Sharing one secret vs. sharing many secrets

A secret sharing scheme is a method for distributing a secret among several parties in such a way that only quali.ed subsets of the parties can reconstruct it and unquali.ed subsets receive no information about the secret. A multi-secret sharing scheme is the natural extension of a secret sharing scheme to the case in which many secrets need to be shared, each with respect to possibly di2erent subsets of quali.ed parties. A multi-secret sharing scheme can be trivially realized by realizing a secret sharing scheme for each of the secrets. In this paper we address the natural questions of whether this simple construction is the most e4cient as well, and, if not, how much improvement is possible over it, with respect to both e4ciency measures used in the literature; namely, the maximum piece of information and the sum of all pieces of information distributed to all parties. We completely answer these questions, as follows. We show the .rst instance for which an improvement is possible; we prove a bound on how much improvement is possible with respect to both measures; and we show instances of multi-secret sharing schemes which achieve this improvement, with respect to both measures, thus showing that the above bound is tight. c © 2002 Published by Elsevier Science B.V.