Additively Homomorphic Encryption with t-Operand Multiplications

Homomorphic encryption schemes are an essential ingredient to design protocols where different users interact in order to obtain some information from the others, at the same time that each user keeps private some of his information. When the algebraic structure underlying these protocols is complicated, then standard homomorphic encryption schemes are not enough, because they do not allow to compute at the same time additions and products of plaintexts through the manipulation of ciphertexts. In this work we define a theoretical object, t-chained encryption schemes, which can be used to compute additions and products of t integer values, by ciphertext manipulation. Efficient solutions have been previously proposed only for the case t = 2. Our solution is not only theoretical: we show that some existing IND-CPA secure (pseudo)homomorphic encryption schemes (some of them based on lattices) can be used to implement in practice the concept of t-chained encryption scheme.