Succinct Computational Secret Sharing

Benny Applebaum, Amos Beimel, Yuval Ishai, Eyal Kushilevitz, Tianren Liu, Vinod Vaikuntanathan

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations


A secret-sharing scheme enables a dealer to share a secret s among n parties such that only authorized subsets of parties, specified by a monotone access structure f:{0,1}n→{0,1}, can reconstruct s from their shares. Other subsets of parties learn nothing about s. The question of minimizing the (largest) share size for a given f has been the subject of a large body of work. However, in most existing constructions for general access structures f, the share size is not much smaller than the size of some natural computational representation of the access structure f, a fact that has often been referred to as the "representation size barrier"in secret sharing. In this work, we initiate a systematic study of succinct computational secret sharing (SCSS), where the secrecy requirement is computational and the goal is to substantially beat the representation size barrier. We obtain the following main results. First, we introduce the notion of a projective PRG, a pseudorandom generator for which any subset of the output bits can be revealed while keeping the other output bits hidden, using a short projective seed. We construct projective PRGs with different levels of succinctness under a variety of computational assumptions, and apply them towards constructing SCSS for graph access structures, monotone CNF formulas, and (less succinctly) useful subclasses of monotone circuits and branching programs. Most notably, under the sub-exponential RSA assumption, we obtain a SCSS scheme that, given an arbitrary access structure f, represented by a truth table of size N=2n, produces shares of size polylog(N)=poly(n) in time Õ(N). For comparison, the share size of the best known information-theoretic schemes is O(N0.58). Secondly, under the (minimal) assumption that one-way functions exist, we obtain a near-quadratic separation between the total share size of computational and information-theoretic secret sharing. This is the strongest separation one can hope for, given the state of the art in secret sharing lower bounds. We also construct SCSS schemes from one-way functions for useful classes of access structures, including forbidden graphs and monotone DNF formulas. This leads to constructions of fully-decomposable conditional disclosure of secrets (also known as privacy-free garbled circuits) for general functions, represented by a truth table of size N=2n, with share size polylog(N) and computation time Õ(N), assuming sub-exponentially secure one-way functions.

Original languageEnglish
Title of host publicationSTOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing
EditorsBarna Saha, Rocco A. Servedio
PublisherAssociation for Computing Machinery
Number of pages14
ISBN (Electronic)9781450399135
StatePublished - 2 Jun 2023
Event55th Annual ACM Symposium on Theory of Computing, STOC 2023 - Orlando, United States
Duration: 20 Jun 202323 Jun 2023

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Conference55th Annual ACM Symposium on Theory of Computing, STOC 2023
Country/TerritoryUnited States


  • Pseudorandom Generators
  • Secret Sharing

ASJC Scopus subject areas

  • Software


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