Exploring the Feasibility of Fully Homomorphic Encryption

In 2010, Gentry and Halevi presented the first FHE implementation. FHE allows the evaluation of arbitrary-functions directly on encrypted data on untrusted servers. However, even for the small setting with 2048 dimensions, the authors reported a performance of 1.8 s for a single bit encryption and 32 s for recryption on a high-end server. Much of the latency is due to computationally intensive multimillion-bit modular multiplications. In this paper, we introduce two optimizations coupled with a novel precomputation technique. In the first optimization called partial FFT, we adopt Strassen´s FFT-based multiplication algorithm along with Barret reduction to speedup modular multiplications. For the encrypt primitive, we employ a window-based evaluation technique along with a modest degree of precomputation. In the full FFT optimization, we delay modular reductions and change the window algorithm, which allows us to carry out the bulk of computations in the frequency domain. We manage to eliminate all FFT conversion except the final inverse transformation drastically reducing the computation latency for all FHE primitives. We implemented the GH FHE scheme on two GPUs to further speedup the operations. Our experimental results with small parameter setting show speedups of 174, 7.6, and 13.5 times for encryption, decryption and recryption, respectively, when compared to the Gentry-Halevi implementation. The speedup is enhanced in the medium setting. However, in the large setting, memory becomes the bottleneck and the speedup is somewhat diminished.