >> > _______________________________________________ Curves mailing list Curves@moderncrypto.org https://moderncrypto.org/mailman/listinfo/curves
