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.
YODL
npub162…atdhy
2025-05-08 22:05:19
in reply to nevent1q…tekk
Author Public Key
npub162zpxufpw8pnuytaf0gfxzkqtvk9rvcwkvppa7x57y3n7qkfpg4shatdhyPublished at
2025-05-08 22:05:19Event JSON
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"content": "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. \n\nYou 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. \n\nSorry for rant\n\n\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e GPT output \u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e\nThe 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.\n\nSince 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.\n\nSo, the number of distinct 256-bit elliptic curves is on the order of 2^{512}, which is an astronomically large number.",
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