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Blind Rotation in Fully Homomorphic Encryption with Extended Keys

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Cyber Security, Cryptology, and Machine Learning (CSCML 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13301))

Abstract

Most solutions for fully homomorphic encryption rely on hard lattice problems. Accordingly, the resulting ciphertexts must contain a certain level of noise to guarantee the security of the encryption. Running homomorphic operations on these noisy ciphertexts in turn further increases the noise level in the resulting ciphertexts. If the noise exceeds a given threshold, the ciphertexts are no longer decryptable. Bootstrapping enables to deal with this issue by resetting the noise present in a ciphertext to a nominal level.

Certain fully homomorphic encryption schemes require the use of binary keys for the bootstrapping operation. This paper describes how to extend the underlying blind rotation so as to efficiently support a wider number of key formats. It also investigates a multi-digit approach wherein multiple key digits are processed concurrently. All in all, the proposed solutions offer more flexibility in the parameter selection and yield a variety of new trade-offs for better performance.

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Notes

  1. 1.

    As originally described, TFHE is defined over the real torus \(\mathbb {R}/\mathbb {Z}\). We rather consider the discretized torus \(q^{-1}\mathbb {Z}/\mathbb {Z}\) and identify its elements with integers modulo q.

  2. 2.

    Starting at \(i=0\).

  3. 3.

    Available at https://bitbucket.org/malb/lwe-estimator/src/master/.

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Acknowledgements

We are grateful to Ben Curtis for his help in compiling Tables 2 and 3. We are also grateful to the anonymous referees for useful comments.

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Authors and Affiliations

  1. Zama, Paris, France

    Marc Joye & Pascal Paillier

Authors
  1. Marc Joye
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  2. Pascal Paillier
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Correspondence to Marc Joye .

Editor information

Editors and Affiliations

  1. Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Shlomi Dolev

  2. University of Maryland, College Park, MD, USA

    Jonathan Katz

  3. Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Amnon Meisels

A Tables

A Tables

Although for higher radices the number of external products remains equal to n, the value of n is a decreasing function of m. Playing with LWE Estimator, at a security level of 128 bits, we get the following tables.

Table 2. LWE dimension n as a function of m for different values for the noise standard deviation \(\sigma \), for \(q = 2^{32}\)
Full size table
Table 3. LWE dimension n as a function of m for different values for the noise standard deviation \(\sigma \), for \(q = 2^{64}\)
Full size table

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Joye, M., Paillier, P. (2022). Blind Rotation in Fully Homomorphic Encryption with Extended Keys. In: Dolev, S., Katz, J., Meisels, A. (eds) Cyber Security, Cryptology, and Machine Learning. CSCML 2022. Lecture Notes in Computer Science, vol 13301. Springer, Cham. https://doi.org/10.1007/978-3-031-07689-3_1

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  • DOI: https://doi.org/10.1007/978-3-031-07689-3_1

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  • Online ISBN: 978-3-031-07689-3

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