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Each member of an $n$-person team has a secret, say a password. The $k$ outof $n$ gruppen secret sharing requires that any group of $k$ members should beable 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 teammember even if other secrets leak out. We prove that when all secrets arechosen independently and have size $s$, then each team member must have a shareof size at least $(n-k)s$, and we present a scheme which achieves this boundwhen $s$ is large enough. This result shows a significant saving over $n$independent applications of Shamir's $k$ out of $n-1$ threshold schemes whichassigns 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 howthe secrets can be recovered, possibly multiple times, without leakinginformation. We also discuss how our scheme fits to the much-investigatedmultiple secret sharing methods.

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