Secret sharing homomorphisms: keeping shares of a secret secret
Proceedings on Advances in cryptology---CRYPTO '86
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STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
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Geometric secret sharing schemes and their duals
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Optimal Black-Box Secret Sharing over Arbitrary Abelian Groups
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CRYPTO '92 Proceedings of the 12th Annual International Cryptology Conference on Advances in Cryptology
Classification of ideal homomorphic threshold schemes over finite Abelian groups
EUROCRYPT'92 Proceedings of the 11th annual international conference on Theory and application of cryptographic techniques
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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 ${\mathcal A}$.