# homomorphic encryption special case of multi party computation?

I read that Fully Homomorphic Encryption schemes are special case of Secure MPC in page no 3. Especially , generalization of two party computation problems stated by Yao

But is there any additional literature showing the connections of the same ?

You've got the relationship a bit muddled. A scheme for fully homomorphic encryption can be used to perform secure multi-party computation (MPC). However, it's not a special case of MPC.

The connection is a standard elementary result, which I will summarize here. If you want to do secure multi-party computation, you can express the computation as a boolean circuit $C$, and you can easily transform any circuit so that it uses only AND gates and NOT gates. Then, it turns out that you can compute $C$ on encrypted data, if the data was encrypted using a fully homomorphic encryption scheme, using the following relationship: when working with $\{0,1\}$, AND can be done by multiplication ($x$ AND $y$ $= xy$), and NOT can be done with addition (NOT($x$) $= 1-x$). Since the fully homomorphic encryption lets you do addition, subtraction, and multiplication on encrypted values, it also lets you do NOT and AND on encrypted values, which is all you need to do secure multi-party computation.

Really the connection is intrinsic. There is at least one other paper I know of that mentions it specifically, however. That is the SPDZ paper. The relevant quotes are below for your convenience.

Recently, another approach has become possible with the advent of Fully Homomorphic Encryption (FHE) by Gentry. In this approach all parties first encrypt their input under the FHE scheme; then they evaluate the desired function on the ciphertexts using the homomorphic properties, and finally they perform a distributed decryption on the final ciphertexts to get the results.

and

As a conceptual contribution we propose what we believe is "the right" way to use FHE/SHE for computationally efficient MPC, namely to use it for implementing a preprocessing phase.

Update
Just came across some newer research you might be interested in:

We present a computationally secure MPC protocol for threshold adversaries which is parametrized by a value L. When L=2 we obtain a classical form of MPC protocol in which interaction is required for multiplications, as L increases interaction is reduced in that one requires interaction only after computing a higher degree function. When L approaches infinity one obtains the FHE based protocol of Gentry, which requires no interaction. Thus one can trade communication for computation in a simple way.

The connection between SMC, FHE techniques has been established recently (1)

We present a computationally secure MPC protocol for threshold adversaries which is parametrized by a value L. When L = 2 we obtain a classical form of MPC protocol in which interaction is required for multiplications, as L increases interaction is reduced in that one requires interaction only after computing a higher degree function. When L approaches infinity one obtains the FHE based protocol of Gentry, which requires nointeraction. Thus one can trade communication for computation in a simple way.

 Choudhury, Ashish, et al. "Between a rock and a hard place: Interpolating between mpc and fhe." Advances in Cryptology-ASIACRYPT 2013. Springer Berlin Heidelberg, 2013. 221-240. APA