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New elliptic curve breaks 18-year-old record

quantamagazine.org

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Re: New elliptic curve breaks 18-year-old record

#41

Post by acer4666 »

fsckboy wrote: post #32
lisper wrote: post #7

Earlier quoted context omitted.

[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…

> Solving that would win you a Fields medal it would not win me a Fields medal: ageism, it's only for under 40s.

Youd probably get the Abel prize (which has a significantly larger cash prize)

Re: New elliptic curve breaks 18-year-old record

#42

Post by ClassyJacket »

jrvieira wrote: post #23
unnouinceput wrote: post #9

Earlier quoted context omitted.

>...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.

you're worried that they'll explain 3rd degree polynomials with a leftist bias?

I mean, look at all the insane places leftists have shoehorned gender crap into lately. I wouldn't put it past them.

Re: New elliptic curve breaks 18-year-old record

#43

Post by smellybigbelly »

ur-whale wrote: post #35

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 ?

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.

Re: New elliptic curve breaks 18-year-old record

#44

Post by QuesnayJr »

ur-whale wrote: post #35

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 ?

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.)

Re: New elliptic curve breaks 18-year-old record

#45

Post by marton78 »

fermigier wrote: post #8
wslh wrote: post #4

Earlier quoted context omitted.

Just saw this, congratulations! Would you mind giving an ELI5 explanation for a wider audience?

Well, the basics, oversimplified, are this: - In general, elliptic curves are solutions of P(x, y) = 0 where P is a polynomial of degree 3 in two variables. "Points" on the curve are solutions of this equation. - 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), t…

This is a fantastic explanation, thank you very much!

Re: New elliptic curve breaks 18-year-old record

#46

Post by tenuousemphasis »

ClassyJacket wrote: post #42
jrvieira wrote: post #23

Earlier quoted context omitted.

you're worried that they'll explain 3rd degree polynomials with a leftist bias?

I mean, look at all the insane places leftists have shoehorned gender crap into lately. I wouldn't put it past them.

Ah yes, leftists shoehorning gender into checks notes elliptic curve math discussion.

Re: New elliptic curve breaks 18-year-old record

#47

Post by throw_a_grenade »

tenuousemphasis wrote: post #46
ClassyJacket wrote: post #42

Earlier quoted context omitted.

I mean, look at all the insane places leftists have shoehorned gender crap into lately. I wouldn't put it past them.

Ah yes, leftists shoehorning gender into checks notes elliptic curve math discussion.

[flagged]

Re: New elliptic curve breaks 18-year-old record

#48

Post by griffzhowl »

jokoon wrote: post #20

I wonder if 3blue1brown could explain this a bit better

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=PL8yHsr3EFj53j51FG6wCb...

Re: New elliptic curve breaks 18-year-old record

#50

Post by AlotOfReading »

fsckboy wrote: post #33
AlotOfReading wrote: post #29

Earlier quoted context omitted.

NIST has deprecated it for government use after 2030, just not today.

well, deprecated does not mean it stops, just that it comes with warnings, so not 2030 either.

It puts it in the same category as triple DES.
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