30 ms·

New elliptic curve breaks 18-year-old record

173 points by calstad 2y ago

perdomon 2y ago
I didn't understand anything in that article, but I'm very excited for the record-breakers and other mathematicians involved. Good job, ya'll.
- commandlinefan 2y ago
  I understood a fair bit of it but only because I've been studying elliptic curves for a while - Quanta does a good job of straddling the line between informing and educating, but they usually err on the side of presenting results rather than proving or explaining them.
  - unnouinceput 2y ago
    >...but they usually err on the side of presenting results rather than proving or explaining them And that's exactly what I like about it. They are a news site, hence they present the news. If the news presenters start to chime in you get what you see at CNN / Fox etc, and that's called propaganda, not news. I want news.
    - jrvieira 2y ago
      you're worried that they'll explain 3rd degree polynomials with a leftist bias?
      - defrost 2y ago
        The overwhelming majority of their publication on organics has an unmistakable bias toward D- sugars ...
      - ClassyJacket 2y ago
        I mean, look at all the insane places leftists have shoehorned gender crap into lately. I wouldn't put it past them.
        tenuousemphasis 2y ago
        Ah yes, leftists shoehorning gender into checks notes elliptic curve math discussion.
        throw_a_grenade 2y ago
        [flagged]
        jrvieira 2y ago
        introducing the concept of mathsplaining
fermigier 2y ago
This discovery was already commented a few months ago: https://news.ycombinator.com/item?id=41475177 https://news.ycombinator.com/item?id=41475177 As I wrote in the comments, I was the record holder, twice, in the 90s: Fermigier, Stéfane - Un exemple de courbe elliptique définie sur Q de rang ≥19. (French) [An example of an elliptic curve defined over Q with rank ≥19] C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 6, 719–722. Fermigier, Stéfane - Une courbe elliptique définie sur Q de rang ≥22. (French) [An elliptic curve defined over Q of rank ≥22] Acta Arith. 82 (1997), no. 4, 359–363.
- wslh 2y ago
  Just saw this, congratulations! Would you mind giving an ELI5 explanation for a wider audience?
  - lisper 2y ago
    [Not the OP but I think I understand it well enough to take a whack at an ELI5.] Elliptic curves are a particular kind of cubic equation, exactly like the quadratic equations you studied in junior high algebra, except with one term being raised to the third power instead of just squared (and a few other conditions). It turns out that these equations have vastly more complicated behavior than quadratics and give rise to a whole host of problems that mathematicians are still working to solve. One of the interesting problems arises when you ask: what are the solutions to the equation if we restrict ourselves only to rational numbers? It turns out that rational solutions to elliptic curve equations can be grouped into families of solutions where each member of the family can be derived from other members by linear operations (addition and multiplication by a constant). The number of such families of solutions is called the rank of the equation. (Note: it's actually a little more complicated than that, but that's the gist of it. See [1] if you want the details.) It is observed empirically (by solving lots of elliptic curve equations) that the rank tends to be small. Indeed, the elliptic curve that made the news did so because it has a rank of 29, the largest rank currently known. But no one knows if this is the biggest possible (almost certainly not) or if there is an upper bound on the possible rank of an elliptic curve. Solving that would win you a Fields medal. (Note: there are results on the upper bound of the average rank of families of elliptic curves [2] but that is not the same as an absolute upper bound.) --- [1]https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve [2] https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve#Upper_bounds_for_the_average_rank https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve#Uppe...
Noumenon72 2y ago
I was going to ask if the math articles from Quanta magazine are a "Matt Levine" situation where only one person can write so well, but I see only six articles by this author there, so maybe it's an editor doing the magic. All I know is this makes math so accessible and that's not easy.
- vessenes 2y ago
  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.
  - neom 2y ago
    I was curious about the rich math guy so I looked it up, leaving this here for the next curious person: https://en.wikipedia.org/wiki/Jim_Simons https://en.wikipedia.org/wiki/Jim_Simons :)
    - DFHippie 2y ago
      > Simons shunned the limelight and rarely gave interviews, citing Benjamin the Donkey in Animal Farm for explanation: "'God gave me a tail to keep off the flies. But I'd rather have had no tail and no flies.' That's kind of the way I feel about publicity." I'm glad to read about billionaires with non-poisonous personalities. I'd prefer a world where no individual held such relative power, but next best is a world in which the dreadful oligarchs have foils to balance them out slightly.
  - fsckboy 2y ago
    > It's funded he died in 2024, did he make arrangements to keep funding it or endow it?
    - vessenes 2y ago
      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.
jokoon 2y ago
I wonder if 3blue1brown could explain this a bit better
- jrvieira 2y ago
  first thing i did when i read "3rd degree polynomial" was search "elliptic curve 3b1b"
- griffzhowl 2y ago
  If you like videos, there are some excellent ones by Richard Borcherds, in very different style to 3b1b but by a Field's medalist This is his algebraic geometry playlist. The whole course is directed at graduate level but the first few videos are very accessible https://www.youtube.com/playlist?list=PL8yHsr3EFj53j51FG6wCbQKjBgpjKa5PX https://www.youtube.com/playlist?list=PL8yHsr3EFj53j51FG6wCb...
syncsynchalt 2y ago
If like me you're interested in the basics of elliptic curves, point addition, and the abelian groups that result then check the first third of my page at https://curves.xargs.org https://curves.xargs.org. It only gets you half way to an understanding of this article but might leave you less mystified. You can also continue through the rest of that page to see how we use this math in cryptography, such as in key exchange.
- arunc 2y ago
  The animations makes it easier to comprehend indeed. Thanks!
- throwaway81523 2y ago
  That sounds great and I'll try to look. I liked Neal Koblitz's book "A Course in Number Theory and Cryptography" a while back, another resource that might be of interest.
ur-whale 2y ago
One thing I've always wondered about elliptic curves is why everything is so centered on degree 3 two variable polynomials. Aren't there rich structures to be explored for curves of degree >3 ? Or is 3 really special ?
- smellybigbelly 2y ago
  I think part of the reason why 3 is special is because you get a lot of bang for your buck. Order 3 is a low order polynomial that is relatively easy to analyse, but already gives tremendous mathematical properties. For example, the points of elliptic curves form groups. The operation of combining the points is described in the article (draw a straight line through two points and mirror in x-axis). That means that all the theorems that are proven for Groups, are also true for elliptic curves. But I think there are many more exciting properties Amateur here (just studying abstract algebra for hobby). I’m also very curious for more reasons.
- QuesnayJr 2y ago
  You can get some higher degree examples (y squared = a degree 4 polynomial, for example), but degree 3 is special. An arbitrary polynomial of degree 4 and higher lack a rich structure (as far as we can tell). You can try to get around it by embedding the curve in a higher dimensional object, but it doesn't get you as far. (This is the idea behind hyperelliptic curve cryptography, for example.)
arunc 2y ago
As a typical software engineer, I'm just curious to know if my curve ed25519 key is safe and for how long. :)
- smellybigbelly 2y ago
  I’m not sure how the discovery of new, exotic elliptic curves has security implications on curves used for cyber security.
deleted 2y ago
[deleted]

173 points by calstad 2y ago

I didn't understand anything in that article, but I'm very excited for the record-breakers and other mathematicians involved. Good job, ya'll.

commandlinefan 2y ago
I understood a fair bit of it but only because I've been studying elliptic curves for a while - Quanta does a good job of straddling the line between informing and educating, but they usually err on the side of presenting results rather than proving or explaining them.
- unnouinceput 2y ago
  >...but they usually err on the side of presenting results rather than proving or explaining them And that's exactly what I like about it. They are a news site, hence they present the news. If the news presenters start to chime in you get what you see at CNN / Fox etc, and that's called propaganda, not news. I want news.
  - jrvieira 2y ago
    you're worried that they'll explain 3rd degree polynomials with a leftist bias?
    - defrost 2y ago
      The overwhelming majority of their publication on organics has an unmistakable bias toward D- sugars ...
    - ClassyJacket 2y ago
      I mean, look at all the insane places leftists have shoehorned gender crap into lately. I wouldn't put it past them.
      - tenuousemphasis 2y ago
        Ah yes, leftists shoehorning gender into checks notes elliptic curve math discussion.
        throw_a_grenade 2y ago
        [flagged]
        jrvieira 2y ago
        introducing the concept of mathsplaining

fermigier 2y ago

This discovery was already commented a few months ago: https://news.ycombinator.com/item?id=41475177 https://news.ycombinator.com/item?id=41475177 As I wrote in the comments, I was the record holder, twice, in the 90s: Fermigier, Stéfane - Un exemple de courbe elliptique définie sur Q de rang ≥19. (French) [An example of an elliptic curve defined over Q with rank ≥19] C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 6, 719–722. Fermigier, Stéfane - Une courbe elliptique définie sur Q de rang ≥22. (French) [An elliptic curve defined over Q of rank ≥22] Acta Arith. 82 (1997), no. 4, 359–363.

wslh 2y ago
Just saw this, congratulations! Would you mind giving an ELI5 explanation for a wider audience?
- lisper 2y ago
  [Not the OP but I think I understand it well enough to take a whack at an ELI5.] Elliptic curves are a particular kind of cubic equation, exactly like the quadratic equations you studied in junior high algebra, except with one term being raised to the third power instead of just squared (and a few other conditions). It turns out that these equations have vastly more complicated behavior than quadratics and give rise to a whole host of problems that mathematicians are still working to solve. One of the interesting problems arises when you ask: what are the solutions to the equation if we restrict ourselves only to rational numbers? It turns out that rational solutions to elliptic curve equations can be grouped into families of solutions where each member of the family can be derived from other members by linear operations (addition and multiplication by a constant). The number of such families of solutions is called the rank of the equation. (Note: it's actually a little more complicated than that, but that's the gist of it. See [1] if you want the details.) It is observed empirically (by solving lots of elliptic curve equations) that the rank tends to be small. Indeed, the elliptic curve that made the news did so because it has a rank of 29, the largest rank currently known. But no one knows if this is the biggest possible (almost certainly not) or if there is an upper bound on the possible rank of an elliptic curve. Solving that would win you a Fields medal. (Note: there are results on the upper bound of the average rank of families of elliptic curves [2] but that is not the same as an absolute upper bound.) --- [1]https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve [2] https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve#Upper_bounds_for_the_average_rank https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve#Uppe...

Noumenon72 2y ago

I was going to ask if the math articles from Quanta magazine are a "Matt Levine" situation where only one person can write so well, but I see only six articles by this author there, so maybe it's an editor doing the magic. All I know is this makes math so accessible and that's not easy.

vessenes 2y ago
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.
- neom 2y ago
  I was curious about the rich math guy so I looked it up, leaving this here for the next curious person: https://en.wikipedia.org/wiki/Jim_Simons https://en.wikipedia.org/wiki/Jim_Simons :)
  - DFHippie 2y ago
    > Simons shunned the limelight and rarely gave interviews, citing Benjamin the Donkey in Animal Farm for explanation: "'God gave me a tail to keep off the flies. But I'd rather have had no tail and no flies.' That's kind of the way I feel about publicity." I'm glad to read about billionaires with non-poisonous personalities. I'd prefer a world where no individual held such relative power, but next best is a world in which the dreadful oligarchs have foils to balance them out slightly.
- fsckboy 2y ago
  > It's funded he died in 2024, did he make arrangements to keep funding it or endow it?
  - vessenes 2y ago
    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.

jokoon 2y ago

I wonder if 3blue1brown could explain this a bit better

jrvieira 2y ago
first thing i did when i read "3rd degree polynomial" was search "elliptic curve 3b1b"
griffzhowl 2y ago
If you like videos, there are some excellent ones by Richard Borcherds, in very different style to 3b1b but by a Field's medalist This is his algebraic geometry playlist. The whole course is directed at graduate level but the first few videos are very accessible https://www.youtube.com/playlist?list=PL8yHsr3EFj53j51FG6wCbQKjBgpjKa5PX https://www.youtube.com/playlist?list=PL8yHsr3EFj53j51FG6wCb...

syncsynchalt 2y ago

If like me you're interested in the basics of elliptic curves, point addition, and the abelian groups that result then check the first third of my page at https://curves.xargs.org https://curves.xargs.org. It only gets you half way to an understanding of this article but might leave you less mystified. You can also continue through the rest of that page to see how we use this math in cryptography, such as in key exchange.

arunc 2y ago
The animations makes it easier to comprehend indeed. Thanks!
throwaway81523 2y ago
That sounds great and I'll try to look. I liked Neal Koblitz's book "A Course in Number Theory and Cryptography" a while back, another resource that might be of interest.

ur-whale 2y ago

One thing I've always wondered about elliptic curves is why everything is so centered on degree 3 two variable polynomials. Aren't there rich structures to be explored for curves of degree >3 ? Or is 3 really special ?

smellybigbelly 2y ago
I think part of the reason why 3 is special is because you get a lot of bang for your buck. Order 3 is a low order polynomial that is relatively easy to analyse, but already gives tremendous mathematical properties. For example, the points of elliptic curves form groups. The operation of combining the points is described in the article (draw a straight line through two points and mirror in x-axis). That means that all the theorems that are proven for Groups, are also true for elliptic curves. But I think there are many more exciting properties Amateur here (just studying abstract algebra for hobby). I’m also very curious for more reasons.
QuesnayJr 2y ago
You can get some higher degree examples (y squared = a degree 4 polynomial, for example), but degree 3 is special. An arbitrary polynomial of degree 4 and higher lack a rich structure (as far as we can tell). You can try to get around it by embedding the curve in a higher dimensional object, but it doesn't get you as far. (This is the idea behind hyperelliptic curve cryptography, for example.)

arunc 2y ago

As a typical software engineer, I'm just curious to know if my curve ed25519 key is safe and for how long. :)

smellybigbelly 2y ago
I’m not sure how the discovery of new, exotic elliptic curves has security implications on curves used for cyber security.

deleted 2y ago

[deleted]