Paper 2017/359

Conditional Disclosure of Secrets via Non-Linear Reconstruction

Tianren Liu, Vinod Vaikuntanathan, and Hoeteck Wee

Abstract

We present new protocols for conditional disclosure of secrets (CDS), where two parties want to disclose a secret to a third party if and only if their respective inputs satisfy some predicate. - For general predicates $\text{pred}:[N]\times [N]\to \{0,1\}$, we present two protocols that achieve $$ communication: the first achieves $$ communication and the second achieves sub-polynomial $$ communication. - As a corollary, we obtain improved share complexity for forbidden graph access structures. Namely, for every graph on $$ vertices, there is a secret-sharing scheme for $$ parties in which each pair of parties can reconstruct the secret if and only if the corresponding vertices in $$ are connected, and where each party gets a share of size $$. Prior to this work, the best protocols for both primitives required communication complexity $$. Indeed, this is essentially the best that all prior techniques could hope to achieve as they were limited to so-called ``linear reconstruction''. This is the first work to break this $$ ``linear reconstruction'' barrier in settings related to secret sharing. To obtain these results, we draw upon techniques for non-linear reconstruction developed in the context of information-theoretic private information retrieval. We further extend our results to the setting of private simultaneous messages (PSM), and provide applications such as an improved attribute-based encryption (ABE) for quadratic polynomials.

Note: Minor revisions.