Is there a 1:1 mapping between private and public EC keys?
Points on secp256k1 fulfill the equation:

$$ y^2=x^3+7 \pmod p$$

Your 257 bit compressed point consists of the sign* of $y$ and the value of $x$. You probably want to restrict $x$ to $0 \leq x &lt; p$ as a canonical representation.

Since $y^2$ is obviously a square,  we need to look at $x^3+7$ to see if it's a square as well:


If $x^3+7$ is a square, then there are two points for that $x$ which only differ by the sign of $y$, which is why the compressed point contains the sign of $y$.
If $x^3+7$ is zero, there is only one solution, since $y=-y$.
If $x^3+7$ is a non-square, then there are no solutions.


Not counting $0$, exactly half the elements of the finite field are square and the other half is non-square. $x^3+7$ complicates matters a bit, so approximately half of the possible $x$ values are square. Hasse's theorem on elliptic curves quantifies this approximation.

You can efficiently check if a field element is square, and thus verify if it is a valid $x$ value of a point on the curve.



* field-elements don't actually have a sign. But since the least significant bit of non-zero values differs between $+y$ and $-y$, we can use it as sign. See Determine if a public key point y is negative or positive, odd or even? for a longer explanation.