EvalRound Algorithm in CKKS Bootstrapping

Seonghak Kim, Minji Park, Jaehyung Kim, Taekyung Kim, Chohong Min

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations


Homomorphic encryption (HE) has opened an entirely new world up in the privacy-preserving use of sensitive data by conducting computations on encrypted data. Amongst many HE schemes targeting computation in various contexts, Cheon–Kim–Kim–Song (CKKS) scheme [8] is distinguished since it allows computations for encrypted real number data, which have greater impact in real-world applications. CKKS scheme is a levelled homomorphic encryption scheme, consuming one level for each homomorphic multiplication. When the level runs out, a special computational circuit called bootstrapping is required in order to conduct further multiplications. The algorithm proposed by Cheon et al. [7] has been regarded as a standard way to do bootstrapping in the CKKS scheme, and it consists of the following four steps: ModRaise, CoeffToSlot, EvalMod and SlotToCoeff. However, the steps consume a number of levels themselves, and thus optimizing this extra consumption has been a major focus of the series of recent research. Among the total levels consumed in the bootstrapping steps, about a half of them is spent in CoeffToSlot and SlotToCoeff steps to scale up the real number components of DFT matrices and round them to the nearest integers. Each scale-up factor is very large so that it takes up one level to rescale it down. Scale-up factors can be taken smaller to save levels, but the error of rounding would be transmitted to EvalMod and eventually corrupt the accuracy of bootstrapping. EvalMod aims to get rid of the superfluous qI term from a plaintext pt+ qI resulting from ModRaise, where q is the bottom modulus and I is a polynomial with small integer coefficients. EvalRound is referred to as its opposite, obtaining qI. We introduce a novel bootstrapping algorithm consisting of ModRaise, CoeffToSlot, EvalRound and SlotToCoeff, which yields taking smaller scale-up factors without the damage of rounding errors.

Original languageEnglish
Title of host publicationAdvances in Cryptology – ASIACRYPT 2022 - 28th International Conference on the Theory and Application of Cryptology and Information Security, Proceedings
EditorsShweta Agrawal, Dongdai Lin
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages27
ISBN (Print)9783031229657
StatePublished - 2022
Event28th International Conference on the Theory and Application of Cryptology and Information Security, ASIACRYPT 2022 - Taipei, Taiwan, Province of China
Duration: 5 Dec 20229 Dec 2022

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume13792 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference28th International Conference on the Theory and Application of Cryptology and Information Security, ASIACRYPT 2022
Country/TerritoryTaiwan, Province of China

Bibliographical note

Publisher Copyright:
© 2022, International Association for Cryptologic Research.


  • Bootstrapping
  • CKKS scheme
  • Homomorphic encryption


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