Paper 2002/036

Optimal Black-Box Secret Sharing over Arbitrary Abelian Groups

Ronald Cramer and Serge Fehr

Abstract

A {\em black-box} secret sharing scheme for the threshold access structure Tt,n is one which works over any finite Abelian group G. Briefly, such a scheme differs from an ordinary linear secret sharing scheme (over, say, a given finite field) in that distribution matrix and reconstruction vectors are defined over the integers and are designed {\em independently} of the group from which the secret and the shares are sampled. This means that perfect completeness and perfect privacy are guaranteed {\em regardless} of which group is chosen. We define the black-box secret sharing problem as the problem of devising, for an arbitrary given , a scheme with minimal expansion factor, i.e., where the length of the full vector of shares divided by the number of players is minimal. Such schemes are relevant for instance in the context of distributed cryptosystems based on groups with secret or hard to compute group order. A recent example is secure general multi-party computation over black-box rings. In 1994 Desmedt and Frankel have proposed an elegant approach to the black-box secret sharing problem based in part on polynomial interpolation over cyclotomic number fields. For arbitrary given with , the expansion factor of their scheme is . This is the best previous general approach to the problem. Using low degree integral extensions of the integers over which there exists a pair of sufficiently large Vandermonde matrices with co-prime determinants, we construct, for arbitrary given with , a black-box secret sharing scheme with expansion factor , which we show is minimal.

Metadata
Available format(s)
PS
Category
Cryptographic protocols
Publication info
Published elsewhere. Unknown where it was published
Contact author(s)
cramer @ daimi aau dk
History
2002-03-22: received
Short URL
https://ia.cr/2002/036
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2002/036,
      author = {Ronald Cramer and Serge Fehr},
      title = {Optimal Black-Box Secret Sharing over Arbitrary Abelian Groups},
      howpublished = {Cryptology {ePrint} Archive, Paper 2002/036},
      year = {2002},
      url = {https://eprint.iacr.org/2002/036}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.