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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

#31

Post by intuitionist »

wbl wrote: post #19
oasisaimlessly wrote: post #16

Earlier 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.

In this case there is a negation missing. If a line intersects three points we have A+B+C=0. To get the group law you have to negate a point.

Of course for this to make sense you have to have a notion of 0, which is traditionally taken to be the point at infinity (so negation is negating the y-coordinate). It’s been a while since my algebraic geometry classes but IIRC this is just a useful convention.

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

#32

Post by fsckboy »

lisper wrote: post #7
wslh wrote: post #4

Earlier quoted context omitted.

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

[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.

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

#33

Post by fsckboy »

AlotOfReading wrote: post #29
CarpaDorada wrote: post #26

Earlier quoted context omitted.

2048bit RSA is not deprecated...

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.

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

#34

Post by fsckboy »

vessenes wrote: post #17
Noumenon72 wrote: post #6

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.

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?

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

#36

Post by btreecat »

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.

I strongly doubt that's the primary factor preventing you winning.

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

#37

Post by 41599799 »

UI_at_80x24 wrote: post #15
fermigier wrote: post #3

This discovery was already commented a few months ago: 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…

As a professional and expert I would love to hear your thoughts and opinions on the use of elliptic curve crypto with SSH. There was a concern (unsure of the validity) that NSA/NIST had compromised the algorithm used and ECC was unfit for 'secure' communication. 2048bit RSA has been deprecated since that declaration and while 4096bit is still viable, the smaller key-size of ed25519 is appealing.

Ever since the DUAL_EC_DBRG backdoor[1], trust in cryptographic algorithms set by NIST has been reduced.

In the case of ECC curves, the NIST curves rely on a number of highly specific but unexplained constants. More info about the safety and security of curves can be found at https://safecurves.cr.yp.to/

For now, Curve25519 is considered a good bet.

[1] https://en.wikipedia.org/wiki/NIST_SP_800-90A#Backdoor_in_Du...

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

#38

Post by arunc »

syncsynchalt wrote: post #25

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 . 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.

The animations makes it easier to comprehend indeed. Thanks!

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

#40

Post by throwaway81523 »

syncsynchalt wrote: post #25

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 . 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.

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.
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