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 ${\mathcal A}$ is G-ideal homomorphic, if there exists an ideal homomorphic SSS realizing the access structure ${\mathcal A}$ over the secret domain G. An access structure ${\mathcal A}$ is universally ideal homomorphic, if for any non-trivial finite abelian group G, ${\mathcal A}$ 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 ${\mathcal A}$ is black-box ideal, if there exists an ideal black-box SSS realizing ${\mathcal A}$. In this paper, we study universally ideal homomorphic and black-box ideal access structures, and prove that an access structure ${\mathcal A}$ is universally ideal homomorphic (black-box ideal) if and only if there is a regular matroid appropriate for ${\mathcal A}$.