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 let \bar{E} be the base change of E to \bar{k}. Miller's algorithm gives an efficient way to compute cup products of normalized first cohomology classes of \bar{E} with coefficients in Z/n or mu_n. This algorithm is an essential tool for key sharing in cryptography. In this talk, I will show the somewhat unexpected result that cup products of normalized first cohomology classes of E are determined by their restrictions to \bar{E}. The same conclusion does not hold in general if E is replaced by a higher genus curve.