To my best knowledge, the notion of non-interactive secure multiparty computation was introduced by Beimel et. al.. In the original paper,
... It (NIMPC) can also be viewed as a simpler
and weaker variant of the recently introduced notion of multi-input functional encryption...
And multi-input functional encryption (MI-FE) is first introduced in this paper.
Multi-input functional is a very general tool for computing
on encrypting data, which allows for mining aggregate information from several different data
sources (rather than just a single source as in single input functional encryption).
For a particular use cas, say $n$ parties, each holding a secret $x_i$, want to perform secure computations on $f(x_1,\ldots, x_n)$. For FHE, we first create a key pair $(\mathsf{sk}, \mathsf{pk})$, and then let each party encrypt its message with the same encryption key $c_i\gets\mathsf{Enc}(\mathsf{pk}, x_i)$, and evaluate the output ciphers.
For MI-FE, we first create a master key $\mathsf{msk}$, and then derivate $n$ different encryption keys (denoted as $\mathsf{ek}_i$). Each party then performs $c_i\gets \mathsf{Enc}(\mathsf{ek}_i, x_i)$, and then performs $\mathsf{output}\gets\mathsf{MIFE}(c_1,\ldots, c_n)$.
