A Note on Zero-Knowledge Proofs of Knowledge and the ZKPOK Ideal Functionality

In this note, the authors provide a formal proof of the fact that any protocol that is a zero-knowledge proof of knowledge for a relation R is also a secure protocol for the zero-knowledge proof of knowledge functionality, where the latter is defined according to the standard framework of stand-alone secure computation. Although this is a well-known fact, to the best of the knowledge, no full proof of this has been published. In order to keep this note brief, they assume familiarity with the definitions of zero-knowledge, zero-knowledge proofs of knowledge, and secure computation in the presence of malicious adversaries.