Semi-homomorphic encryption and multiparty computation

TLDR: In this article, the authors define the relaxed notion of a semi-homomorphic encryption scheme, where the plaintext can be recovered as long as the computed function does not increase the size of the input "too much".

Abstract: An additively-homomorphic encryption scheme enables us to compute linear functions of an encrypted input by manipulating only the ciphertexts. We define the relaxed notion of a semi-homomorphic encryption scheme, where the plaintext can be recovered as long as the computed function does not increase the size of the input "too much". We show that a number of existing cryptosystems are captured by our relaxed notion. In particular, we give examples of semi-homomorphic encryption schemes based on lattices, subset sum and factoring. We then demonstrate how semi-homomorphic encryption schemes allow us to construct an efficient multiparty computation protocol for arithmetic circuits, UC-secure against a dishonest majority. The protocol consists of a preprocessing phase and an online phase. Neither the inputs nor the function to be computed have to be known during preprocessing. Moreover, the online phase is extremely efficient as it requires no cryptographic operations: the parties only need to exchange additive shares and verify information theoretic MACs. Our contribution is therefore twofold: from a theoretical point of view, we can base multiparty computation on a variety of different assumptions, while on the practical side we offer a protocol with better efficiency than any previous solution.