Secret Sharing, Visual Cryptography, and Voting

Abstract

Suppose you have n people, with whom you wish to share a secret. However, you do not wish to entrust any of them individually with the secret. Rather, you only want to allow them to learn what the secret is when a significant coalition of the people, at least k, say, work together to learn it. Furthermore, you don’t want to allow smaller coalitions to learn bits of the secret; you want any coalition of at most $k-1$ people not to be able to learn anything at all. How do you do it?