General Bootstrapping Approach for RLWE-Based Homomorphic Encryption

Homomorphic Encryption (HE) makes it possible to compute on encrypted data without decryption. In lattice-based HE, a ciphertext contains noise, which accumulates along with homomorphic computations. Bootstrapping refreshes the noise and it is possible to perform arbitrary-depth computations on HE with bootstrapping, which we call Fully Homomorphic Encryption (FHE). In this article, we propose a new general bootstrapping technique for RLWE-based schemes and its practical instantiation for FHE. It can be applied to all three RLWE-based leveled FHE schemes: Brakerski-Gentry-Vaikuntanathan (BGV), Brakerski/Fan-Vercauteren (BFV), and Cheon-Kim-Kim-Song (CKKS) with minor deviations in the algorithms. Our new construction of bootstrapping extracts a noiseless ciphertext for a part of the input, scales it, and finally removes it. In contrast with previous bootstrapping algorithms, the proposed method consumes only 1-2 levels and uses smaller parameters. For BGV and BFV, our new bootstrapping does not have any restrictions on a plaintext modulus unlike typical cases of the previous methods. The error introduced by our approach for CKKS is comparable to a rescaling error, allowing us to preserve a large amount of precision after bootstrapping.