A Full RNS Variant of Approximate Homomorphic Encryption

  • Jung Hee Cheon
  • Kyoohyung Han
  • Andrey Kim
  • Miran Kim
  • Yongsoo SongEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11349)


The technology of Homomorphic Encryption (HE) has improved rapidly in a few years. The newest HE libraries are efficient enough to use in practical applications. For example, Cheon et al. (ASIACRYPT’17) proposed an HE scheme with support for arithmetic of approximate numbers. An implementation of this scheme shows the best performance in computation over the real numbers. However, its implementation could not employ a core optimization technique based on the Residue Number System (RNS) decomposition and the Number Theoretic Transformation (NTT).

In this paper, we present a variant of approximate homomorphic encryption which is optimal for implementation on standard computer system. We first introduce a new structure of ciphertext modulus which allows us to use both the RNS decomposition of cyclotomic polynomials and the NTT conversion on each of the RNS components. We also suggest new approximate modulus switching procedures without any RNS composition. Compared to previous exact algorithms requiring multi-precision arithmetic, our algorithms can be performed by using only word size (64-bit) operations.

Our scheme achieves a significant performance gain from its full RNS implementation. For example, compared to the earlier implementation, our implementation showed speed-ups 17.3, 6.4, and 8.3 times for decryption, constant multiplication, and homomorphic multiplication, respectively, when the dimension of a cyclotomic ring is 32768. We also give experimental result for evaluations of some advanced circuits used in machine learning or statistical analysis. Finally, we demonstrate the practicability of our library by applying to machine learning algorithm. For example, our single core implementation takes 1.8 min to build a logistic regression model from encrypted data when the dataset consists of 575 samples, compared to the previous best result 3.5 min using four cores.


Homomorphic encryption Approximate arithmetic Residue number system 



This work was partially supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIT) (No. 2017R1A5A1015626). M. Kim was supported by the National Institute of Health (NIH) under award number U01EB023685 and R01GM118574 as well as Cancer Prevention Research Institute of Texas (CPRIT) grant RR180012.


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Authors and Affiliations

  1. 1.Seoul National UniversitySeoulSouth Korea
  2. 2.University of Texas, Health Science Center at HoustonHoustonUSA
  3. 3.University of California, San DiegoLa JollaUSA

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