This reminds me of authentication protocols in cryptology, where the one-way-encryption keys are never exchanged.

A sends a sequence of random values and B must reply with the correct encryption or just a property of it. After n correct answers A will be statistically "convinced" that B had the correct key.

Because of the asymmetric nature the correct key can't be deduced by a listener in the middle.

I doubt it helps, but the question allows so many wild interpretations...