Classification of universally ideal homomorphic secret sharing schemes and ideal black-box secret sharing schemes

Abstract

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 A is G-ideal homomorphic, if there exists an ideal homomorphic SSS realizing the access structure A over the secret domain G. An access structure A is universally ideal homomorphic, if for any non-trivial finite abelian group G, 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 A is blackbox ideal, if there exists an ideal black-box SSS realizing A. In this paper, we study universally ideal homomorphic and black-box ideal access structures, and prove that an access structure A is universally ideal homomorphic (black-box ideal) if and only if there is a regular matroid appropriate for A. © Springer-Verlag Berlin Heidelberg 2005.