Secret-Sharing for NP from Indistinguishability Obfuscation

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 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 schemes. 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 P). Around 1990 Rudich raised the possibility of obtaining secret-sharing for all monotone functions in NP: In order to reconstruct the secret a set of parties must be “qualified” and provide a witness attesting to this fact. Recently, there has been much excitement regarding the possibility of obtaining program obfuscation satisfying the “indistinguishability obfuscation” requirement: A transformation that takes a program and outputs an obfuscated version of it so that for any two functionally equivalent programs the output of the transformation is computationally indistinguishable. Our main result is a construction of a computational secret-sharing scheme for any monotone function in NP assuming the existence of an efficient indistinguishability obfuscator for P and one-way functions. Furthermore, we show how to get the same result but relying on a weaker obfuscator: an efficient indistinguishability obfuscator for 3CNF formulas. Research supported in part by a grant from the I-CORE Program of the Planning and Budgeting Committee, the Israel Science Foundation and the Citi Foundation. Incumbent of the Judith Kleeman Professorial Chair.