On Defining Proofs of Knowledge in the Bare Public Key Model

One contribution provided by the groundbreaking concept of interactive proofs is the notion of proofs of knowledge, where a prover can convince a verifier that she knows a secret related to a public statement. This notion was formalized in the conventional complexity-theoretic model of interactive protocols and showed to be very useful for cryptographic applications, such as entity authentication schemes. Motivated by these applicability considerations, in this paper, we consider proofs of knowledge in a cryptographic model, called the bare public-key model (BPK model in short), where round-efficient interactive proofs with strong variants of security against provers (i.e., soundness) and security against verifiers (i.e., zero-knowledge) have been presented. We formally define notions of proofs of knowledge in the BPK model, and show that there are 4 distinct such notions for each of the previously studied four known notions of soundness. Finally, under the existence of any homomorphic one-way function family, (a generalization of) a 4-round argument system for all NP languages from the literature is a proof of knowledge that is secure against concurrent attacks from provers or verifiers.