I am quite new to cryptography low-level mathematic details, though had worked in the crypto area for 2.5 years before. So if I am wrong about any of below part, please correct me without a facepalm gesture ;)
There are already some discussion about this along with questions about what is a valid private key per say, but none of those answers is convincing. At least I don't agree with their arguments that any valid private key is a good key.
EdDSA (Ed25519) is any random number sufficient for a good private key? Yes .... answered Oct 28 '17 at 9:03 Frank Denis
Are all possible EC private keys valid? @Fozi: 1 is neither better or worse than 0x3b6ddba1f4b325cee4505084bc507d2019e86539f8d4be027004b69f9aa0bc74, as long as they have equal probability of occurring, namely 1/n or about ∼1/2256, so that the adversary has no better probability of success by guessing one or the other first. Note that the probability of getting either one of them individually is negligible, just like any other possible secret scalar. – Squeamish Ossifrage Sep 11 '17 at 18:28
Here is my argument:
AFAIK, a public key is calculated by multiplying
n times of generator point G, where
n is a random number generated as private key.
The notion is
pub = n * G
Then for any known curve, I can calculate a million public keys as a rainbow-alike table, where n ranges from 1 to 1,000,000. That would take only 32 MiB space in case of a 256-bit curve. If somebody use any value in that range to generate a public key, I can easily nail it by comparing it to the table. In practice, test vectors for a given curve normally starts from n = 1, 2, 3, ... e.g. https://crypto.stackexchange.com/a/21206/95843
In a wilder scale, someone can store even a 1TiB or 1PiB(oh money!) this type of table. So in that sense, any private key less than 2^20 or even 2^50 is not good!
Now from above deduction the lower bound of private key is settled. But is there a Upper bound? Or any valid number above the lower bound is good and bullet proof enough?