Looking at just EC Public Key parameters, how can you tell if it is invalid?

Question

I am trying to handle when a parsers goes off the rails and reads an EC public keys wrong (just the X and Y components, I know the curve prior).

Right now I check for the following (false means invalid):

1. Is the key even on the Curve?
2. Is the Public Key X component less than the Curve's Base Point X?

I am pretty confident in #1, but not as confident in #2. #2 was just made by never seeing a case that contradicted it.

So my question is, 1) are these two checks correct? 2) are there other checks that can detect bad public keys?

Answer 1

The general rule for curves is given in;

• 2003 - Validation of Elliptic Curve Public Keys by Adrian Antipa,Daniel Brown, Alfred Menezes, and René StruikScott Vanstone

  They defined a point is valid if

  1. $P \neq \mathcal{O}$
  2. The $x$ and $y$ coordinates of $P$, $x(P),y(P)$ are valid elements of the field.
  3. $P$ satisfies the curve equation - against the twist attack
  4. Check $[n]P = \mathcal{O}$ for prime curves ($h=1$) and check $[h]P \neq \mathcal{O}$ for non-prime curves ($h>1$) where $h$ is the cofactor $h = \#E(k)/n$

if 1,2, and 3 are verified and $h=1$ (i.e. prime curve) then the 4th is already satisfied.