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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

#11

Post by jjice »

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…

This is a fantastic ELI5, thank you!

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

#12

Post by eddd-ddde »

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…

For the longest time I thought elliptic curves where quadratic curves.

Wouldn't it had been more accurate to name them elliptic surfaces?

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

#13

Post by lisper »

eddd-ddde wrote: post #12
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…

For the longest time I thought elliptic curves where quadratic curves. Wouldn't it had been more accurate to name them elliptic surfaces?

The name derives from the fact that they originally arose in connection with trying to determine the arc length of an ellipse. See:

https://people.math.rochester.edu/faculty/doug/mypapers/wayn...

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

#14

Post by lisper »

jjice wrote: post #11
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…

This is a fantastic ELI5, thank you!

Thanks! I try hard to produce quality technical pedagogy, so you just made my day.

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

#15

Post by UI_at_80x24 »

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.

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

#16

Post by oasisaimlessly »

Sniffnoy wrote: post #10
fermigier wrote: post #8

Earlier quoted context omitted.

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…

> - 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.

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

#17

Post by vessenes »

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.

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

#18

Post by QuesnayJr »

eddd-ddde wrote: post #12
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…

For the longest time I thought elliptic curves where quadratic curves. Wouldn't it had been more accurate to name them elliptic surfaces?

Just to be clear, an ellipse is a quadratic curve. Ellipses are not elliptic curves. (They are still curves, though, as long as you restrict to plugging in real numbers, not complex.) The terminology is unfortunate.

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

#19

Post by wbl »

oasisaimlessly wrote: post #16
Sniffnoy wrote: post #10

Earlier 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.

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.
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