On matroids and non-ideal secret sharing

Amos Beimel, Noam Livne

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

15 Scopus citations


Secret-sharing schemes are a tool used in many cryptographic protocols. In these schemes, a dealer holding a secret string distributes shares to the parties such that only authorized subsets of participants can reconstruct the secret from their shares. The collection of authorized sets is called an access structure. An access structure is ideal if there is a secret-sharing scheme realizing it such that the shares are taken from the same domain as the secrets. Brickell and Davenport (J. of Cryptology, 1991) have shown that ideal access structures are closely related to matroids. They give a necessary condition for an access structure to be ideal - the access structure must be induced by a matroid. Seymour (J. of Combinatorial Theory B, 1992) showed that the necessary condition is not sufficient: There exists an access structure induced by a matroid that does not have an ideal scheme. In this work we continue the research on access structures induced by matroids. Our main result in this paper is strengthening the result of Seymour. We show that in any secret sharing scheme realizing the access structure induced by the Vamos matroid with domain of the secrets of size k, the size of the domain of the shares is at least k + Ω(√k). Our second result considers non-ideal secret sharing schemes realizing access structures induced by matroids. We prove that the fact that an access structure is induced by a matroid implies lower and upper bounds on the size of the domain of shares of subsets of participants even in non-ideal schemes (this generalized results of Brickell and Davenport for ideal schemes).

Original languageEnglish
Title of host publicationTheory of Cryptography
Subtitle of host publicationThird Theory of Cryptography Conference, TCC 2006, Proceedings
Number of pages20
StatePublished - 7 Jul 2006
Event3rd Theory of Cryptography Conference, TCC 2006 - New York, NY, United States
Duration: 4 Mar 20067 Mar 2006

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume3876 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference3rd Theory of Cryptography Conference, TCC 2006
Country/TerritoryUnited States
CityNew York, NY

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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