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On a Relation Between Verifiable Secret Sharing Schemes and a Class of Error-Correcting Codes

  • Ventzislav Nikov
  • Svetla Nikova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3969)

Abstract

In this paper we try to shed a new insight on Verifiable Secret Sharing Schemes (VSS). We first define a new “metric” (with slightly different properties than the standard Hamming metric). Using this metric we define a very particular class of codes that we call error-set correcting codes, based on a set of forbidden distances which is a monotone decreasing set. Next we redefine the packing problem for the new settings and generalize the notion of error-correcting capability of the error-set correcting codes accordingly (taking into account the new metric and the new packing). Then we consider burst-error interleaving codes proposing an efficient burst-error correcting technique, which is in fact the well known VSS and Distributed Commitments (DC) pair-wise checking protocol and we prove the error-correcting capability of the error-set correcting interleaving codes.

Using the known relationship, due to Van Dijk, between a Monotone Span Program (MSP) and a generator matrix of the code generated by the suitable set of vectors, we prove that the error-set correcting codes in fact has the allowed (opposite to forbidden) distances of the dual access structure of the access structure that the MSP computes. We give an efficient construction for them based on this relation and as a consequence we establish a link between Secret Sharing Schemes (SSS) and the error-set correcting codes.

Further we give a necessary and sufficient condition for the existence of linear SSS (LSSS), to be secure against (Δ,Δ A )-adversary expressed in terms of an error-set correcting code. Finally, we present necessary and sufficient conditions for the existence of a VSS scheme, based on an error-set correcting code, secure against (Δ,Δ A )-adversary.

Our approach is general and covers all known linear VSS/DC. It allows us to establish the minimal conditions for security of VSSs. Our main theorem states that the security of a scheme is equivalent to a pure geometrical (coding) condition on the linear mappings describing the scheme. Hence the security of all known schemes, e.g. all known bounds for existence of unconditionally secure VSS/DC including the recent result of Fehr and Maurer, can be expressed as certain (geometrical) coding conditions.

Keywords

Secret Sharing Linear Code Access Structure Parity Check Matrix Weight Enumerator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ventzislav Nikov
    • 1
  • Svetla Nikova
    • 2
  1. 1.Department of Mathematics and Computing ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Department Electrical Engineering, ESAT/COSICKatholieke Universiteit LeuvenHeverlee-LeuvenBelgium

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