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On the Download Rate of Homomorphic Secret Sharing

Authors Ingerid Fosli, Yuval Ishai, Victor I. Kolobov, Mary Wootters

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Author Details

Ingerid Fosli
  • Google, Houston, TX, USA
Yuval Ishai
  • Technion, Haifa, Israel
Victor I. Kolobov
  • Technion, Haifa, Israel
Mary Wootters
  • Stanford University, CA, USA

Funding

  • Ishai, Yuval: Research partially supported by ERC Project NTSC (742754), ISF grant 2774/20, and BSF grant 2018393.
  • Kolobov, Victor I.: Same as Yuval Ishai.
  • Wootters, Mary: Research partially supported by NSF grants CCF-1844628 and CCF-1814629, and by a Sloan Research Fellowship.

Acknowledgements

We thank Elette Boyle and Tsachy Weissman for helpful conversations and the ITCS reviewers for useful comments.

Cite As Get BibTex

Ingerid Fosli, Yuval Ishai, Victor I. Kolobov, and Mary Wootters. On the Download Rate of Homomorphic Secret Sharing. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 71:1-71:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.71

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 𝓁 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 𝓁 (polynomial in all problem parameters), the optimal rate can be realized using Shamir’s scheme, even with secrets over 𝔽₂. 
- 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^{-Ω(𝓁)} error probability.

Subject Classification

ACM Subject Classification
  • Theory of computation → Cryptographic primitives
Keywords
  • Information-theoretic cryptography
  • homomorphic secret sharing
  • private information retrieval
  • secure multiparty computation
  • regenerating codes

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