# Difference between Non-Interactive Secure Multiparty Computation (NISMPC) and Fully Homomorphic Encryption (FHE)

Until recently, I only knew about SMPC* and FHE, but now I just encountered the term NISMPC. I was wondering what is their difference and what is their difference in their use cases?

• Sometimes in bibliography, SMPC is referred to just as MPC referring to the same thing.
• Can you link where you've seen this term? Commented Apr 11, 2022 at 3:40

## 1 Answer

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 $$\mathsf{output}\gets\mathsf{MIFE}(c_1,\ldots, c_n)$$.