--- Log opened Wed Dec 01 00:00:48 2021
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06:14 < roconnor> The isomorphism exists because y^2 = x^3 + 7 if and only if (y*t^3)^2 = (x*t^2)^3 + 7*t^6 (for t != 0).
06:23 < roconnor> well that an the fact that (x,y) -> (x*t^2, y*t^3) is a linear transformation that preserves verticle lines.  Linear transformations map lines to lines, and lines (and verticle lines) is what defines point addition on elliptic curves.
06:24 < roconnor> thus the transformation also preserves point addition.
06:42 < sipa> interesting, i had not thought about it that way
06:42 < roconnor> sipa: We had this discussion when I wrote the PR that robot-dreams is reviewing.
06:44 < sipa> Then apparently i have thought about this this way, but just can't remember.
06:51 < roconnor> [Tuesday, June 22, 2021] [1:35:31 PM EDT] <roconnor> I see, it is a linear transformation, so it maps lines to lines, and lines are what define point addition.
06:51 < roconnor> [Tuesday, June 22, 2021] [1:36:07 PM EDT] <sipa> i hadn't thought about it that way, but indeed!
06:51 < roconnor> *lol* Now I don't even believe you when you first said you hadn't thought about it that way before.
06:57 < sipa> LOL
07:25 < robot-dreams> Nice :)
07:25 < robot-dreams> I just checked algebraically that for a curve with y^2 = x^3 + Ax + B where A = 0, the group law still holds under (x, y) -> (x*t^2, y*t^3).
07:27 < robot-dreams> As for the geometric argument, what would happen if we consider a different linear transformation like (x, y) -> (x*t^2, y*t^2)? Naively I would think this still maps lines to lines, and also preserves vertical lines, but I don't think this is an isomorphism.
07:27 < sipa> robot-dreams: it maps lines to lines, but doesn't map curve points to curve points :)
07:28 < sipa> that's obviously a constraint for it to be valid curve morphism
07:28 < robot-dreams> Yes that makes sense, haha
07:29 < robot-dreams> OK, that's a very nice geometric argument!
13:27 < roconnor> I don't understand why I don't see robot-dreams's comments on https://github.com/bitcoin-core/secp256k1/pull/900#pullrequestreview-820493747
13:29 < robot-dreams> I deleted it because I accidentally left it on an outdated part, I'll repost it in a bit when I finish reviewing the whole PR
13:35 < roconnor> ah okay
13:36 < roconnor> BTW, pr #900 is quite a bit more dubious than #899.  So if your conclusion ends up being: this is stupid, I won't be offended.
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--- Log closed Thu Dec 02 00:00:49 2021