Attacks on elliptic-curve based cryptosystems through solving the Decisional Diffie-Hellman Problem with the Weil Pairing
Proposition  One-wayness of ElGamal encryption holds under the Computational Diffie-Hellman (CDH), and conversely. 

CDH Problem (informally) Given $(P, [r]P, [s]P)$ find $[rs]P$.

Let the public key be $Q = [s]P$ where $s$ is the corresponding secret key.  Let also a ciphertext $C = (C_1, C_2)$ with $C_1  = [r]P$ and $C_2 = M + [r]Q$ for a message $M$ (viewed as a point on an elliptic curve).

Proof (of the proposition):


If you can solve the CDH then you can break the one-wayness of the encryption.
Suppose you are given a ciphertext $C = (C_1, C_2)$, the goal is to recover the corresponding plaintext. On input $(P,C_1,Q)$, you first obtain $[rs]P$ by solving the CDH. Next you recover the plaintext as $M = C_2 - [rs]P$.
If you can break the one-wayness of the encryption then you can solve the CDH.
Suppose you are given a DH triple $(P,[r]P,[s]P)$, the goal is to obtain $[rs]P$.  You define the ciphertext $C = (C_1,C_2)$ where $C_1 = [r]P$ and $C_2$ is a random point.  You define the public key as $Q := [s]P$.  You recover the corresponding message $M$ (because you can break the one-wayness) and find $[rs]P$ as $[rs]P = M - C_2$.