Multiparty Computation to calculate addition of shares without revealling individual shares I have $n$ persons, each holding a secret integer $x_i$ ($i$ from $1$ to $n$) and I'm looking for a way for them to jointly compute the sum of these secrets without revealing to each other their individual secrets.
Verification of the sum isn't important in this problem, as an error will cause an abort at a later point. The important issue is that they not get information about the numbers held by the other parties.
My other question is can this still be done / does it require a different solution if the addition is over a finite field?
I have $n$ persons, each holding a secret integer $x_i$ ($i$ from $1$ to $n$) and I'm looking for a way for them to jointly compute the sum of these secrets without revealing to each other their individual secrets.

A simple application of arithmetic secret-sharing based secure multi-party computation (&quot;arithmetic GMW&quot;) can do that.
The protocol for that goes as follows:

(One-Time Setup) Pick a prime $p$ that is guaranteed to be larger than the sum of all input values (you can safely overshoot the sume by a large margin here, e.g. pick a prime larger than $2^{96}$ for 64-bit inputs). This prime will be used to define the finite field we will work in. If you already have a preferred one, use it here. All arithmetic that follows will be carried out in this field.
Have every party $P_i$ choose $n-1$ random integers uniformly at random from the finite field. Call them $s_{i,j}$ and let $s_{i,n}=x_i-\sum_{j=1}^{n-1}s_{i,j}$. This creates an arithmetic $n$-out-of-$n$ sharing of $x_i$
Have every party $P_i$ send $s_{i,j}$ to $P_{j}$.
Have every party $P_i$ compute $r_i=\sum_{\ell=1}^{n}s_{\ell,i}$, that is, have them all sum up all the shares they receive.
Have every party $P_i$ broadcast their result $r_i$ to all other parties (or only to the one needing to know the sum).
Have every party with with all $r_j$ compute $\sigma=\sum_{j=1}^{n}r_j$. This value is the desired sum.