TITLE: Problems known to be in both NP and coNP, but not known to be in P
QUESTION [43 upvotes]: One such problem I know is integer factorization.
What are other interesting cases?

REPLY [4 votes]: Since you mentioned integer factoring, an analogous problem is the discrete log problem.
Given the cyclic group $G = \mathbb{Z}^*_p$ for a prime $p$ and any generator $g$ of $G$ along with another $h \in G$ (which will also be a generator), the discrete log asks to find $x \in \mathbb{Z}_{p-1}$ such that $g^x = h$. To convert to decision problem, let $k$ be any positive integer $< \log_2(p-1)$. The problem is then: 
Is the $k^{th}$ bit of $x$ true? 
This is also in NP $\cap$ coNP but not known to be in P.