Optimal Black-Box Secret Sharing over Arbitrary Abelian Groups
Ronald Cramer
February 2002 |

## Abstract:
A
black-box secret sharing scheme for the threshold
access structure is one which works over any finite Abelian group
. 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 and are designed independently of the group from which the secret and the shares are
sampled. This means that perfect completeness and perfect privacy are
guaranteed 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 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
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