Paper 2024/1379

EvalRound+ Bootstrapping and its Rigorous Analysis for CKKS Scheme

Abstract

Bootstrapping stands as a fundamental component of fully homomorphic encryption (FHE) schemes, facilitating an infinite number of operations by recovering the ciphertext modulus. This work is aimed at significantly reducing the consumption of modulus in bootstrapping, thereby enhancing the efficiency of FHE performance, specifically for the Cheon--Kim--Kim--Song (CKKS) scheme proposed by Cheon et al. Building on the EvalRound bootstrapping method proposed by Kim et al., which includes the steps of ModRaise, CoeffToSlot, EvalRound and SlotToCoeff, we introduce $\textrm{EvalRound}^{+}$ bootstrapping. This bootstrapping inherits the advantage of EvalRound bootstrapping in CoeffToSlot and resolves its disadvantage in SlotToCoeff. Furthermore, we conduct a set of rigorous and comprehensive analyses to precisely determine the optimal choices of the parameters. The implementation of $\textrm{EvalRound}^{+}$ bootstrapping, along with optimal choices, has achieved a reduction in modulus consumption by over $40\%$ for CoeffToSlot and SlotToCoeff. Additionally, it has increased the number of levels for general multiplication by 2-4 in the most widely used bootstrapping parameter sets.