What are elliptic curve pairings? | Hacker News Reader

What are elliptic curve pairings?

zellic.io

64 pointslisper14 comments

I don't know nearly as much abstract algebra as I would like to, but when I read "A cyclic abelian group" I thought, "hold on, aren't all cyclic groups abelian?" And I remembered correctly, they are.

So there's that suspicion, never far away when dealing with something cryptocoin-related, that maybe they're trying to bedazzle more than explain. Honestly, who can follow this who doesn't already know it?

I think they meant it the other way around. They wanted to talk about abelian groups, and here specifically a cyclic (abelian) group.

It would be silly to talk about a cyclic group that is specifically abelian, all cyclic groups are abelian. But not all abelian groups are cyclic.

They could have kept it implicit, but not everyone might know that fact, so it doesn't really hurt

> that maybe they're trying to bedazzle more than explain

I don't know much abstract algebra as well but I have studied elliptic curve pairings before, so from what I can tell, every sentence in the article counts (maybe not every word, though). The article may sound bedazzling to a reader not experienced in the subject, but its complexity can totally be justified by the complexity of technical details that it tries to explain clearly.

That said, it's indeed hard to distinguish between a bedazzling article and an article that actively tries to explain concepts as clearly and concisely as possible. I do have such feelings with many AI papers these days, but not this one.

It doesn't help if you explain things accurately, if you're explaining it in such a way that no one would understand it who didn't already understand it.

One key concept this text skips over, that I at least can't remember from any of the stuff I've read, is elliptic curves. Sure, you learn about abelian groups in an abstract algebra course. And you certainly learn about multiplicative subgroups of the integers modulo a prime. But do you learn about elliptic curves in a basic abstract algebra class these days? I mean maybe we should since they've become so important in cryptography, but I don't think we do.

> "hold on, aren't all cyclic groups abelian?" And I remembered correctly, they are.

Yes, but not all abelian groups are cyclic.

Pictures or demonstrations would make this easier to digest.

A good challenge; no idea why you are being downvoted*. I am using SageMath for plotting complicated algebraic objects, and I am sure I can combine its output with TikZ for nice illustrations of pairings and other EC constructions. I just need to find the time, but it should be time well-spent.

*In fact... every comment in this thread seems to be getting downvotes? wth?

Pictures can be found here! https://x.com/zellic_io/status/1745874206695596427?s=20

> Pictures or demonstrations would make this easier to digest.

The commutative diagrams in the text are a kind of picture. :-)

They are "Bodacious New Assumptions", in the words of Neal Koblitz:

https://www.ams.org/notices/201003/rtx100300357p.pdf

We've moved on to much more exciting new assumptions in the 14 years since that was written. Pairing-related assumptions are practically respectable now.

I submitted this paper before, but it did not get any eyeballs; I am submitting it again because I am genuinely amused by that classic debate between Koblitz and the applied cryptographers. Personally, I think both sides have their merits, but Koblitz tends to overstate his case by far.

The paper is a really good (and fun) read, though.

Just remember to use the proper Dual_EC_DRBG curve parameters :)

Too mathy for my CS degree. It'd take me hours of googling to understand an article written like this

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