> there is concern that the NIST curves are backdoored and should be disfavored and replaced with Curve25519 and curves of similar construction.
Of course, "there is concern" is pretty vague, but it should be made clear that such concerns are in the realm of pure speculation at this point. There is simply no known way of constructing a "backdoored" elliptic curve of prime order over a prime field (in particular, the closest thing resembling such a backdoor, namely Teske's key escrow technique based on isogenies from GHS-weak curves, cannot work over a prime field). Scientifically speaking, I don't see more reasons to believe the assertion that "NIST parameters are backdoored because they aren't rigid" than the (equally unfounded) speculation that "Curve25519 may be weak because it has small parameters/a special base field/composite order/etc.".
Moreover, to say that the NSA has backdoored the NIST curve parameters is to assume that they have known, for quite a long time now, a serious weakness affecting a significant fraction of all elliptic curves of prime order over a given base field that has so far escaped the scrutiny of all mathematicians and cryptographers not working for a TLA. Being leaps and bounds ahead of the academic community in an advanced, pure mathematical subject doesn't quite align with what we know about NSA capabilities.
Don't take this the wrong way: there are good reasons to favor Curve25519 and other implementation-friendly elliptic curves (namely, they are faster, and they are fewer ways of shooting yourself in the foot if you implement them), but "NIST curves are backdoored" is not a very serious one.
The issue with the NIST P- curves is that there's no good reason to trust them. And, for what it's worth, being ahead of academia on pure math isn't science fiction; NSA employs a lot of mathematicians. But the notion of a backdoor in the NIST curves is totally speculative.
Here's what I was trying to capture:
Despite its very weird submission as a story to HN, what you'd been reading was just a very long HN comment; I wrote it in a single draft and in the style I would use when writing a comment.