[HN Gopher] New elliptic curve breaks 18-year-old record
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New elliptic curve breaks 18-year-old record
Author : calstad
Score : 72 points
Date : 2024-11-11 16:08 UTC (2 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| perdomon wrote:
| 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 wrote:
| 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 wrote:
| >...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.
| fermigier wrote:
| 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, Stefane - Un exemple de courbe elliptique definie sur
| Q de rang >=19. (French) [An example of an elliptic curve defined
| over Q with rank >=19] C. R. Acad. Sci. Paris Ser. I Math. 315
| (1992), no. 6, 719-722.
|
| Fermigier, Stefane - Une courbe elliptique definie sur Q de rang
| >=22. (French) [An elliptic curve defined over Q of rank >=22]
| Acta Arith. 82 (1997), no. 4, 359-363.
| wslh wrote:
| Just saw this, congratulations! Would you mind giving an ELI5
| explanation for a wider audience?
| lisper wrote:
| [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
|
| [2] https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve#U
| ppe...
| jjice wrote:
| This is a fantastic ELI5, thank you!
| lisper wrote:
| Thanks! I try hard to produce quality technical pedagogy,
| so you just made my day.
| eddd-ddde wrote:
| For the longest time I thought elliptic curves where
| quadratic curves.
|
| Wouldn't it had been more accurate to name them elliptic
| surfaces?
| lisper wrote:
| 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/w
| ayn...
| QuesnayJr wrote:
| 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.
| fermigier wrote:
| 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), 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
| every pair of points associates a third point. This can be
| explicitly calculated.
|
| - It can be proven (again, by explicit calculation) that this
| operation is associative and commutative, and that there is a
| "zero" element, i.e. that this operation forms a "group".
|
| Now we want to study these elliptic curves and their
| associated groups with one additional condition: that the
| points are rational, i.e. have coordinates that are rational
| numbers (a/b). For each curve with rational parameters (i.e.
| the coefficients of the polynomial are rational), we want to
| study the rational points of this curve.
|
| For some elliptic curves, there is a finite number of points,
| so the associated group is a finite commutative group.
|
| For other elliptic curves, however, there are infinitely many
| rational points, and mathematicians have wanted to classify
| their structure.
|
| A foundational result in number theory known as the Mordell-
| Weil theorem states that the group of rational points on an
| elliptic curve over a number field (such as the rationals, Q)
| is finitely generated. In other words, although there may be
| infinitely many points, they can be expressed as a finite set
| of points (known as "generators") combined under the group
| operation. This structure forms what is called a "finitely
| generated abelian group", which can be decomposed into a
| direct sum of a finite subgroup (called the "torsion") and a
| free part of rank r, where r is called the "rank" of the
| elliptic curve.
|
| This rank "r" essentially measures the "size" of the free
| part of the group and has deep implications in both
| theoretical and computational number theory. For example, if
| r=0, the group is finite, meaning that the set of rational
| points on the curve is limited to a finite collection. When
| r>0, there are infinitely many rational points, which can be
| generated by combining a finite number of points.
|
| So the challenge is to find a curve with a large number of
| generators. All of these computations (for a given curve at
| least) are quite explicit, and can be carried out with a
| bignum library (the numbers tend to get quite large quickly).
| I used PARI/GP for my thesis.
| Sniffnoy wrote:
| > - 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 every pair of points associates a third
| point. This can be explicitly calculated.
|
| > - It can be proven (again, by explicit calculation) that
| this operation is associative and commutative, and that
| there is a "zero" element, i.e. that this operation forms a
| "group".
|
| I feel like it's worth clarifying here that this operation
| is actually _not_ the group operation, although the group
| operation is defined in terms of it.
| oasisaimlessly wrote:
| 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.
| wbl wrote:
| 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.
| UI_at_80x24 wrote:
| 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.
| Noumenon72 wrote:
| 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 wrote:
| 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.
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