[1] https://words.filippo.io/dispatches/parameters/#fn1
(Yes, I hide too much stuff in the footnotes.)
Meanwhile, a big part of current cryptographic design is recognizing that many of our current primitives have far larger margins than we really need, and that we can squeeze performance out of our schemes by taking a more evidence-based approach to parameter selection/round counts/etc.[1].
That author's subtext, as I understood it, was that curves should be selected by functional optimization; pick the curve parameters that best solve the engineering problems. It was a useful argument given their preferred curve, 25519, which does indeed have a lot of attractive engineering features! But it's not hard to see that engineering excellence is a selection principle with even more degrees of freedom than mathematical constants; moreover, you're only ever considering optimality at a fixed point in time, but the optimums change over time --- cofactors might be less important when we lack complete additional formulae for a Weierstrass curves, which makes them hard to implement in constant time, but much more important once we do (as is the case now).
There's really no way around just doing the computer science and cryptological work of working out curve attacks. That's the Menezes and Koblitz argument from the "Enigma" paper: that the reason to trust the NIST P-curves is that we'd know by now if they were weak (not because NSA doesn't have secret attacks, but because the mechanism by which they were generated would result in whole large classes of broken curves that academic cryptography and mathematics would have caught by now).
If you don't find Alfred Menezes and Neil Koblitz persuasive surveyors of elliptic curve security, that's fine, but my response would be that you can't really trust any parameters at all at that point. Certainly, investing trust in the cryptographers best known to the generalist programmer audience seems like a bad alternative strategy.
PDF:
> In August 2015 the U.S. National Security Agency (NSA) released a major policy statement on the need for post-quantum cryp- tography (PQC). This announcement will be a great stimulus to the development, standardization, and commercialization of new quantum- safe algorithms. However, certain peculiarities in the wording and tim- ing of the statement have puzzled many people and given rise to much speculation concerning the NSA, elliptic curve cryptography (ECC), and quantum-safe cryptography. Our purpose is to attempt to evaluate some of the theories that have been proposed.
* https://eprint.iacr.org/2015/1018.pdf
Then-coverage:
* https://blog.cryptographyengineering.com/2015/10/22/a-riddle...
... ok. stock one is probably easier to influence by suborning the exchange :) - all world exchanges?
The bitcoin block hash thing is interesting, but if you're talking about an attacker with that kind of unknown but sophisticated capability, how do you know it actually thwarts their plans? Maybe all they need is any large number with high entropy or maybe the leading zeros characteristic of bitcoin block hashes is actually key to their attack? Once you imbue your adversary with unknown powers, it's by definition difficult to know what helps or hurts them.
But, yeah, magic powers, sure, anything goes. I was just thinking there must be something more reliable than NYT headlines.
Maybe get physical measurements, stock exchange data, newspaper headlines, and bitcoin blocks, all in 6 months in the future, and XOR all of them.
If I remember correctly, part of the reason for the cofactor in 25519 is because Montgomery curves have to have a cofactor and only those curves have the nice x-only Montgomery Ladder, which basically rules out invalid curve attacks (so long as the curve is twist-secure — not all the NIST curves are IIRC). Do the complete addition formulas for short Weierstrass curves also fix this? Invalid curve attacks are a major danger for NIST curves.
I know you can use compressed point representation to get the same benefit, but that seems very rare in NIST curve implementations (because old now-expired patents). The landscape of NIST prime-order curve implementations is not great. If all those libraries were going to actually switch to complete addition formulas and compressed public keys then I might feel better about them enjoying a renaissance.
Might such discoveries by academics be forcibly suppressed by government agencies?
That sounds like an argument against standard parameters. Even if we trust an algorithm, we can't trust others to choose parameters for them? The solution seems obvious - find a better way to choose (non-standard) parameters.
The k1 curve has parameters like 15, whereas the "standard" one has curves with parameters chosen as an "arbitrary" number in the billions for some reason. People believe the NSA may have iterated through the previous ones until they found a vulnerable curve, and NIST recommended that. Take a look:
https://cointelegraph.com/news/this-researcher-says-bitcoins...
“(1) The Koblitz curve is specially designed for faster scalar multiplications. Hence the (signing, verifying and key generation) operations on Secp256k1 are faster than those on Secp256r1. (2) Although the Secp256r1 curve was announced to be randomly selected, there could still exist some suspicion that some backdoor might be secretly set up in the curve parameters. In contrast, the Koblitz curve parameters are mathematically determined, and there is little possibility for setting such a backdoor.”
However, given the prevalence of the r1 curve, Ethereum devs might want to add a precompiled contract so that people can sign into the blockchain without trusting web sites and wallet software:
https://ethereum-magicians.org/t/eip-7212-precompiled-for-se...
https://security.stackexchange.com/questions/256088/is-the-e...