Classification of Universally Ideal Homomorphic Secret Sharing Schemes and Ideal Black-Box Secret Sharing Schemes

  • Zhanfei Zhou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3822)


A secret sharing scheme (SSS) is homomorphic, if the products of shares of secrets are shares of the product of secrets. For a finite abelian group G, an access structure \({\mathcal A}\) is G-ideal homomorphic, if there exists an ideal homomorphic SSS realizing the access structure \({\mathcal A}\) over the secret domain G. An access structure \({\mathcal A}\) is universally ideal homomorphic, if for any non-trivial finite abelian group G, \({\mathcal A}\) is G-ideal homomorphic.

A black-box SSS is a special type of homomorphic SSS, which works over any non-trivial finite abelian group. In such a scheme, participants only have black-box access to the group operation and random group elements. A black-box SSS is ideal, if the size of the secret sharing matrix is the same as the number of participants. An access structure \({\mathcal A}\) is black-box ideal, if there exists an ideal black-box SSS realizing \({\mathcal A}\).

In this paper, we study universally ideal homomorphic and black-box ideal access structures, and prove that an access structure \({\mathcal A}\) is universally ideal homomorphic (black-box ideal) if and only if there is a regular matroid appropriate for \({\mathcal A}\).


Universally ideal homomorphic secret sharing scheme black-box secret sharing scheme chain group regular matroid 


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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Zhanfei Zhou
    • 1
  1. 1.State Key Laboratory of Information SecurityGraduate School of the Chinese Academy of SciencesBeijingChina

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