Secret sharing over infinite domains

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.