Succinct Homomorphic Secret Sharing

Abstract

This work introduces homomorphic secret sharing (HSS) with succinct share size. In HSS, private inputs are shared between parties, who can then homomorphically evaluate a function on their shares, obtaining a share of the function output. In succinct HSS, a portion of the inputs can be distributed using shares whose size is sublinear in the number of such inputs. The parties can then locally evaluate a function f on the shares, with the restriction that f must be linear in the succinctly shared inputs. We construct succinct, two-party HSS for branching programs, based on either the decisional composite residuosity assumption, a DDH-like assumption in class groups, or learning with errors with a superpolynomial modulus-to-noise ratio. We then give several applications of succinct HSS, which were only previously known using fully homomorphic encryption, or stronger tools: Succinct vector oblivious linear evaluation (VOLE): Two parties can obtain secret shares of a long, arbitrary vector \(\boldsymbol{x}\) , multiplied by a scalar \(\varDelta \) , with communication sublinear in the size of the vector. Batch, multi-party distributed point functions : A protocol for distributing a batch of secret, random point functions among N parties, for any polynomial N , with communication sublinear in the number of DPFs. Sublinear MPC for any number of parties: Two new constructions of MPC with sublinear communication complexity, with N parties for any polynomial N : (1) For general layered Boolean circuits of size s , with communication \(O(N s/\log \log s)\) , and (2) For layered, sufficiently wide Boolean circuits, with communication \(O(N s/\log s)\) .

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