Paper 2024/1105

A New CRT-based Fully Homomorphic Encryption

Abstract

The idea of computing on encrypted data without decryption dates back to the notion of privacy homomorphisms introduced by Rivest, Adleman, and Dertouzos (1978). Their proposals built using the elegant structure of Chinese Remainder Theorem (CRT), were later shown to be insecure under simple known-plaintext attacks. Subsequent CRT-based fully homomorphic encryption (FHE) over the integers addresses this algebraic transparency by injecting noise and basing security on approximate common divisor–type assumptions, but the resulting designs are burdened by large public keys and costly ciphertext refresh procedures. In this work, we develop a new CRT-based FHE scheme whose security relies on the Ring-LWE (RLWE) hardness assumption. For this purpose, we introduce the CRT-RLWE problem. We show that the problem is at least as hard as the RLWE, thereby positioning our construction within the established post-quantum security landscape of RLWE-based cryptography. Our scheme retains an explicit CRT embedding, separating a message component modulo a prime-power plaintext modulus and an auxiliary CRT component, while using RLWE-style key and ring arithmetic for compactness and efficiency. Finally, we make it fully homomorphic by using a new bootstrapping procedure, that adopts the recryption paradigm for BGV/BFV schemes utilizing the linear transformation and digit extraction techniques.