According to my understanding there are some pretty solid seeming roadblocks to carrying out an index calculus on an elliptic curve over a finite field. The general strategy is to take points over $E(\mathbb{F}_p)$, lift them to points over $E(\mathbb{Q})$, and then hope that the curve over $\mathbb{Q}$ has high rank. The two major obstructions seem to be
It is very hard to find elliptic curves over $\mathbb{Q}$ with high ranks, and
It is highly unlikely to find a reasonable factor basis - that is, a collection of points on the curve with a bounded height.
This approach assumes that we lift the problem to one over the rationals $\mathbb{Q}$. I was wondering if anyone has any idea whether such an algorithm could be carried out by lifting the curve to an elliptic surface, i.e. an elliptic curve over the function field $\mathbb{F}_p(T)$ of some curve? At the very least we seem to know more about how the ranks of these things behave better than in the number field case.
I am not sure what I am asking even makes sense, so an explanation of its nonsense is welcome as well.