Is it possible to perform an MPC protocol to compute a product in a polynomial ring? I hope not to be asking much, but I have N parties, each one holding a polynomial in with 0-1 coefficients and fixed degree $n-1$. I was wandering if it is possible (I mean feasible) to compute the product of all of these polynomials with an MPC protocol.

Actually, for security I would need the result modulo $X^n+1$, so my actual request is the following : Given $N$ polynomials $P_1,P_2,\dots,P_N$ with $0-1$ coefficients and degree $n-1$, compute $(\prod P_i)\mod(X^n+1)$ with a multiparty computation protocol where each party holds a $P_i$.

My question regards the complexity of such a product ($O(n \log n\log N)$ integer multiplications if an FFT algorithm is used ?) 

It is my first question here, so please attach a welcome message to you answer :-)
You mention a ring but you are only using the multiplication operation. So it's really just a protocol for a group operation that you want. If the group is abelian, then there is a simple classical protocol for $n$-party group product:


Each party performs $n$-out-of-$n$ multiplicative secret sharing of their input. In more detail, party $i$ has input $P_i$ and chooses random polynomials $P_{i,j}$ so that $\prod_j P_{i,j} = P_i$, then privately sends $P_{i,j}$ to party $j$.
Each party $j$ now has $\{ P_{i,j} \mid i \in [n] \}$. He/she computes their product $P'_j = \prod_i P_{i,j}$ and publicly announces $P'_j$.
If the group is abelian, then $\prod_j P'_j = \prod_{i,j} P_{i,j} = \prod_i P_i$, so everyone can compute the final answer from the public $P'_j$ values.


Furthermore, the view of any number of colluding semi-honest parties leaks no more than the final result.