TY - GEN

T1 - Secret sharing over infinite domains

AU - Chor, Benny

AU - Kushilevitz, Eyal

N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1990.

PY - 1990

Y1 - 1990

N2 - A (k, n) secret sharing scheme is a probabilistic mapping of a secret to n shares, such that The secret can be reconstructed from any k shares. No subset of k − 1 shares reveals any partial information about the secret. Various secret sharing schemes have been proposed, and applications in diverse con- texts 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 do- mains. 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 encryp- tion 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.

AB - A (k, n) secret sharing scheme is a probabilistic mapping of a secret to n shares, such that The secret can be reconstructed from any k shares. No subset of k − 1 shares reveals any partial information about the secret. Various secret sharing schemes have been proposed, and applications in diverse con- texts 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 do- mains. 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 encryp- tion 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.

UR - http://www.scopus.com/inward/record.url?scp=38149087056&partnerID=8YFLogxK

U2 - 10.1007/0-387-34805-0_27

DO - 10.1007/0-387-34805-0_27

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AN - SCOPUS:38149087056

SN - 9780387973173

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 299

EP - 306

BT - Advances in Cryptology — CRYPTO 1989, Proceedings

A2 - Brassard, Gilles

PB - Springer Verlag

Y2 - 20 August 1989 through 24 August 1989

ER -