In this note, we show the existence of \emph{constant-round} computational zero-knowledge \emph{proofs of knowledge} for all $\NP$. The existence of constant-round zero-knowledge proofs was proven by Goldreich and Kahan (Journal of Cryptology, 1996), and the existence of constant-round zero-knowledge \emph{arguments} of knowledge was proven by Feige and Shamir (CRYPTO 1989). However, the existence of constant-round zero-knowledge proofs of knowledge for all $\NP$ is folklore, to the best of our knowledge, since no proof of this fact has been published.

Fixed some small errors; simplified the proof a little.

In this note, we show the existence of \emph{constant-round} computational zero-knowledge \emph{proofs of knowledge} for all $\cal NP$. The existence of constant-round zero-knowledge proofs was proven by Goldreich and Kahan (Journal of Cryptology, 1996), and the existence of constant-round zero-knowledge \emph{arguments} of knowledge was proven by Feige and Shamir (CRYPTO 1989). Although it is widely believed that there exist constant-round zero-knowledge proofs of knowledge for all $\cal NP$, to the best of our knowledge, no proof of this fact has been published.