Threshold Fully Homomorphic Encryption Over the Torus

Fully homomorphic encryption (FHE) enables arithmetic operations to be performed over plaintext by operations on undecrypted ciphertext. The Chillotti-Gama-Georgieva-Izabachene (CGGI) scheme is a typical FHE scheme, has attracted attention because of its fast bootstrapping and the availability of open-source implementation software. A threshold FHE (ThFHE) scheme has protocols for distributed key generation and distributed decryption that are executed cooperatively among the parties while keeping the decryption key distributed among them. It is useful for secure computations with inputs from multiple parties. However, a ThFHE scheme based on CGGI has yet to be proposed. In this paper, we propose a client-aided ThFHE scheme based on CGGI. Our scheme achieves the same bootstrapping as CGGI without affecting the noise analysis or any CGGI parameter. Therefore, existing open-source software implementing CGGI can easily be extended to our scheme, a ThFHE variant of the CGGI scheme, without changing the implementation part regarding homomorphic operations.