We are working on some detailed estimates for the time required for a quantum computer to break ECC, and are using secp256k1 as an example. This question does not seem to be directly addressed in standard cryptography literature. What is the probability when selecting a random point on the curve that the selected point could be a valid public key (e.g. n*G, for any n and G is the base point)?
In other words, which of the following apply: (A) essentially all points on the curve are potentially a valid public key (e.g. ~>99%), (B) only about half of all point on the curve are potentially valid public keys, or (C) some smaller fraction of points are potentially valid public keys?
If you could provide reasoning or a citation, that would be helpful. If the answer depends on the curve, assume secp256k1.
