Secret sharing schemes for very dense graphs

Amos Beimel, Oriol Farràs, Yuval Mintz

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

6 Scopus citations


A secret-sharing scheme realizes a graph if every two vertices connected by an edge can reconstruct the secret while every independent set in the graph does not get any information on the secret. Similar to secret-sharing schemes for general access structures, there are gaps between the known lower bounds and upper bounds on the share size for graphs. Motivated by the question of what makes a graph "hard" for secret-sharing schemes, we study very dense graphs, that is, graphs whose complement contains few edges. We show that if a graph with n vertices contains ( n 2 - n 1+β edges for some constant 0 ≤ β < 1, then there is a scheme realizing the graph with total share size of Õ(n 5/4+3β/4). This should be compared to O(n 2/logn) - the best upper bound known for general graphs. Thus, if a graph is "hard", then the graph and its complement should have many edges. We generalize these results to nearly complete k-homogeneous access structures for a constant k. To complement our results, we prove lower bounds for secret-sharing schemes realizing very dense graphs, e.g., for linear secret-sharing schemes we prove a lower bound of Ω(n 1 + β/2).

Original languageEnglish
Title of host publicationAdvances in Cryptology, CRYPTO 2012 - 32nd Annual Cryptology Conference, Proceedings
Number of pages18
StatePublished - 3 Sep 2012
Event32nd Annual International Cryptology Conference, CRYPTO 2012 - Santa Barbara, CA, United States
Duration: 19 Aug 201223 Aug 2012

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume7417 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference32nd Annual International Cryptology Conference, CRYPTO 2012
Country/TerritoryUnited States
CitySanta Barbara, CA

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science (all)


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