Cryptology ePrint Archive: Report 2021/1532

Ingerid Fosli and Yuval Ishai and Victor I. Kolobov and Mary Wootters

Abstract: A homomorphic secret sharing (HSS) scheme is a secret sharing scheme that supports evaluating functions on shared secrets by means of a local mapping from input shares to output shares. We initiate the study of the download rate of HSS, namely, the achievable ratio between the length of the output shares and the output length when amortized over $\ell$ function evaluations. We obtain the following results.

* In the case of linear information-theoretic HSS schemes for degree-$d$ multivariate polynomials, we characterize the optimal download rate in terms of the optimal minimal distance of a linear code with related parameters. We further show that for sufficiently large $\ell$ (polynomial in all problem parameters), the optimal rate can be realized using Shamir's scheme, even with secrets over $\mathbb{F}_2$.

* We present a general rate-amplification technique for HSS that improves the download rate at the cost of requiring more shares. As a corollary, we get high-rate variants of computationally secure HSS schemes and efficient private information retrieval protocols from the literature.

* We show that, in some cases, one can beat the best download rate of linear HSS by allowing nonlinear output reconstruction and $2^{-\Omega(\ell)}$ error probability.

Category / Keywords: cryptographic protocols / Information theoretic cryptography, homomorphic secret sharing, private information retrieval, secure multiparty computation

Original Publication (with major differences): ITCS 2022

Date: received 19 Nov 2021, last revised 19 Nov 2021

Contact author: tkolobov at cs technion ac il

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2021/1532

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