On the Isolation of a Common Secret

Summary

Two parties are said to “share a secret” if there is a question to which only they know the answer. Since possession of a shared secret allows them to communicate a bit between them over an open channel without revealing the value of the bit, shared secrets are fundamental in cryptology.

We consider below the problem of when two parties with shared knowledge can use that knowledge to establish, over an open channel, a shared secret. There are no issues of complexity or probability; the parties are not assumed to be limited in computing power, and secrecy is judged only relative to certainty, not probability. In this context the issues become purely combinatorial and in fact lead to some curious results in graph theory.

Applications are indicated in the game of bridge, and for a problem involving two sheriffs, eight suspects and a lynch mob.