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.