Conditional Disclosure of Secrets via Non-linear Reconstruction

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 P : [N ]×[N ] → {0,1}, we present two protocols that achieve o(N 1/2) communication: the first achieves O(N 1/3) communication and the second achieves sub-polynomial 2O( p log N loglog N ) = N o(1) communication. • As a corollary, we obtain improved share complexity for forbidden graph access structures. Namely, for every graph on N vertices, there is a secret-sharing scheme for N parties in which each pair of parties can reconstruct the secret if and only if the corresponding vertices in G are connected, and where each party gets a share of size 2O( p log N loglog N ) = N o(1). Prior to this work, the best protocols for both primitives required communication complexity Õ(N 1/2). 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 O(N 1/2) “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. *E-mail: [email protected]. Research supported in part by NSF Grants CNS-1350619 and CNS-1414119, and by the Defense Advanced Research Projects Agency (DARPA) and the U.S. Army Research Office under contracts W911NF-15-C-0226 and W911NF-15-C-0236. †E-mail: [email protected]. Research supported in part by NSF Grants CNS-1350619 and CNS-1414119, Alfred P. Sloan Research Fellowship, Microsoft Faculty Fellowship, the NEC Corporation, a Steven and Renee Finn Career Development Chair from MIT. This work was also sponsored in part by the Defense Advanced Research Projects Agency (DARPA) and the U.S. Army Research Office under contracts W911NF-15-C-0226 and W911NF-15-C-0236. ‡E-mail: [email protected]. Research supported in part by ERC Project aSCEND (H2020 639554) and NSF Award CNS1445424.