How to share secrets simultaneously

Each member of a team consisting of n person has a secret. The k out of n simultaneous threshold secret sharing requires that any group of k members should be able to recover the secret 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 members. We show that when all secrets are independent and have size s then each team member must receive a share of size at least (n − k)s, and we present a scheme which achieves this bound. This result shows a significant saving over n independent applications of the k out of n− 1 threshold schemes which assigns shares of size (n− 1)s to each team member independently of k.