# Black-Box Secret Sharing from Primitive Sets in Number Fields

Ronald Cramer, Serge Fehr, Martijn Stam, Black-Box Secret Sharing from Primitive Sets in Number Fields. *Advances in Cryptology -- CRYPTO'05*, pp. 344–360. August 2005. No electronic version available.
## Abstract

A {\em black-box} secret sharing scheme (BBSSS) for a given access
structure works in exactly the same way over any finite Abelian
group, as it only requires black-box access to group operations and
to random group elements. In particular, there is no dependence on
e.g.\ the structure of the group or its order. The expansion factor
of a BBSSS is the length of a vector of shares (the number of group
elements in it) divided by the number of players $n$.
At CRYPTO 2002 Cramer and Fehr proposed a threshold BBSSS with an
asymptotically minimal expansion factor $\Theta(\log n)$.
In this paper we propose a BBSSS that is based on a new paradigm,
namely, {\em primitive sets in algebraic number fields}. This leads
to a new BBSSS with an expansion factor that is absolutely minimal
up to an additive term of at most~2, which is an improvement by a
constant additive factor.

We provide good evidence that our scheme is considerably more
efficient in terms of the computational resources it requires.
Indeed, the number of group operations to be performed is
$\tilde{O}(n^2)$ instead of $\tilde{O}(n^3)$ for sharing and
$\tilde{O}(n^{1.6})$ instead of $\tilde{O}(n^{2.6})$ for
reconstruction.
Finally, we show that our scheme, as well as that of Cramer and
Fehr, has asymptotically optimal randomness efficiency.

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