As a typical software engineer, I'm just curious to know if my curve ed25519 key is safe and for how long. :)
I’m not sure how the discovery of new, exotic elliptic curves has security implications on curves used for cyber security.
New elliptic curve breaks 18-year-old record
51–55 of 55 posts
Re: New elliptic curve breaks 18-year-old record
#52Re: New elliptic curve breaks 18-year-old record
#53Earlier quoted context omitted.
I too love Quanta. It's funded by an extremely wealthy math guy as a public service; they have the luxury of affording excellent journalists who all seem to me to have graduate degrees in the area they cover, but have not lost the power of communication in exchange. Just a very nice gift to the world.
> It's funded he died in 2024, did he make arrangements to keep funding it or endow it?
Renaissance was characterized by, not least, it's fucking fantastic perspicacity and foresight. I'm sure if he wanted it to continue then it will. And I'd take the lack of fundraising banners on the site to be good news. A number of years ago I reached out and offered to add some funding for a paper magazine, and they were like "we're good bro, thanks" -- that's when I looked up who was actually financing it, and I was like "...yep, makes sense". Still wish they had a paper magazine, though. I wanted something to leave around for my then-teenagers to read.
Re: New elliptic curve breaks 18-year-old record
#54Earlier quoted context omitted.
> - If you intersect an elliptic curve with a straight line, you end up with a polynomial in one variable, of degree 3 (in general). Since a polynomial of degree 3 has 3 solutions (in the appropriate context), this means that if you have two points on the curve, and you draw a line through these two points, there is a third aligned with them which belongs to the curve. So we have an operation on the curve, which to e…
If you going to contradict someone, be specific about it. What is your " the group operation" and how is this not it? A given mathematical object can have more than one group operation defined for it.
"The" group operation is the standard one for elliptic curves, that the article discusses. "The" here means "the one we're talking about" and also "the ones that mathematicians mean if they're not specifying otherwise". The vast majority of possible group structures on a given set are not particularly noteworthy and aren't under discussion.
I didn't bother specifying the correct one because I didn't think it was important enough to the point to be worth the effort to describe. But for completeness, if we use # to denote the operation described by fermigier above, then the group law a+b is given by a+b=(a#b)#O, where O here denotes the vertical point at infinity.
...and at this point we get into a whole can of worms, because that's right this whole time this was actually all taking place in the projective plane, not the affine plane, a complication the article didn't get into, meaning there's this hidden point you didn't know about. And actually we could have used any point as the basepoint and gotten a group law (although they all end up being isomorphic!), which is why technically an elliptic curve is (despite the name) defined to be not just a curve of genus 1, but rather a curve of genus 1 together with a choice of basepoint; the use of the vertical point at infinity as basepoint is just the default convention when you're doing things in this equational way rather than more abstractly, etc... and now you see why I didn't want to get into it.
Also, # isn't a group law because it doesn't satisfy the requirements of one. For instance, there's no identity (maybe barring some weird degenerate cases? I'm not an algebraic geometer so I'm not too familiar with the details here). For # to have an identity P, all of the curve's tangent lines would have to pass through P (and P would have to be an inflection point). Again, not an algebraic geometer but I think that's impossible! (It certainly isn't typical.) I also don't think # is associative but I don't really want to check that right now. Regardless it definitely is not typically a group structure.
Re: New elliptic curve breaks 18-year-old record
#55Earlier quoted context omitted.
If you going to contradict someone, be specific about it. What is your " the group operation" and how is this not it? A given mathematical object can have more than one group operation defined for it.
"The" group operation is the standard one for elliptic curves, that the article discusses. "The" here means "the one we're talking about" and also "the ones that mathematicians mean if they're not specifying otherwise". The vast majority of possible group structures on a given set are not particularly noteworthy and aren't under discussion. I didn't bother specifying the correct one because I didn't think it was impo…
Actually, one addition here: I said that there are lots of possible group laws on an elliptic curve, but IINM, the ones described here are the only ones that are algebraic (and as for what exactly that means, I'll definitely skip going into that because I would probably get it wrong :P ).