Secret sharing over infinite domains
Various schemes for secret-sharing, i.e. distributing a secret among \(n\) parties in a way that allows ``legal'' sets of parties to reconstruct the secret while preventing ``illegal'' sets of parties to get information about the secret, are already known. However, these schemes were designed for cases where the set of secrets and the set of shares are finite. In the article the existence of secret-sharing schemes over infinite domains is studied. It is shown that infinity alone is not sufficient to decide the (im)possibility of secret-sharing schemes, as the cardinality of the domain must be taken into account as well. Particularly, if the sets of secrets and shares are countable then no secret-sharing scheme exists. This result contrasts with another one stating that if the sets of secrets and shares have the cardinality of the reals then perfect (i.e. no illegal set of shares reveals any partial information about the secret) secret-sharing schemes do exist. As a consequence it is shown that perfect private-key encryption (i.e. the one where an eavesdropper gets no partial information about the plaintext by examining the ciphertext) is not possible over a countable domain. On the other hand, over the reals a perfect private-key encryption scheme does exist.
scientific article; zbMATH DE number 399377
scientific article