Gruppen secret sharing or how to share several secrets if you must?

Each member of an n -person team has a secret, say a password. The k out of n gruppen secret sharing requires that any group of k members should be able to recover the secrets of the other n − k members, while any group of k − 1 or less members should have no information on the secret of other team member even if other secrets leak out . We prove that when all secrets are chosen independently and have size s , then each team member must have a share of size at least ( n − k ) s , and we present a scheme which achieves this bound when s is large enough. This result shows a significant saving over n independent applications of Shamir’s k out of n − 1 threshold schemes which assigns shares of size ( n − 1) s to each team member independently of k . We also show how to set up such a scheme without any trusted dealer, and how the secrets can be recovered, possibly multiple times, without leaking information. We also discuss how our scheme fits to the much-investigated multiple secret sharing methods.