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2019 | OriginalPaper | Buchkapitel

Homomorphic Secret Sharing from Lattices Without FHE

verfasst von : Elette Boyle, Lisa Kohl, Peter Scholl

Erschienen in: Advances in Cryptology – EUROCRYPT 2019

Verlag: Springer International Publishing

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Abstract

Homomorphic secret sharing (HSS) is an analog of somewhat- or fully homomorphic encryption (S/FHE) to the setting of secret sharing, with applications including succinct secure computation, private manipulation of remote databases, and more. While HSS can be viewed as a relaxation of S/FHE, the only constructions from lattice-based assumptions to date build atop specific forms of threshold or multi-key S/FHE. In this work, we present new techniques directly yielding efficient 2-party HSS for polynomial-size branching programs from a range of lattice-based encryption schemes, without S/FHE. More concretely, we avoid the costly key-switching and modulus-reduction steps used in S/FHE ciphertext multiplication, replacing them with a new distributed decryption procedure for performing “restricted” multiplications of an input with a partial computation value. Doing so requires new methods for handling the blowup of “noise” in ciphertexts in a distributed setting, and leverages several properties of lattice-based encryption schemes together with new tricks in share conversion.

The resulting schemes support a superpolynomial-size plaintext space and negligible correctness error, with share sizes comparable to SHE ciphertexts, but cost of homomorphic multiplication roughly one order of magnitude faster. Over certain rings, our HSS can further support some level of packed SIMD homomorphic operations. We demonstrate the practical efficiency of our schemes within two application settings, where we compare favorably with current best approaches: 2-server private database pattern-match queries, and secure 2-party computation of low-degree polynomials.

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Nächstes Kapitel Improved Bootstrapping for Approximate Homomorphic Encryption

Namely, solving the Discrete Logarithm in a Interval problem with interval length R in time \(o(\sqrt{R})\).

Although the cost of bootstrapping has fallen dramatically in recent years [16, 17, 23, 32], the efficiency is still orders of magnitude worse than low-depth somewhat homomorphic encryption using SIMD operations.

So-called “third generation” SHE schemes based on GSW [28] have simpler homomorphic multiplication, but much larger ciphertexts that grow with \(\varOmega (N \log ^2 q)\) instead of \(O(N \log q)\), for (R)LWE dimension N and modulus q.

Note that nearly linear decryption generically implies existence of a public-key encryption procedure.

This can be decreased to \((d-1)\) \(R_q\)-elements communicated, as \(s_1=1 \in R_q\).

To simplify the analysis, we restrict the definition to 2-power cyclotomic rings. However, our construction can be generalized to arbitrary cyclotomics.

Choosing a sparse secret like this does incur a small loss in security, and only gives us a small gain in parameters for the HSS. The main reason we choose s like this is to allow a fair comparison with SHE schemes, which typically have to use sparse secrets to obtain reasonable parameters.

Using S/FHE alone instead of HSS allows for the stronger setting of single-server PIR. However, a major advantage of HSS with additive reconstruction is that shares across many rows can easily be combined, allowing more expressive queries with simpler computation.

Actually, these works use function secret-sharing [8] for point functions, which in this case is equivalent to HSS for the same class of functions.

Other queries such as returning the record identifier, or min/max and range queries can easily be supported with similar techniques, as previously shown in [6, 40].

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Titel: Homomorphic Secret Sharing from Lattices Without FHE
verfasst von: Elette Boyle
Lisa Kohl
Peter Scholl
Verlag: Springer International Publishing
Buch: Advances in Cryptology – EUROCRYPT 2019
Print ISBN: 978-3-030-17655-6

Electronic ISBN: 978-3-030-17656-3

Copyright-Jahr: 2019
DOI: https://doi.org/10.1007/978-3-030-17656-3_1

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