Function Secret Sharing.

Motivated by the goal of securely searching and updating distributed data, we introduce and study the notion of function secret sharing (FSS). This new notion is a natural generalization of distributed point functions (DPF), a primitive that was recently introduced by Gilboa and Ishai (Eurocrypt 2014). Given a positive integer p >= 2 and a class F of functions f : {0, 1}(n) -> G, where G is an Abelian group, a p-party FSS scheme for F allows one to split each f is an element of F into p succinctly described functions f(i) : {0, 1}(n). G -> 1 <= i <= p, such that: (1) Sigma(p)(i=1) f(i) = f, and (2) any strict subset of the f(i) hides f. Thus, an FSS for F can be thought of as method for succinctly performing an "additive secret sharing" of functions from F. The original definition of DPF coincides with a two-party FSS for the class of point functions, namely the class of functions that have a nonzero output on at most one input. We present two types of results. First, we obtain efficiency improvements and extensions of the original DPF construction. Then, we initiate a systematic study of general FSS, providing some constructions and establishing relations with other cryptographic primitives. More concretely, we obtain the following main results: - IMPROVED DPF. We present an improved (two-party) DPF construction from a pseudorandom generator (PRG), reducing the length of the key describing each f(i) from O(lambda.n(log23)) to O(lambda n), where lambda is the PRG seed length. - MULTI-PARTY DPF. We present the first nontrivial construction of a p-party DPF for p >= 3, obtaining a near-quadratic improvement over a naive construction that additively shares the truth-table of f. This constrcution too can be based on any PRG. - FSS for simple functions. We present efficient PRG-based FSS constructions for natural function classes that extend point functions, including interval functions and partial matching functions. - A study of general FSS. We show several relations between general FSS and other cryptographic primitives. These include a construction of general FSS via obfuscation, an indication for the implausibility of constructing general FSS from weak cryptographic assumptions such as the existence of one-way functions, a completeness result, and a relation with pseudorandom functions.