Modern Cryptography Through the Lens of Secret Sharing

Secret sharing is a mechanism by which a trusted dealer holding a secret “splits” the secret into many “shares” and distributes the shares to a collections of parties. Associated with the sharing is a monotone access structure, that specifies which parties are “qualified” and which are not: any qualified subset of parties can (efficiently) reconstruct the secret, but no unqualified subset can learn anything about the secret. In the most general form of secret sharing, the access structure can be any monotone NP language. In this work, we consider two very natural extensions of secret sharing. In the first, which we call distributed secret sharing, there is no trusted dealer at all, and instead the role of the dealer is distributed amongst the parties themselves. Distributed secret sharing can be thought of as combining the features of multiparty non-interactive key exchange and standard secret sharing, and may be useful in settings where the secret is so sensitive that no one individual dealer can be trusted with the secret. Our second notion is called functional secret sharing, which incorporates some of the features of functional encryption into secret sharing by providing more fine-grained access to the secret. Qualified subsets of parties do not learn the secret, but instead learn some function applied to the secret, with each set of parties potentially learning a different function. Our main result is that both of the extensions above are equivalent to several recent cuttingedge primitives. In particular, general-purpose distributed secret sharing is equivalent to witness PRFs, and general-purpose functional secret sharing is equivalent to indistinguishability obfuscation. Thus, our work shows that it is possible to view some of the recent developments in cryptography through a secret sharing lens, yielding new insights about both these cutting-edge primitives and secret sharing. ∗Weizmann Institute of Science, Israel. Email: [email protected]. Supported in part by a grant from the I-CORE Program of the Planning and Budgeting Committee, the Israel Science Foundation, BSF and the Israeli Ministry of Science and Technology. †Stanford University and MIT. Email: [email protected]. Supported by the DARPA PROCEED program.