Paper 2011/386 Original: eprint.iacr.org/2011/386 ↗

How to share secrets simultaneously

Abstract

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 $k$.