  The key-agreement problem (finding a private key to use for secret messages,
otherwise referred to as the public-key distribution problem), was introduced
by Diffie and Hellman in 1976. An approach to structuring key-agreement
protocols via the use of one-way associative functions was proposed in 1993 by
Rabi and Sherman. We propose here a provably strong associative one-way
function based upon knot composition (answering an open problem proposed by
Rabi and Sherman whether any such associative one-way functions exist). We also
introduce and solve a game, exploring its relation to problems in graph and
braid theory and develop a new technique for computing whether a graph is
n-colorable. En route we look at estimator and prediction problems raised in
Classical Probability Theory using Urn problems.
