A Full RNS Variant of FV Like Somewhat Homomorphic Encryption Schemes

Abstract

Since Gentry’s breakthrough work in 2009, homomorphic cryptography has received a widespread attention. Implementation of a fully homomorphic cryptographic scheme is however still highly expensive. Somewhat Homomorphic Encryption (SHE) schemes, on the other hand, allow only a limited number of arithmetical operations in the encrypted domain, but are more practical. Many SHE schemes have been proposed, among which the most competitive ones rely on Ring Learning With Errors (RLWE) and operations occur on high-degree polynomials with large coefficients. This work focuses in particular on the Chinese Remainder Theorem representation (a.k.a. Residue Number Systems) applied to the large coefficients. In SHE schemes like that of Fan and Vercauteren (FV), such a representation remains hardly compatible with procedures involving coefficient-wise division and rounding required in decryption and homomorphic multiplication. This paper suggests a way to entirely eliminate the need for multi-precision arithmetic, and presents techniques to enable a full RNS implementation of FV-like schemes. For dimensions between \(2^{11}\) and \(2^{15}\), we report speed-ups from \(5{\times }\) to \(20{\times }\) for decryption, and from \(2{\times }\) to \(4{\times }\) for multiplication.