Generalized linear threshold scheme
Proceedings of CRYPTO 84 on Advances in cryptology
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Secret sharing homomorphisms: keeping shares of a secret secret
Proceedings on Advances in cryptology---CRYPTO '86
A combinatorial approach to threshold schemes
SIAM Journal on Discrete Mathematics
Completeness theorems for non-cryptographic fault-tolerant distributed computation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
On hiding information form an oracle
Journal of Computer and System Sciences
Communications of the ACM
Generalized Secret Sharing and Monotone Functions
CRYPTO '88 Proceedings of the 8th Annual International Cryptology Conference on Advances in Cryptology
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
Shared Generation of Authenticators and Signatures (Extended Abstract)
CRYPTO '91 Proceedings of the 11th Annual International Cryptology Conference on Advances in Cryptology
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A (k, n) secret sharing scheme is a probabilistic mapping of a secret to n shares, such that 驴 The secret can be reconstructed from any k shares. 驴 No subset of k - 1 shares reveals any partial information about the secret.Various secret sharing schemes have been proposed, and applications in diverse contexts were found. In all these cases, the set of secrets and the set of shares are finite.In this paper we study the possibility of secret sharing schemes over infinite domains. The major case of interest is when the secrets and the shares are taken from a countable set, for example all binary strings. We show that no (k, n) secret sharing scheme over any countable domain exists (for any 2 k n).One consequence of this impossibility result is that no perfect private-key encryption schemes, over the set of all strings, exist. Stated informally, this means that there is no way to perfectly encrypt all strings without revealing information about their length.We contrast these results with the case where both the secrets and the shares are real numbers. Simple secret sharing schemes (and perfect private-key encryption schemes) are presented. Thus, infinity alone does not rule out the possibility of secret sharing.