Designs, Codes and Cryptography

, Volume 74, Issue 2, pp 495–510 | Cite as

Extending Brickell–Davenport theorem to non-perfect secret sharing schemes

  • Oriol Farràs
  • Carles Padró


One important result in secret sharing is the Brickell–Davenport theorem: every ideal perfect secret sharing scheme defines a matroid that is uniquely determined by the access structure. We present a generalization of the Brickell–Davenport theorem to the general case, in which non-perfect schemes are also considered. After analyzing that result under a new point of view and identifying its combinatorial nature, we present a characterization of the (not necessarily perfect) secret sharing schemes that are associated with matroids. Some optimality properties of such schemes are discussed.


Secret sharing Non-perfect secret sharing scheme Matroid Polymatroid 

Mathematics Subject Classification

94A62 05B35 



We thank the anonymous reviewers. Their valuable suggestions greatly improved the presentation of the paper. The first author’s work was partially supported by the EU through the project FP7-ICT-317731, by the Spanish Government through the projects Consolider Ingenio 2010 CSD2007-00004 and TIN2011/27076-C03-01, and by the Catalan Government through the Grant 2009 SGR 1135. The second author’s work was supported by the Singapore National Research Foundation under Research Grant NRF-CRP2-2007-03.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Universitat Rovira i VirgiliTarragonaSpain
  2. 2.Nanyang Technological UniversitySingaporeSingapore

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