Quote from: hosseinimr93 on Today at 02:08:06 PM

128 or 256?

128.

Quote from: hosseinimr93 on Today at 02:08:06 PM

Isn't that any integer between 1 and 2²⁵⁶ (or any 64 character number in hexadecimal format) can be turned into a valid private key?
I know the exact number is a little smaller due to secp256k1 ECDSA standard, but the number of valid private keys should be much bigger than 2¹²⁸.

Correct.

Quote from: hosseinimr93 on Today at 02:08:06 PM

What I am missing here?

That a length of 256 bits does not equate to 256 bits of security.

The best known attack against a private key is not random brute force (which would indeed equate to around 256 bits of security), but rather attempting to solve the ECDLP, which provides 128 bits of security.

This can be seen in Standards for Efficient Cryptography. SEC 2: Recommended Elliptic Curve Domain Parameters. (Table at the bottom of page 4.)