2005 | OriginalPaper | Chapter
Classification of Universally Ideal Homomorphic Secret Sharing Schemes and Ideal Black-Box Secret Sharing Schemes
Author : Zhanfei Zhou
Published in: Information Security and Cryptology
Publisher: Springer Berlin Heidelberg
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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
${\mathcal A}$
.