Homomorphic Encryption for Multiple Users With Less Communications

Abstract

Keeping privacy for every entity in outsourced computation is always a crucial issue. For efficient secure computation, homomorphic encryption (HE) can be one of nice solutions. Especially, multikey homomorphic encryption (MKHE) which allows homomorphic evaluation on encrypted data under different keys can be one of the simplest solutions for a secure computation which handles multiple users' data. However, the current main problem of MKHE is that the dimension of its evaluated ciphertext relies on the number of users. To solve this problem, there are several variants of multikey homomorphic encryption schemes to keep the size of ciphertext constant for a fixed number of users. However, users interact one another before computation to provide their inputs, which increases setup complexity. Moreover, all the existing MKHE schemes and their variants have unique benefits which cannot be easily achieved at the same time in one scheme. In other words, each type of scheme has a suitable computational scenario to put its best performance. In this paper, we suggest more efficient evaluation key generation algorithms (relinearization key and bootstrapping key) for the existing variants of MKHE schemes which have no ciphertext expansion for a fixed number of users. Our method only requires a very simple and minor pre-processing; distributing public keys, which is not counted as a round at all in many other applications. Regarding bootstrapping, we firstly provide an efficient bootstrapping for multiple users which is the same as the base single-key scheme thanks to our simplified key generation method without a communication. As a result, participants have less communication, computation, and memory cost in online phase. Moreover, we provide a practical conversion algorithm between the two types of schemes in order to efficiently utilize both schemes' advantages together in more various applications. We also provide detailed comparison among similar results so that users can choose a suitable scheme for their homomorphic encryption based application scenarios.