Breaking barriers in secret sharing

Abstract

In this thesis, we study secret sharing schemes for general (non-threshold) access functions. In a secret sharing scheme for n parties associated to a monotone function [mathematical formula], a dealer distributes shares of a secret among n parties. Any subset of parties [mathematical formula] can jointly reconstruct the secret if F(T) = 1, and should have no information about the secret if F(T) = 0. One of the major long-standing questions in information-theoretic cryptography is to determine the minimum size of the shares in a secret-sharing scheme for an access function F. There is an exponential gap between lower and upper bounds for share size: the best known scheme for general monotone functions has shares of size 2[superscript n-o(n)], while the best lower bound is n² / log(n). In this thesis, we improve this more-than-30-year-old upper bound by construct- ing a secret sharing scheme for any access function with shares of size 2[superscript 0.994n] and a linear secret sharing scheme for any access function with shares of size 2[superscript 0.994n]. As a contribution of independent interest, we also construct a secret sharing scheme with shares of size [mathematical formula] for a family of [mathematical formula] monotone access functions, out of a total of [mathematical formula] of them. As an intermediate result, we construct the first conditional disclosure of secrets (CDS) with sub-exponential communication complexity. CDS is a variant of secret sharing, in which a group of parties want to disclose a secret to a referee the parties' respective inputs satisfy some predicate. The key conceptual contribution of this thesis is a novel connection between secret sharing and CDS, and the notion of (2-server, information-theoretic) private information retrieval.