The Diffie-Hellman key exchange is a cryptographic protocol for symmetric key exchange. There are various implementations of this protocol. The following interchange between Alice and Bob demonstrates the Elliptic Curve Diffie-Hellman key exchange.

• 1) Alice and Bob publicly agree on an elliptic curve $E$ over a large finite field $F$ and a point $P$ on that curve.
• 2) Alice and Bob each privately choose large random integers, denoted $a$ and $b$
• 3) Using elliptic curve point-addition, Alice computes $aP$ on $E$ and sends it to Bob. Bob computes $bP$ on $E$ and sends it to Alice.
• 4) Both Alice and Bob can now compute the point $abP$ Alice by multipliying the received value of $bP$ by her secret number $a$ and Bob vice-versa.
• 5) Alice and Bob agree that the $x$ coordinate of this point will be their shared secret value.

An evil interloper Eve observing the communications will be able to intercept only the objects $E$ $P$ $aP$ and $bP$ She can succeed in determining the final secret value by gaining knowledge of either of the values $a$ or $b$ Thus, the security of the exchange depends on the hardness of that problem, known as the elliptic curve discrete logarithm problem. For large $a$ and $b$ it is a computationally ``difficult'' problem.

As a side note, some care has to be taken to choose an appropriate curve $E$ Singular curves and ones with ``bad" numbers of points on it (over the given field) have simplified solutions to the discrete log problem.