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
