Secret sharing over infinite domains

Abstract

Let ℱ _n be a monotone, nontrivial family of sets over {1, 2, …,n}. An ℱ _n perfect secret-sharing scheme is a probabilistic mapping of a secret ton shares, such that:

• • The secret can be reconstructed from any setT of shares such thatT ∈ ℱ _n .

• • No subsetT ∉ ℱ _n of 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 overinfinite domains. The major case of interest is when the secrets and the shares are taken from acountable set, for example all binary strings. We show that no ℱ _n secret-sharing scheme over any countable domain exists (for anyn ≥ 2).

One consequence of this impossibility result is that noperfect private-key encryption schemes, over the set of all strings, exist. Stated informally, this means that there is no way to encrypt all strings perfectly without revealing information about their length. These impossibility results are stated and proved not only for perfect secret-sharing and private-key encryption schemes, but also for wider classes—weak secret-sharing and private-key encryption schemes.

We constrast these results with the case where both the secrets and the shares are real numbers. Simple perfect secret-sharing schemes (and perfect private-key encryption schemes) are presented. Thus, infinity alone does not rule out the possibility of secret sharing.