Let $E$ be an elliptic curve over a finite field $k$, and let $n$ be a positive integer not divisible by the characteristic of $k$. Suppose $\bar{k}$ is an algebraic closure of $k$, and $\bar{E}=\bar{k}\otimes_k E$. Miller's algorithm gives an efficient way to compute cup products of normalized classes of $\bar{E}$ with coefficients in $\mathbb{Z}/n$ or $\mu_n$. This algorithm is an essential tool for key sharing in cryptography. In this talk, we will discuss a recent extension of Miller's algorithm to the cup products of normalized classes of $E$. This result cannot be generalized to higher genus curves. This is joint work with Ted Chinburg.
