In this paper we extend the notion of interactive proofs of assertions to interactive proofs of knowledge. This leads to the definition of unrestricted input zero-knowledge proofs of knowledge in which the prover demonstrates possession of knowledge without revealing any computational information whatsover (not even the one bit revealed in zero-knowledge proofs of assertions). We show the relevance of these notions to identification schemes, in which parties prove their identity by demonstrating their knowledge rather than by proving the validity of assertions. We describe a novel scheme which is provably secure if factoring is difficult and whose practical implementations are about two orders of magnitude faster than RSA-based identification schemes. The advantages of thinking in terms of proofs of knowledge rather than proofs of assertions are demonstrated in two efficient variants of the scheme: unrestricted input zero-knowledge proofs of knowledge are used in the construction of a scheme which needs no directory; a version of the scheme based on parallel interactive proofs (which are not known to be zero knowledge) is proved secure by observing that the identification protocols are proofs of knowledge.