Fully homomorphic encryption over the [discretized] torus

Fully Homomorphic Encryption (FHE) enables computations on encrypted data, and its adoption is growing across public and private sectors due to recent advancements. A paper presented at CHES 2022 explores TFHE, a torus-based FHE scheme, and its implementation on a discretized torus, emphasizing programmable bootstrapping techniques. The study categorizes FHE schemes into two families based on plaintext encoding positions and highlights the use of the discretized torus for LWE-based schemes, which supports various message formats and clarifies the absence of native ciphertext multiplication. The paper suggests using a scheme like TGSW for integer-based plaintexts and stresses the importance of efficient external product operations to manage noise propagation. Additionally, it provides a detailed explanation of bootstrapping, referencing Gentry’s 'recryption' technique, and identifies the rounding operation during LWE decryption as a potential area for optimization to enhance bootstrapping performance, ultimately facilitating functional circuits through programmable bootstrapping.