How many "256 bit curves" are out there, I wonder

No need to wonder you can check here https://www.secg.org/sec2-v2.pdf so looks like 2 😉 Though the 256 thing is aaa well quite nuanced.

That's a lot of reading, and it only specifies that two are _recommended_ (will look it over more though, as seems interesting). ChatGPT informs me there are close to 2^512 of them (see attached gpt detail output if you're interested), but I think it's wrong in that it only considered counting for a single choice of p; there are several (many?) candidates for primes yielding 256-bit curves I would think. Anyway... TLDR, many many exist out there, and I guess many considerations go into choosing one. You always have to wonder though, in a world with so many choices, why we end up with the ones we do... Recently read that it's an argument for choosing as many "nice" properties a curve could have (e.g. ED25519 has additional nice properties, allegedly), because requiring that would make it harder for a bad actor to pick a curve they have a backdoor to. Sorry for rant >>>>>>>>>>>>>>>>>>> GPT output >>>>>>>>>>>>>>>>>>> The number of distinct 256-bit elliptic curves depends on the specific parameters and the field over which the curve is defined. For a prime field of size p, which is approximately 2^{256} for a 256-bit curve, the number of distinct elliptic curves is roughly the number of pairs (a, b) that satisfy the curve equation y^2 = x^3 + ax + b over that field, with the condition that 4a^3 + 27b^2 \neq 0 to avoid singularities. Since a and b can each take any value in the field F_p, and the field has p elements, the total number of pairs (a, b) is p^2. Excluding the pairs that make the curve singular removes a negligible fraction, so the number of distinct curves is approximately p^2, which is about 2^{512} distinct curves. So, the number of distinct 256-bit elliptic curves is on the order of 2^{512}, which is an astronomically large number.