TITLE: Does the difficulty of discrete logarithm depend on the difficulty of integer factorization?
QUESTION [5 upvotes]: The security of many (most? all?) public-key cryptography systems are based on the difficulty of the discrete logarithm or integer factorization. Are these two problems related at all?
With the discrete logarithm problem I mean the following problem:

Let $G$ be a group and let $g$ and $h$ be elements of $G$ such that $h \in \langle g \rangle$. Find an integer $x$ such that $g^x = h$.

In particular I am interested in the case where $G = \mathbb{Z}_p^*$. If we know an efficient way to factor integers, does this help us in solving the discrete logarithm problem for a given $\mathbb{Z}_p^*$? 
Conversely, if we know how to solve  the discrete logarithm problem efficiently for any group $G$ (or just $\mathbb{Z}_p^*$), does this help with factoring integers?

REPLY [2 votes]: Are these two problems related at all?

The answer is yes, and it is at many levels. The general number field sieve, which is generally thought of as a factoring algorithm, is also very useful for solving discrete logs.
Theoretically:
Solving the discrete log with a composite modulus is exactly as hard as factoring [3]. However, it is not known if solving discrete log with a prime modulus leads to efficient factoring [4].

If we know an efficient way to factor integers, does this help us in solving the discrete logarithm problem for a given $\mathbb{Z}_p^*$?
Conversely, if we know how to solve the discrete logarithm problem efficiently for any group G  (or just $\mathbb{Z}_p^*$), does this help with factoring integers?

For composite modulus, the answer is yes. For prime modulus, we don't know.
A good reference for a lot of this information is John Gregg's thesis page 43.