Secret sharing over infinite domains

Benny Chor, Eyal Kushilevitz

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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.

Original languageEnglish
Title of host publicationAdvances in Cryptology — CRYPTO 1989, Proceedings
EditorsGilles Brassard
PublisherSpringer Verlag
Number of pages8
ISBN (Print)9780387973173
StatePublished - 1990
Externally publishedYes
EventConference on the Theory and Applications of Cryptology, CRYPTO 1989 - Santa Barbara, United States
Duration: 20 Aug 198924 Aug 1989

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume435 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


ConferenceConference on the Theory and Applications of Cryptology, CRYPTO 1989
Country/TerritoryUnited States
CitySanta Barbara


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