Batched Fully Homomorphic Encryption from TFHE

A surge of interest in fully homomorphic encryption (FHE) has produced various FHE constructions in recent years. One of the main goals of these FHE schemes is to improve the efficiency when using the bootstrapping technique. Particularly, TFHE, a fast FHE scheme over the torus, decreases the running time of the bootstrapping significantly. However, it only supports a single-bit bootstrapping, and thus still has high computation cost. In this paper, we propose a batched FHE from TFHE, which can simultaneously bootstrap n torus-based LWE (TLWE) ciphertexts. In more detail, we employ a binary packing tree to pack several TLWE ciphertexts into one TLWE ciphertext, and further optimize the bootstrapping procedure with the homomorphic number theoretic transform (NTT). As a consequence, the amortized computation cost of our batched FHE scheme is reduced to O(logN) from O(N) in the original TFHE scheme, where N is the dimension of the TLWE assumption. Moreover, we provide an implementation of the proposed FHE scheme with specific parameters, so as to demonstrate its practicability.