A potential use for such systems is for passwords--one can prove one has the knowledge without actually producing it (by typing in a password, for example). I don't know that anyone is actually exploring this application, yet, but I expect it'll come.
Look at "Strongbox: A System for Self-Securing Programs" by J. D. Tygar and B. S. Yee in the "CMU Computer Science 25th Anniversary Commemorative" proceedings (from 1991). As the paper describes: ``Strongbox uses an authentication protocol derived from Rabin's observation about the square root operation: if one can extract square roots modulo n where n=p*q , p and q primes, then one can factor n . [That should be `if and only if', i.e., finding the square roots is too hard unless you created n in the first place.] Both our protocol and FFS are *zero-knowledge authentication protocols_* [. . .] And in contrast to Needham and Schroeder's authentication protocol, zero-knowledge authentication protocols require no central authentication server and thus there is no single point of failure that would cripple the entire system.'' In addition to zero-knowledge authentication, the paper provides an algorithm for the secure exchange of sessional symmetric encryption keys, and ways of combining authentication and key-exchange steps. I managed to get the key-exchange working some months back (in C++, using GMP to handle the number-crunching), but it was hampered by my incredibly slow 386 on one side and odd bugs in the Sun4 environment on the other. Contact me if you want to hack around on it. I also know where to find unreleased GMP 1.9 sources for some additional, probably more reliable, functions for calculating the Legendre symbol (which the whole system depends upon). Derek