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2020 | OriginalPaper | Chapter

Boosting Verifiable Computation on Encrypted Data

Authors : Dario Fiore, Anca Nitulescu, David Pointcheval

Published in: Public-Key Cryptography – PKC 2020

Publisher: Springer International Publishing

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Abstract

We consider the setting in which an untrusted server stores a collection of data and is asked to compute a function over it. In this scenario, we aim for solutions where the untrusted server does not learn information about the data and is prevented from cheating. This problem is addressed by verifiable and private delegation of computation, proposed by Gennaro, Gentry and Parno (CRYPTO’10), a notion that is close to both the active areas of homomorphic encryption and verifiable computation (VC). However, in spite of the efficiency advances in the respective areas, VC protocols that guarantee privacy of the inputs are still expensive. The only exception is a protocol by Fiore, Gennaro and Pastro (CCS’14) that supports arithmetic circuits of degree at most 2. In this paper we propose new efficient protocols for VC on encrypted data that improve over the state of the art solution of Fiore et al. in multiple aspects. First, we can support computations of degree higher than 2. Second, we achieve public delegatability and public verifiability whereas Fiore et al. need the same secret key to encode inputs and verify outputs. Third, we achieve a new property that guarantees that verifiers can be convinced about the correctness of the outputs without learning information on the inputs. The key tool to obtain our new protocols is a new SNARK that can efficiently handle computations over a quotient polynomial ring, such as the one used by Ring-LWE somewhat homomorphic encryption schemes. This SNARK in turn relies on a new commit-and-prove SNARK for proving evaluations on the same point of several committed polynomials. We propose a construction of this scheme under an extractability assumption over bilinear groups in the random oracle model.

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previous chapter Witness Indistinguishability for Any Single-Round Argument with Applications to Access Control

next chapter Lossy CSI-FiSh: Efficient Signature Scheme with Tight Reduction to Decisional CSIDH-512

In particular, relevant to our work is the compiler that shows that commit-and-prove SNARKs for Pedersen-like commitments can be made compatible with one another.

We consider linearly homomorphic commitment schemes \(\mathsf{MPoly\text {-}}{\mathcal Com}\) and we commit in \(C_T\) and C to vectors of \(n+1 \leqslant \ell \) polynomials \((C_{T}, \tau ) \leftarrow \mathsf{MPoly.}\mathsf{Com}(T, 0, 0 \dots 0)\) and

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-45388-6_5/498015_1_En_5_IEq209_HTML.gif

with an appropriate number of 0’s, i.e., \((T, \{P_j\}) = (T, \{0\}) + (0, \{P_j\})\), such that computing \(C_{T} \times C\) results in a commitment \((C_{T} \times C, \tau +\rho ) \leftarrow \mathsf{MPoly.}\mathsf{Com}(T, \{P_j\}_{j=1}^{n})\) to the concatenation of T, \(\{P_j\}\).

The commitment key \(\mathsf{ck}\) can have some special property for optimization, for example, it may consist of two keys, one for committing to polynomials \(P_j \in {\mathbb R_q}\) of degrees \(\leqslant d\) and another longer key to commit to polynomials \(T\in {\mathbb Z}_q [X]\) of degree \( \nu \).

Note that, although Q is a uni-variate polynomial in Y, it can also be seen as a bivariate polynomial.

Note that W can be computed by aggregating the results of \(\ell \) polynomial divisions of degree d.

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Title: Boosting Verifiable Computation on Encrypted Data
Authors: Dario Fiore
Anca Nitulescu
David Pointcheval
Publisher: Springer International Publishing
Book: Public-Key Cryptography – PKC 2020
Print ISBN: 978-3-030-45387-9

Electronic ISBN: 978-3-030-45388-6

Copyright Year: 2020
DOI: https://doi.org/10.1007/978-3-030-45388-6_5

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