Secret-Sharing for NP

Abstract

A computational secret-sharing scheme is a method that enables a dealer, that has a secret, to distribute this secret among a set of parties such that a "qualified" subset of parties can efficiently reconstruct the secret while any "unqualified" subset of parties cannot efficiently learn anything about the secret. The collection of "qualified" subsets is defined by a monotone Boolean function. It has been a major open problem to understand which (monotone) functions can be realized by a computational secret-sharing scheme. Yao suggested a method for secret-sharing for any function that has a polynomial-size monotone circuit (a class which is strictly smaller than the class of monotone functions in $${\mathsf {P}}$$P). Around 1990 Rudich raised the possibility of obtaining secret-sharing for all monotone functions in $${\mathsf {NP}}$$NP: in order to reconstruct the secret a set of parties must be "qualified" and provide a witness attesting to this fact. Recently, Garg et al. (Symposium on theory of computing conference, STOC, pp 467---476, 2013) put forward the concept of witness encryption, where the goal is to encrypt a message relative to a statement $$x\in L$$xźL for a language $$L\in {\mathsf {NP}}$$LźNP such that anyone holding a witness to the statement can decrypt the message; however, if $$x\notin L$$xźL, then it is computationally hard to decrypt. Garg et al. showed how to construct several cryptographic primitives from witness encryption and gave a candidate construction. One can show that computational secret-sharing implies witness encryption for the same language. Our main result is the converse: we give a construction of a computational secret-sharing scheme for any monotone function in $${\mathsf {NP}}$$NP assuming witness encryption for $${\mathsf {NP}}$$NP and one-way functions. As a consequence we get a completeness theorem for secret-sharing: computational secret-sharing scheme for any single monotone $${\mathsf {NP}}$$NP-complete function implies a computational secret-sharing scheme for every monotone function in $${\mathsf {NP}}$$NP.