A history of elliptic curves in tweets | Hacker News Reader

A history of elliptic curves in tweets

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54 pointsbeefhash16 comments

This is cute, but I would much rather see this history written out as a serious non-tweety article. There is a ton of fascinating and useful information hiding underneath the humor.

Go and read a maths textbook then?

I'm not being facetious, some books are much less Theorem Lemma Proof (and nothing else) than others.

You may not be facetious, but you are being condescending by implying that I have never read a math textbook, or that I have made no attempt to find this information. In fact, I have read fairly extensively about elliptic curves and I have never seen any source that describes the history. To the contrary, many sources say that elliptic curves have nothing to do with ellipses, but this series of tweets seems to imply that this isn't true. So if you want to point me to an actual source, that would be much appreciated. But all you have to say is RTFM then my response is STFU.

I think you are over-reacting. I didn't read mhh__'s comment as directed at you specifically. It's more of "When in want of X, go read a book on X".

You're meant to pick up the history as you learn the techniques. You're supposed to learn about Galois when you study groups, Cardano when studying complex numbers, etc. We explicitly invoke the names of mathematicians not to try to summon some saintly Great Men, but to imagine their mindsets and how their various discoveries fit together. Directly examining your textual argument, it's not possible to learn about elliptic curves without learning at least a tiny fragment of the history, where one of Weierstrauss, Montgomery, or Edwards has had their name directly associated with the concrete representation of the equation defining a curve.

I'd like to think that I'm pretty good with elliptic curves, given that I explain them to other folks. However, this tweet-thread still showed me things that I didn't know about the history of maths. Some of these tweets directly led to minutes (hours) on Wikipedia and chasing down various PDFs to read.

Honestly, you could stand to be less self-aggrandizing. It's okay if you don't get every joke immediately; you don't have to know every bit of historical background.

Elliptic curves aren't interesting because of the answers that we have. They're interesting because of the questions. One open question [0] is on the Millennium Prize list. Another open question is almost never written down because it's so simple; "why 1728?" [1] Coming to understand these questions is the goal of our entire process, so there's not much that I can really do other than to ask you to join the process of exploration.

[0] https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conj...

[1] https://en.wikipedia.org/wiki/J-invariant

One detailed discussion of elliptic curves with lots of history is: https://www.maa.org/sites/default/files/pdf/upload_library/2...

To answer your specific question: "Elliptic curves have almost nothing to do with ellipses at all. Why then are they called elliptic curves? The answer lies in the word almost. There is a connection between ellipses and elliptic curves, but it’s not at all obvious and is the result of a connected but distinctly nonlinear sequence of mathematical events."

Math textbooks are really bad at giving a high-level overview of a whole lineage of ideas like this. I would love to read the long-form article on this subject for a mathematically-competent reader, but have no desire to actually study the subject in detail.

If you really want to know, you could cover most of it with [1,2].

[1] https://www.maa.org/sites/default/files/pdf/upload_library/2...

[2] https://www.maa.org/sites/default/files/pdf/upload_library/2...

Thanks!

This is great! Thank you.

There are a lot of other mediums that explain it differently. Not everything has to be written in a dry textbook flavour.

> NSA: Use our curves. They were selected randomly. Promise, wink wink.

Here's a better explanation about this cryptic tweet:

Backdoors in NIST elliptic curves

Of particular concern are the NIST standard elliptic curves. There is a concern that these were some-how “cooked” to facilitate an NSA backdoor into elliptic curve cryptography. The suspicion is that while the vast majority of elliptic curves are secure, these ones were deliberately chosen as having a mathematical weakness known only to the NSA.

https://miracl.com/blog/backdoors-in-nist-elliptic-curves/

I’m not a bernsteintheist, but I must be a little bit of one after all, because I was like “hey, he went to a lot of work to show he had nothing up his sleeve..”

Lisper here says that he wished for a serious writeup, I agree — right now this is a set of in-jokes; it would make a fantastic Quanta Magazine article, or two, or n..

Shor: Are you sure?

NSA: Antoine Joux > Quantum Computers, Like him!

Crown Sterling: We sell CADO-NFS™ for breaking ECDSA of Nakamoto's funny money.

To be continued ...

I wish I could find a good primer on elliptic integrals. They come up constantly in magnetic fields and are impractically hard to solve. I mean you can do it, and I've seen it done, but give myself 5% odds of pulling it off myself. My feeling is that this is why only a handful of magnetic field equations are provided in textbooks.

Utterly brilliant!

A great mathematical mind -- meets a great ability to summarize the works of previous mathematicians (not easy to do!)!

Worth re-reading in the future!

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