The Knowledge Complexity of Interactive Proof Systems

Abstract. Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.

In this paper a computational complexity theory of the "knowledge " contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and quadratic nonresiduosity.

These are the first examples of zeroknowledge proofs for languages not known to be efficiently recognizable. Key words, cryptography, zero knowledge, interactive proofs, quadratic residues AMS(MOS) subject classifications. 68Q15, 94A60 1.

Introduction. It is often regarded that saying a language L is in NP (that is, acceptable in nondeterministic polynomial time) is equivalent to saying that there is a polynomial time "proof system " for L. The proof system we have in mind is one where on input x, a "prover " creates a string a, and the "verifier " then computes on x and a in time polynomial in the length of the binary representation of x to check that