Paper 2025/888

Bootstrapping GBFV with CKKS

Abstract

Generalized BFV (GBFV) replaces the integer plaintext modulus of BFV with a polynomial plaintext modulus, yielding a flexible tradeoff between precision and SIMD parallelism. This framework encompasses both BFV and CLPX, with CLPX corresponding to the extreme large-precision endpoint where very large plaintext moduli can be supported without increasing the underlying RLWE dimension. However, bootstrapping remains a bottleneck throughout the large-precision GBFV regime: existing GBFV bootstrapping methods rely on BFV-style linear transformations, whose noise growth becomes prohibitive as the plaintext precision grows. We introduce a GBFV bootstrapping method that treats the noise of a GBFV ciphertext as a CKKS plaintext and bootstraps it through CKKS. The main obstacle is that the BFV-from-CKKS bootstrapping technique of Kim et al. (CCS'24) relies on integer plaintext scaling and does not directly apply to GBFV, whose scaling factor involves a polynomial plaintext modulus. We overcome this by developing a noise-extraction and recovery procedure tailored to polynomial plaintext moduli. Our implementation bootstraps large-precision GBFV instances, including CLPX for the first time, supporting plaintext moduli of more than $500{,}000$ bits in less than $20$ seconds. We also introduce GBFV-to-CKKS and CKKS-to-GBFV conversions, interpreting the former as an approximate digit-decomposition toolkit for non-arithmetic operations over large GBFV plaintext spaces.