ElGamal with elliptic curves I've searched some information on ECC, but so far I have only found Diffie-Hellman key-exchange implementations using ECC, but I don't want to exchange keys, I want to encrypt &amp; decrypt data like in ElGamal.
I know that ElGamal with elliptic curves should be possible (Since ElGamal is based on DH), but I have no idea how.
So, could anyone tell me how to implement ElGamal using elliptic curves.
I think I do not need to much background information,


What is the private, what is the public key?
How to encrypt messages? and
How to decrypt messages?


should be enough.
Your answer is in the paper Elliptic curve cryptosystems from Neal Koblitz:


Set up an elliptic curve $E$ over a field $\mathbb{F}_q$ and a point $P$ of order $N$ just the same as for EC-DDH  as system parameters.
You need a public known function $f : m \mapsto P_m$, which maps messages $m$ to points $P_m$ on $E$. It should be invertible, and one way is to use $m$ in the curve's equation as $x$ and calculate the according $y$.
Choose a secret key $x \in_R [1,N-1]$ randomly, publish the point $Y=x P$ as public key.
Encryption: choose random $k\in_R [1,N-1]$ , then calculate $C=kP$ and $C'=kY$ and calculate $P_m = f(m)$. The ciphertext is the tuple $(C, C'+P_m)$.
Decryption: From a ciphertext $(C,D)$ calculate $C' = xC$, and retrieve the point $P_m$ with $P_m = D-C' = (k(xP)+P_m)-(x(kP))$. Then calculate the message $m$ with $f^{-1}(P_m)$.