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A Full RNS Variant of Approximate Homomorphic Encryption.

Jung Hee Cheon1, Kyoohyung Han1, Andrey Kim1

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|April 19, 2021
PubMed
Summary

This study introduces an optimized approximate homomorphic encryption scheme, enabling faster computations on encrypted real numbers using Residue Number System (RNS) and Number Theoretic Transform (NTT) optimizations. Practical applications, including machine learning, show significant speedups.

Keywords:
Homomorphic encryptionapproximate arithmeticresidue number system

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Area of Science:

  • Cryptography and Information Security
  • Applied Mathematics and Computer Science

Background:

  • Homomorphic encryption technology has advanced, with recent schemes supporting approximate number arithmetic.
  • Existing approximate homomorphic encryption schemes, while performant, lack optimizations like Residue Number System (RNS) decomposition and Number Theoretic Transform (NTT).

Purpose of the Study:

  • To present a variant of approximate homomorphic encryption optimized for standard computer systems.
  • To enable efficient computation over encrypted real numbers by integrating RNS and NTT techniques.

Main Methods:

  • Introduced a novel ciphertext modulus structure facilitating RNS decomposition of cyclotomic polynomials and NTT on RNS components.
  • Developed new approximate modulus switching procedures that eliminate the need for RNS composition.
  • Implemented algorithms using only 64-bit word-size operations, avoiding multi-precision arithmetic.

Main Results:

  • Achieved significant performance gains through full RNS implementation, with speed-ups of 17.3x for decryption, 6.4x for constant multiplication, and 8.3x for homomorphic multiplication.
  • Demonstrated practical utility by applying the scheme to machine learning tasks, such as building a logistic regression model from encrypted data in 1.8 minutes (single core).
  • Outperformed previous methods, notably reducing the time for logistic regression model building by nearly half compared to a multi-core implementation.

Conclusions:

  • The proposed approximate homomorphic encryption scheme offers substantial performance improvements over prior methods.
  • The integration of RNS and NTT optimizations makes homomorphic encryption practical for real-world applications, including machine learning on encrypted data.
  • The scheme's efficiency allows for computations on encrypted data using standard hardware with significant speed advantages.