Attacks on schemes based on elliptic curves when the transmitted points are not on the curve
In this state we have well known attack that is called invalid-curve attack.

Let $E:y^2=x^3+ax+b$ and $E':y^2=x^3+ax+b'$ be two elliptic curves with reduced Weierstrass form. $E'$ is called an invalid curve relative to $E$.

Since formulae for adding and doubling points on $E$ does not involve coefficient $b$ thus addition law for $E$ and $E'$ is same. In this attack, we can send several low order points and find the private key using the chinese remainder theorem.

For more detail you can see "Guide to Elliptic Curve Cryptography" page 182.