A Characterization of Optimal-Rate Linear Homomorphic Secret Sharing Schemes, and Applications.
CoRR(2023)
摘要
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 each
server. Recent work (Fosli, Ishai, Kolobov, and Wootters, ITCS 2022)
established a fundamental limitation on the download rate of linear HSS schemes
for computing low-degree polynomials, and gave an example of HSS schemes that
meet this limit.
In this paper, we further explore optimal-rate linear HSS schemes for
polynomials. Our main result is a complete characterization of such schemes, in
terms of a coding-theoretic notion that we introduce, termed optimal
labelweight codes. We use this characterization to answer open questions about
the amortization required by HSS schemes that achieve optimal download rate. In
more detail, the construction of Fosli et al. required amortization over $\ell$
instances of the problem, and only worked for particular values of $\ell$. We
show that -- perhaps surprisingly -- the set of $\ell$'s for which their
construction works is in fact nearly optimal, possibly leaving out only one
additional value of $\ell$. We show this by using our coding-theoretic
characterization to prove a necessary condition on the $\ell$'s admitting
optimal-rate linear HSS schemes. We then provide a slightly improved
construction of optimal-rate linear HSS schemes, where the set of allowable
$\ell$'s is optimal in even more parameter settings. Moreover, based on a
connection to the MDS conjecture, we conjecture that our construction is
optimal for all parameter regimes.
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