Sparsity and Privacy in Secret Sharing: A Fundamental Trade-Off

TL;DR

This paper explores the fundamental tradeoff between sparsity and privacy in secret sharing schemes, proposing optimal schemes that balance minimal information leakage with computational efficiency for distributed matrix multiplication.

Contribution

It introduces a fundamental tradeoff between sparsity and privacy in secret sharing, and constructs optimal schemes minimizing information leakage for given sparsity levels.

Findings

Sparsity in shares leads to weaker privacy for matrices with i.i.d. entries.

Optimal sparse secret sharing schemes can minimize information leakage.

Application to distributed matrix multiplication with no colluding workers.

Abstract

This work investigates the design of sparse secret sharing schemes that encode a sparse private matrix into sparse shares. This investigation is motivated by distributed computing, where the multiplication of sparse and private matrices is moved from a computationally weak main node to untrusted worker machines. Classical secret-sharing schemes produce dense shares. However, sparsity can help speed up the computation. We show that, for matrices with i.i.d. entries, sparsity in the shares comes at a fundamental cost of weaker privacy. We derive a fundamental tradeoff between sparsity and privacy and construct optimal sparse secret sharing schemes that produce shares that leak the minimum amount of information for a desired sparsity of the shares. We apply our schemes to distributed sparse and private matrix multiplication schemes with no colluding workers while tolerating stragglers. For the setting of two non-communicating clusters of workers, we design a sparse one-time pad so that no private information is leaked to a cluster of untrusted and colluding workers, and the shares with bounded but non-zero leakage are assigned to a cluster of partially trusted workers. We conclude by discussing the necessity of using permutations for matrices with correlated entries.