Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Fast Implementation of Functional Bootstrapping and Private Information Retrieval

Guangsheng Ma, Hongbo Li

TL;DR

It is shown that employing a single quantum server can significantly enhance both the efficiency and security of privacy-preserving computation, and an efficient quantum algorithm is proposed for functional bootstrapping of large-precision plaintexts, reducing the time complexity from exponential to polynomial in plaintext-size compared to classical algorithms.

Abstract

Classical privacy-preserving computation techniques safeguard sensitive data in cloud computing, but often suffer from low computational efficiency. In this paper, we show that employing a single quantum server can significantly enhance both the efficiency and security of privacy-preserving computation. We propose an efficient quantum algorithm for functional bootstrapping of large-precision plaintexts, reducing the time complexity from exponential to polynomial in plaintext-size compared to classical algorithms. To support general functional bootstrapping, we design a fast quantum private information retrieval (PIR) protocol with logarithmic query time. The security relies on the learning with errors (LWE) problem with polynomial modulus, providing stronger security than classical ``exponentially fast'' PIR protocol based on ring-LWE with super-polynomial modulus. Technically, we extend a key classical homomorphic operation, known as blind rotation, to the quantum setting through encrypted conditional rotation. Underlying our extension are insights for the quantum extension of polynomial-based cryptographic tools that may gain dramatic speedups.

Quantum Fast Implementation of Functional Bootstrapping and Private Information Retrieval

TL;DR

It is shown that employing a single quantum server can significantly enhance both the efficiency and security of privacy-preserving computation, and an efficient quantum algorithm is proposed for functional bootstrapping of large-precision plaintexts, reducing the time complexity from exponential to polynomial in plaintext-size compared to classical algorithms.

Abstract

Classical privacy-preserving computation techniques safeguard sensitive data in cloud computing, but often suffer from low computational efficiency. In this paper, we show that employing a single quantum server can significantly enhance both the efficiency and security of privacy-preserving computation. We propose an efficient quantum algorithm for functional bootstrapping of large-precision plaintexts, reducing the time complexity from exponential to polynomial in plaintext-size compared to classical algorithms. To support general functional bootstrapping, we design a fast quantum private information retrieval (PIR) protocol with logarithmic query time. The security relies on the learning with errors (LWE) problem with polynomial modulus, providing stronger security than classical ``exponentially fast'' PIR protocol based on ring-LWE with super-polynomial modulus. Technically, we extend a key classical homomorphic operation, known as blind rotation, to the quantum setting through encrypted conditional rotation. Underlying our extension are insights for the quantum extension of polynomial-based cryptographic tools that may gain dramatic speedups.
Paper Structure (15 sections, 7 theorems, 27 equations, 1 figure, 5 tables)

This paper contains 15 sections, 7 theorems, 27 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Classical Fully Homomorphic Encryption
  4. Instance of Fully Homomorphic Encryption Based on the Learning With Errors
  5. Paillier Partially Homomorphic Encryption Scheme
  6. Quantum Fully Homomorphic Encryption
  7. Instance of Quantum Fully Homomorphic Encryption based on Pauli One-time Pad
  8. Hybrid Quantum-classical Private Information Retrieval
  9. Quantum Random Access Memory
  10. Private Information Retrieve with Classical Client and Single Quantum Server.
  11. Quantum Fast Implementation of Bootstrapping
  12. Quantum Blind Rotation
  13. Quantum Implementation of Bootstrapping
  14. Quantum Implementation of Functional Bootstrapping
  15. Paillier-based Quantum Private Information Retrieve with Classical Communication

Key Result

Proposition 2.6

The Paillier encryption map, PHE.Enc$(m,r):\mathbb{Z}_n \times Z^{*}_n\ \rightarrow Z^{*}_{n^2}$, is bijective .

Figures (1)

  • Figure 1: Quantum fast bootstrapping and functional bootstrapping. An input LWE encryption, after quantum blind rotation (Algorithm \ref{['123']}), can be converted into a combination of quantum and classical ciphertexts. These ciphertexts can then be merged into an encryption of either the original input message (Subsection \ref{['101']}) or its function value (Algorithm \ref{['909']}). Bootstrapping, with the parameters set to $L'=L$ and $m'=m$, ensures that the output LWE encryption matches the input parameters.

Theorems & Definitions (19)

  • Definition 2.1: Classical homomorphic encryption scheme
  • Definition 2.2: Classical pure FHE and leveled FHE
  • Definition 2.3: Learning with errors (LWE) problem regev2009lattices
  • Definition 2.4: Standard LWE encryption scheme
  • Definition 2.5: Paillier homomorphic encryption scheme
  • Proposition 2.6: paillier1999public
  • Definition 2.7: Quantum homomorphic encryption scheme
  • Definition 2.8: Pauli group and Clifford gates
  • Definition 2.9: Pauli one-time pad encryption
  • Lemma 2.10: Encrypted conditional rotation ma2022quantum
  • ...and 9 more