The 521 curve is the only one using a Mersenne prime, which has significant advantages when you do calculations, but also feels less arbitrary.

This vulnerability has very little to do with P-521 per se. The issue is with ECDSA: any use of ECDSA with biased nonce generation, regardless of the elliptic curve it's implemented over, immediately causes secret key leakage.

(Rant: All these years later, we're all still doing penance for the fact that Schnorr signatures were patented and so everyone used ECDSA instead. It's an absolute garbage fire of a signature scheme and should be abandoned yesterday for many reasons, e.g., no real proof of security, terrible footguns like this.)

Schnorr wouldn't have helped in this specific case, since Schnorr is equally vulnerable to biased nonces (https://ecc2017.cs.ru.nl/slides/ecc2017-tibouchi.pdf).

EdDSA, which is essentially deterministic Schnorr, does solve the problem.

Also, the use of P-521 didn't specifically cause the vulnerability, but the bad interaction between SHA512 and P-521 did play a role. It is unfortunate that nature conspired against us to make 2^511 - 1 a composite number. The fact that you have to go up to 521 bits to get a Mersenne prime whereas the natural target length for a hash output is 512 bits is the fatal interaction here.

Excellent points all around, and thank you for the pointer to the ECC slides :)

(And indeed, nature could have been kinder to us and given us a Mersenne between 127 and 521...)

Shouldn't there be another close enough prime? Like 2^510-1 or 2^511-19?

> Schnorr signatures

Never heard of (which probably demonstrates that I know pretty much nothing about cryptography?), so seeing a name spelled like "Schn...r" in this context makes at least me think of an entirely different luminary in the area. Thought it was a typo at first.

I’m over here wondering why someone would want deterministic nonces.

Isn’t it kind of the point to just roll random numbers? When would you calculate?

It says in the OP. Windows at the time did not provide a cryptographic quality random number source.

I saw that, and wondered why PuTTY didn't contain it's own good CSPRNG, something like Fortuna, if Windows didn't offer one.

You still need a source of entropy, which is easier for an OS. An app has to resort to the user moving the mouse or bashing keys, which is a worse UX, although I guess they did that for actual key generation (if PuTTY did that) but it would be annoying to do it every time you made a connection.