Computing the permanent $\bmod p$  of an $n\times n$ $0/1$-matrix is $\#P$-complete if $p$ is a prime $p>n$. We have an FPTAS for approximating the $0/1$ integer matrix permanent over the reals. Moreover we can compute the permanent of an $n\times n$ $0/1$-matrix in polynomial time if its value is bounded by $n^c$ for a fixed $c>0$.Is it possible to compute the permanent $\bmod p$ of a $0/1$ $n\times n$ integer matrix in polynomial time if its value is bounded by $(\log p)^c$ for a fixed $c>0$, where $p>n$ holds.  
On the permanent mod $p$
cc.complexity theory;permanent