Bootstrapping in approximate fully homomorphic encryption: a research survey

Abstract Fully homomorphic encryption (FHE) has emerged as a prominent area of cryptographic research in recent years, offering the capability to perform computations on ciphertext without compromising data privacy. Among various FHE schemes, the Cheon–Kim–Kim–Song (CKKS) algorithm for approximate homomorphic encryption has gained prominence due to its efficient handling of floating-point operations. Bootstrapping, a critical technique that enables unlimited homomorphic operations by refreshing noisy ciphertexts, represents both the most complex and essential component of practical FHE implementations. This survey provides a comprehensive analysis of bootstrapping techniques in CKKS, examining their evolution from the original proposal to current state-of-the-art methods. Recent literature has witnessed a proliferation of novel bootstrapping schemes for CKKS, these diverse approaches often emphasize different performance aspects, leading to a lack of a unified quantitative framework for comparative analysis. To address this gap, we systematically categorize existing approaches into three main directions: optimization of homomorphic modular reduction, optimization of encoding/decoding operations, and development of alternative constructions using blind rotation techniques. Through detailed comparative analysis, we identify that current schemes can achieve either high throughput (processing over 1000 ciphertexts per second) or high precision (up to 400 bits), but exhibit limitations in concurrent optimization of both parameters. Furthermore, potential directions for future optimizations are explored and discussed, contributing to the ongoing development of efficient and practical FHE systems.