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 → (image found), where (image found) is an Abelian group, a p-party FSS scheme for F allows one to split each f ∈ F into p succinctly described functions f_i: {0, 1}ⁿ →(image found), 1 ≤ i ≤ p, such that: (1) ∑^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 twoparty 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 fi from O(λ ・ n^{log2 3}) to O(λn), where λ 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.