Paper 2025/350

Bootstrapping with RMFE for Fully Homomorphic Encryption

Abstract

There is a heavy preference towards instantiating BGV and BFV homomorphic encryption schemes where the cyclotomic order $m$ is a power of two, as this admits highly efficient fast Fourier transformations. Field Instruction Multiple Data (FIMD) was introduced to increase packing capacity in the case of small primes and improve amortised performance, using reverse multiplication-friendly embeddings (RMFEs) to encode more data into each SIMD slot. However, FIMD currently does not admit bootstrapping. In this work, we achieve bootstrapping for RMFE-packed ciphertexts with low capacity loss. We first adapt the digit extraction algorithm to work over RMFE-packed ciphertexts, by applying the recode map after every evaluation of the lifting polynomial. This allows us to follow the blueprint of thin bootstrapping, performing digit extraction on a single ciphertext. To achieve the low capacity loss, we introduce correction maps to the Halevi-Shoup digit extraction algorithm, to remove all but the final recode of RMFE digit extraction. We implement several workflows for bootstrapping RMFE-packed ciphertexts in HElib, and benchmark them against thin bootstrapping for $m=32768$. Our experiments show that the basic strategy of recoding multiple times in digit extraction yield better data packing, but result in very low remaining capacity and latencies of up to hundreds of seconds. On the other hand, using correction maps gives up to $6$ additional multiplicative depth and brings latencies often below $10$ seconds, at the cost of lower packing capacity.