Requirements for group independent linear threshold secret sharing schemes

Abstract

In a t out of n threshold scheme, any subset of t or more participants can compute the secret key k, while subsets of t – 1 or less participants cannot compute k. Some schemes are designed for specific algebraic structures, for example finite fields. Whereas other schemes can be used with any finite abelian group. In [24], the definition of group independent sharing schemes was introduced.In this paper, we develop bounds for group independent t out of n threshold schemes. The bounds will be lower bounds which discuss how many subshares are required to achieve a group independent linear threshold scheme. In particular, we will show that our bounds for the n – 1 out of n threshold schemes are tight for infinitely many n.