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Improved Trade-Offs Between Amortization and Download Bandwidth for Linear HSS

Authors Keller Blackwell , Mary Wootters

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Author Details

Keller Blackwell
  • Department of Computer Science, Stanford University, CA, USA
Mary Wootters
  • Departments of Computer Science and Electrical Engineering, Stanford University, CA, USA

Funding

  • Blackwell, Keller: KB is supported by a National Science Foundation Graduate Research Fellowship and by a graduate fellowship award from Knight-Hennessy Scholars at Stanford University. KB’s work is supported partially supported by NSF grant CCF-2231157.
  • Wootters, Mary: MW’s work was partially supported by NSF grants CCF-1844628, CCF-2133154, and CCF-2231157.

Acknowledgements

We thank the anonymous referees for helpful feedback.

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Keller Blackwell and Mary Wootters. Improved Trade-Offs Between Amortization and Download Bandwidth for Linear HSS. In 5th Conference on Information-Theoretic Cryptography (ITC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 304, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ITC.2024.7

Abstract

A Homomorphic Secret Sharing (HSS) scheme is a secret-sharing scheme that shares a secret x among s servers, and additionally allows an output client to reconstruct some function f(x) using information that can be locally computed by each server. A key parameter in HSS schemes is download rate, which quantifies how much information the output client needs to download from the servers. Often, download rate is improved by amortizing over 𝓁 instances of the problem, making 𝓁 also a key parameter of interest. 
Recent work [Fosli et al., 2022] established a limit on the download rate of linear HSS schemes for computing low-degree polynomials and constructed schemes that achieve this optimal download rate; their schemes required amortization over 𝓁 = Ω(s log(s)) instances of the problem. Subsequent work [Blackwell and Wootters, 2023] completely characterized linear HSS schemes that achieve optimal download rate in terms of a coding-theoretic notion termed optimal labelweight codes. A consequence of this characterization was that 𝓁 = Ω(s log(s)) is in fact necessary to achieve optimal download rate.
In this paper, we characterize all linear HSS schemes, showing that schemes of any download rate are equivalent to a generalization of optimal labelweight codes. This equivalence is constructive and provides a way to obtain an explicit linear HSS scheme from any linear code. Using this characterization, we present explicit linear HSS schemes with slightly sub-optimal rate but with much improved amortization 𝓁 = O(s). Our constructions are based on algebraic geometry codes (specifically Hermitian codes and Goppa codes).

Subject Classification

ACM Subject Classification
  • Theory of computation → Cryptographic primitives
  • Theory of computation → Error-correcting codes
Keywords
  • Error Correcting Codes
  • Homomorphic Secret Sharing

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