Elliptic Curve Analogues of Diffie-Hellman

The Diffie-Hellman key exchange from Section 3.1 works well on an elliptic curve with no serious modification. Michael and Nikita agree on a secret key as follows:

1. Michael and Nikita agree on a prime $p$ , an elliptic curve $E$ over $\mathbb{Z}/p\mathbb{Z}{}$ , and a point $P\in E(\mathbb{Z}/p\mathbb{Z}{})$ .
2. Michael secretly chooses a random $m$ and sends $m P$ .
3. Nikita secretly chooses a random $n$ and sends $nP$ .
4. The secret key is $nmP$ , which both Michael and Nikita can compute.

Presumably, an adversary can not compute $nmP$ without solving the discrete logarithm problem (see Problem 3.1.2 and Section 6.4.3 below) in $E(\mathbb{Z}/p\mathbb{Z}{})$ . For well-chosen $E$ , $P$ , and $p$ experience suggests that the discrete logarithm problem in $E(\mathbb{Z}/p\mathbb{Z}{})$ is much more difficult than the discrete logarithm problem in $(\mathbb{Z}/p\mathbb{Z}{})^*$ (see Section 6.4.3 for more on the elliptic curve discrete log problem).