New elliptic curve breaks 18-year-old record (opens in new tab)

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.)

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[1]https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve

[2] https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve#Uppe...

- 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.

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...

he died in 2024, did he make arrangements to keep funding it or endow it?

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.