The seminal work of Goldwasser, Micali and Rackoff put forward a computational approach to knowledge in interactive systems, providing the foundation of modern Cryptography. Their notion bounds the knowledge of a player in terms of his potential computational power (technically defined as polynomial-time computation).
We put forward a stronger notion that precisely bounds the knowledge gained by a player in an interaction in
terms of the actual computation he has performed (which can be considerably less than any arbitrary polynomial-time computation).
Our approach not only remains valid even if P= NP, but is most meaningful when modeling knowledge of computationally easy properties. As such, it broadens the applicability of Cryptography and weakens the complexity theoretic assumptions on which Cryptography can be based.
Joint work with Silvio Micali