On the Download Rate of Homomorphic Secret Sharing

TL;DR

This paper studies the download efficiency of homomorphic secret sharing schemes, providing optimal rates for linear schemes, a rate-amplification method, and scenarios surpassing linear limitations with nonlinear reconstruction.

Contribution

It characterizes the optimal download rate for linear HSS schemes, introduces a rate-amplification technique, and demonstrates potential improvements with nonlinear methods.

Findings

Optimal download rate characterized by linear code parameters.

Rate-amplification technique improves download efficiency.

Nonlinear reconstruction can outperform linear schemes with low error.

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.