ReSBM: Region-based Scale and Minimal-Level Bootstrapping Management for FHE via Min-Cut

The RNS-CKKS scheme in Fully Homomorphic Encryption (FHE) supports crucial features for privacy-preserving machine learning, such as fixed-point arithmetic and SIMDstyle vectorization. Yet, managing the escalation of ciphertext scales from homomorphic multiplications, which risks capacity overflow, along with bootstrapping, presents significant challenges. These complexities are exacerbated by the need to efficiently handle scale and bootstrapping at compile time while ensuring rapid encrypted inference.

In this paper, we present ReSBM, a novel compiler technique that simultaneously optimizes scale and bootstrapping for encrypted inference under RNS-CKKS. By partitioning a program's data flow graph (DFG) into regions with a uniform multiplicative depth of one, ReSBM ensures that placements of Scale Management Operations (SMOs) and bootstraps affect only the latency of a region, not the scales and levels of its live-out ciphertexts. Our region-based approach tackles the NP-hard challenge of optimal bootstrapping placement with hierarchical strategies: (1) optimal intra-region SMO and bootstrapping placement using min-cut, (2) bootstrappingguided rescaling region identification across a sequence of regions, culminating in tentative bootstrapping at two terminal regions, and (3) minimal-level bootstrap placement across the DFG, elevating ciphertexts only to the necessary minimal