Sharing one secret vs. sharing many secrets: Tight bounds for the max improvement ratio

Abstract

Asecret sharing scheme is a method for distributing a secret among several parties in such a way that only qualified subsets of the parties can reconstruct it and unqualified subsets receive no information about the secret. Am ulti 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 different subsets of qualified parties. Am ulti secret sharing scheme can be trivially realized by realizing a secret sharing scheme for each of the secrets. An atural question in the area is whether this simple construction is the most efficient as well, and, if not, how much improvement is possible over it. In this paper we address and answer this question, with respect to the most widely used efficiency measure, that is, the maximum piece of information distributed among all the parties. Although no improvement is possible for several instances of multi secret sharing, we present the first instance for which some improvement is possible, and, in fact, we show that for this instance an improvement factor equal to the number of secrets over the above simple construction is possible. The given improvement is also proved to be the best possible, thus showing that the achieved bound is tight.