What is point compression?
If we know the x-coordinate of a point, we can find the y-coordinate by solving the curve equation. Since it is a quadratic equation, there will be two possible results (or none), so we need an extra bit to choose the correct solution. This technique can also be used to choose random points on the curve (a retry will be needed if the quadratic equation has no solution). Point compression is nice because it reduces the size of public keys from say 510 bits to 256 bits. Solving quadratic equations over finite fields is a reasonably cheap operation.
