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?