Goldreich-Krawczyk Revisited: A Note on the Zero Knowledge of Proofs of Knowledge

The seminal work of Goldreich and Krawczyk (SIAM Journal on Computing) shows that any constant-round public-coin interactive proof for languages not in BPP cannot be black-box zero knowledge. Their result says nothing, however, about proofs (or arguments) of knowledge for languages in BPP. As a special case, their work leaves open the question of whether Schnorr's protocol for proving knowledge of discrete logarithms in cyclic groups is black-box zero knowledge.

In this work we focus on the zero knowledge of proofs of knowledge, centering on Schnorr's protocol as a prominent example.