Relationship between Elliptic Curve Discrete Log, Integer Discrete Log, and Integer Factorization
Actually, there are two known reductions among these three problems:


If you can solve discrete logs in $Z^*_n$ for composite $n$, you can use that to efficiently factor $n$
If you can solve discrete logs in $GF(p^k)$, you can compute discrete logs in an Elliptic Curve over $GF(p)$.  That's because there is a known mapping of Elliptic Curves over $GF(p)$ into $GF(p^k)$ (for an integer $k$ that depends on the curve) that preserves the group operation; hence you can solve discrete logs over an Elliptic Curves by mapping the generator and the target into $GF(p^k)$, and solving the discrete log there.  Now, for curves used in practice, $k$ is large enough that, with the current discrete log algorithms, this method is actually slower than just using a generic discrete log algorithm in the elliptic curve.


Of course, neither of the above reductions would apply to the recently discovered algorithms for doing discrete logs over small characteristic fields; for the former, we're not talking about a field at all ($Z^*_n$ is a ring, not a field), and for prime elliptic curves (which is what we use in practice), $GF(p^k)$ has a very large characteristic.