Your post names sense to me, but I don't think it's too crazy to allow for the possibility that e.g. the NSA has backdoored some of the "recommended" crypto constants in current use.
To back door the NIST curves would require that the NSA knows a secret attack against ECC and used brute force search to find seed hashes to generate the curves to be vulnerable. It would have to be a secret attack since nobody else has found it.
Thing is… if this is true it means there is a secret attack against some elliptic curves.
Using 1990s technology they couldn’t have brute forced all that many curves, meaning some non-trivial percentage of curves must be vulnerable.
How do we know curve25519 isn’t vulnerable to this secret attack? We don’t.
The ultimate conclusion is that if NIST curves are backdoored using secret math we shouldn’t use ECC at all, at least unless NSA discloses the math. But they couldn’t do that without blowing up the Internet since so much uses the NIST curves. It would be an argument to phase out all ECC.
1. The NSA is one of the world's only institutions that has any use for the otherwise irrelevant specialism of designing asymmetrically backdoored cryptography algorithms. They also have (or had) lots of maths PhDs writing internal papers. It's reasonable to assume they know things about kleptography that others don't, as it's not a well funded sub-field of cryptography outside the signals intelligence world. So if there was an attack they'd discovered it wouldn't be surprising if others didn't find it.
2. A good protection against kleptographic attacks is to use "nothing up my sleeve numbers", where the constants that go into an algorithm are derived from some well known source that isn't suspicious.
3. The NIST curves know about this sort of risk and attempt to use NUMS numbers. The constants are derived from the output of SHA1, which at the time was considered secure.
4. But the input to SHA1 is a giant random-looking number, not something obvious and above suspicion. Thus the setup fails to provide the assurance it was supposed to create because the NSA could have searched for a weak curve (if there was such a thing to begin with).
The argument that curve25519 wouldn't be susceptible is simply that curve25519 uses NUMS numbers properly, and so there's no wiggle room where djb or anyone else could have done a secret scan to find curves with certain properties.
As may be clear, how strong you think the above argument / problem is, will depend entirely on your intuition about how well explored kleptography is by non-secret research. Unfortunately as people generally don't publish negative results that's very hard to judge.
NSA required the use of "Suite B Cryptography" for commercial vendors of government systems, which in its latest revision meant ECC. However, vendors were (and are) slow to adopt ECC from the previously used RSA. If you want public evidence of how slow such transitions can be, check any other commercial crypto like certificate authorities and see which trust chains are entirely elliptic curve. Mostly people are stuck on RSA, even though elliptic curves broadly offer better speed and smaller keys/signatures for the same or better levels of security. There's also still plenty of deployed DES and SHA-1, even though the former is inadvisable and the latter inexcusable. In fact, from what I read in the response to the NSA proposing to drop SHA-3 in favour of SHA-2 in the new PQ standards, vendors were a bit frustrated at the change because of the short timescale involved in the migration. I interpret the demanding schedule for adoption of PQC as a deliberate choice by NSA - a somewhat passive aggressive response to vendors to tell them to get their acts together, based on their experience of trying to roll out ECC.
What NSA said is "if you haven't migrated to elliptic curve cryptography, you should now wait for post quantum and then start on that". You can read that message here: https://web.archive.org/web/20151123081120/https://www.nsa.g... and here is the exact quote:
> For those partners and vendors that have not yet made the transition to Suite B elliptic curve algorithms, we recommend not making a significant expenditure to do so at this point but instead to prepare for the upcoming quantum resistant algorithm transition.
I don't think there's much mystery here - basically it amounts to a bunch of procurement rules and guidelines.
They are now pushing PQ crypto because they think, probably reasonably, that we are one or two breakthroughs from a scalable quantum computer.
My favourite also reasonable explanation for the mysterious Dual EC DBRG constants is that some senior person picked their favourite numbers (birthday of a niece, phone number of a friend, whatever) not realising that these should be Nothing Up My Sleeve Numbers. Later when a subordinate says "For documentation we need to explain these numbers" it was late to change them and so the agency can't do better than insist they were chosen at random.
If this was crucial technology we should do the work to re-make it with Nothing Up My Sleeve Numbers, but it's garbage, indeed that's one reason for the suspicion, this is a very complicated machine, well suited to hiding a backdoor, why do this at all?
Cryptographic common sense is that if you use an "algebraic" generator, you feed the output through a "chaotic" one at the end. This can't possibly harm security (as long as the output transformation doesn't depend on secret state) as there's a reduction in which the adversary just does the transformation themselves. This is even more important if the algebraic transformation is efficiently invertible in principle, for example if someone has extra secret knowledge (such as the dlog of the base point in use).
If they'd used Dual-EC followed by SHA1 or something that would have not only been better according to folk wisdom, and demonstrably no worse for security (and costing very little compared to the EC operations) but it would also have shut down a lot of conjectured attacks that one could do with twiddled constants.
Yet for some reason, Dual-EC decided to go with an algebraic approach without a "chaotic" output transformation, which is either extreme incompetence or strong evidence that someone is up to no good.
Hard disagree. Dual_EC_DRBG more-or-less encrypts the next state of the random number generator to a certain elliptic curve public key, cuts off a few bits, and outputs the truncated ciphertext as a "pseudorandom number". Given this framing, the backdoor is obvious: guess the truncated bits, decrypt the ciphertext, and check whether the decrypted next state is consistent with later data that depend on that DRBG.
This is a pretty fucking weird design unless you're intending a backdoor. It's orders of magnitude slower and more complex than just symmetric-key encryption or hashing, which are traditionally used for DRBGs, and elliptic curve ciphertexts aren't uniformly pseudorandom so it's slightly biased as a DRBG as well. The claimed justification is that it's secure based on properties of elliptic curves, but even aside from the backdoor this is false (since it's biased), and anyway you'd be going for provable security here but there isn't any proof (even aside from the bias AND the backdoor... except maybe in the generic group model, which is too strong to be relevant here). Also, the backdoor potentials of Dual_EC_DRBG were discussed both before and after standardization, but it was standardized and used anyway.
It is conceivable that NSA chose this design through a firm but misguided belief that it has superior security properties, and that they don't have the private key corresponding to that public key, and that they paid RSA security $10 million to make it the default DRBG in BSAFE because they were just that proud of their generator, and they didn't notice the backdoor before publication, and they didn't consider the need for nothing-up-my-sleeve numbers because they weren't paying any attention. But IMHO it's much more reasonable to believe that Dual_EC_DRBG's backdoor is intentional, and that NSA does know the private key corresponding to their published recommended public key.
Also note that even if NSA doesn't have the backdoor key, a malicious actor can substitute their own backdoor key and everything just keeps working. With this change of 64 bytes of code, fully permitted by the spec, that actor (and nobody else) now controls the backdoor. This happened to Juniper networks.
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None of this is apparently the case for the NIST elliptic curves, though. While it is possible that NSA knows a secret weakness in them (or, indeed, in all elliptic curves!), even 25 years later there is no publicly known class of weak curves that could include the NIST curves. Furthermore, the spec does include the values that hash to the curves' coefficients: while this doesn't guarantee that those values were really generated randomly, it significantly constrains how a hypothetical backdoor could work, and what the families of vulnerable curves could look like. If there were a backdoor, it would also in some sense be a less powerful backdoor than in Dual_EC_DRBG: the Dual_EC backdoor is only exploitable to someone who has the private key, but given the constraints on the NIST elliptic curves, it is likely that anyone who figured out the math of a hypothetical backdoor could exploit it.
IMHO it is still reasonable to believe that the NIST curves are backdoored. But IMHO they are very likely not backdoored, and I think my view also matches the consensus of the crypto community.
That said, I still think the finite-field inversion in ECDSA signatures (instead of doing vanilla Schnorr) is dumb and risky. (Bernstein's deterministic signature improvement is even better, and would probably have prevented the PS3 hack, but I can't blame NIST for this because it was not known at the time. I think you're allowed to do it with the NIST curves too nowadays.)
FIPS 182 gives a non-constant-time implementation of point multiplication with the note "The algorithm given below is for reference purposes. Other (constant time) algorithms that produce an equivalent result may be used." which should be a MUST not a MAY, and in my opinion the non-constant-time one should never have been in the standard in the first place because that (and the accompanying note's wording) encourages people to go with it.
Checking for invalid curve points is also a solved problem in theory, but as far as I know not always implemented correctly in practice.
I also didn't notice when originally replying, but maybe it's worth tacking on:
> ... even if the NIST curves are not backdoored mathematically, they're designed in a way to strongly coerce implementors towards a route which leaves side-channels open ...
[emphasis mine]
The NIST curves are short Weierstrass curves with a=-3. I believe that before 2007 (Edwards), these were generally considered the best, fastest, easiest-to-implement elliptic curves for general operations. Yes, Montgomery curves are faster and simpler for ECDH, at the cost of having a cofactor, but not for signatures.
So it's worth noting that NIST/NSA/Certicom did not choose this family of curves as a trap for implementers. It was the best choice at the time, but Montgomery or Edwards curves are arguably a better choice now.
But yeah, he also argues that Montgomery / Edwards curves are easier to implement securely. IMHO he's right, but he exaggerates the difference. Montgomery and Edwards curves still have plenty of footguns, and some of these are not present in prime-order short Weierstrass curves like the NIST curves. In particular, the fact that Montgomery and Edwards curves must have a cofactor of at least 4 leads to problems: side-channel attacks like "May the 4th be with you", EdDSA signature validity not actually being defined, needing additional complexity (Ristretto/Decaf) to avoid protocol changes, etc.