[nullc]

Here is a suggestion: If you're going to write yet another tutorial on elliptic curve digital signatures and you find yourself explaining the operation of the group law: you've probably messed up.

ECDSA and friends work fine in an abstract group model. Virtually everything interesting about the signature algorithm is indifferent to the details of the group. Moreover, the group operator will remaining largely opaque except in a false-knowledge monkey-see-monkey-do sense without a substantial amount of number theory education that these sorts of tutorials never provide; and which aren't really that useful except for implementers (and if an implementer is learning it from one of these tutorials I fear for the security of the result).

So I recommend: just present the idea of an abstract group and its relevant properties, then continue on with the high level algebra. For most people this would be a lot more informative.

[royalfork]

For RSA, most non-experts have an intuitive feel for why the cryptosystem is secure: we all know that factoring numbers is hard. Do you think it's possible to give a concise explanation for why ECC is secure aside from "hand wavey black box" + understandable algebra = security?

Total aside: when I was writing this, you answered a lot of my questions in #bitcoin. Thanks for helping me out :)

[defen]

I'll describe the discrete logarithm problem; the elliptic curve discrete logarithm problem is like it, but harder (different group operation).

Pick a prime number p.

Pick a positive integer x < p

Then every positive integer < p can be written in the form: x^n (mod p), for some positive integer n < p.

For example if p is 5 and x is 2: 2^1 ≡ 2 (mod 5); 2^2 ≡ 4 (mod 5); 2^3 ≡ 3 (mod 5); 2^4 ≡ 1 (mod 5);

Try it with larger p or different x's and try to find a pattern. The discrete logarithm problem says: given c, x, and p, find y such that x^y ≡ c (mod p). In English, what power do I have to raise x to, in order to get a number that is equivalent to c, modulo p. The "modulo p" thing is what makes it hard - without that you just have x^y = c, so ln(x^y) = ln(c) and therefore y = ln(c)/ln(x)

[apo]

So I recommend: just present the idea of an abstract group and its relevant properties, then continue on with the high level algebra. For most people this would be a lot more informative.

Are there any treatments like this that you'd recommend?

[tptacek]

The best single-web-page treatment I've read is AGL's:

https://www.imperialviolet.org/2010/12/04/ecc.html

It describes the group mechanics but only briefly and then moves on to how, once you stipulate the group exists and has the operations you expect, you'd actually use it.

Another way to frame the critique above is: the tutorial here has pretty basic coverage of the group mechanics and almost no coverage at all of how signing schemes with curves work --- and curve signing is actually pretty interesting! So: you'd kind of want the tutorial to pick one of those things --- the fundamental mechanism of elliptic curves as an abstract algebra concept, or curve crypto signing --- and do a better job on just that thing.

For signing, the best I've read on a single page is DJB's:

https://blog.cr.yp.to/20140323-ecdsa.html

Someone could probably do some good fleshing out each of these sections for laypersons.

(I'm spitballing).

[matthewdgreen]

Those are two very detailed explanations of how elliptic curves work, with a side helping of side-channel attack resistance just to make things more confusing for the beginner.

A better explanation to me would take you through the basic group theory, the overall design of a Schorr signature and then maybe a sketch of the security argument. Unfortunately doing that requires random oracles and so it’s messy. But what’s the point otherwise?

[tptacek]

What are three practical goals you'd want to achieve --- at a higher level than "demonstrating the design of a Schnorr signature" --- in a curve sig tutorial for beginners?

[matthewdgreen]

First off, I don't understand why anyone would want to talk about "curve sigs". Unless there's some fundamental new property being conveyed by the use of elliptic curves (e.g., pairing-based signatures) then you don't want to start with elliptic curves at all. All of these signatures were initially designed to work in (e.g., Schnorr-type) finite-field groups. The use of elliptic curves instead of FF is at most an efficiency/security optimization that you tack on to the existing algorithms. It's interesting -- and you can get lost in the details -- but it's not fundamental.

The three goals I would aim for are:

* Explain groups and group operations, exponentiation and DL * Describe the interactive Schnorr identification protocol and explain why/how it works * Show how this can be flattened into a (non-interactive) signature using a hash function, and why that works * (Optionally, show how [EC]DSA is just a bastardization of Schnorr/Elgamal) * (Optionally, describe the proof techniques and the Forking Lemma, but maybe nobody really cares.)

Then as icing on the cake you could explain how elliptic curve subgroups are instantiated, and why they represent an improvement over the Schnorr groups. But unless you're actually developing new EC software (and you probably shouldn't) that's more informational. Besides, there are a ton of tutorials on that out on the Internet, and not a lot of explanations that cover the operation and security of actual signature schemes.