# Can zero knowledge proof and zero knowledge proof of knowledge transfer to each other?

Recently I'm studying learning with errors crypto systems and I'm running into a problem. I try to prove that the plaintext is in some specific range(for example 0~10) using zero knowledge proof.

However, almost all the papers I find in Google are talking about constructing "zero knowledge proof of knowledge" schemes, but what I want is zero knowledge proof, can these two primitives transfer to each other?

A zero-knowledge proof $\rho$ demonstrates that an instance $\phi$ is in a relation $R$, i.e. there exists $w$ such that $(\phi, w) \in R$.
A zero-knowledge proof of knowledge $\pi$ for an instance $\phi$ demonstrates that the random tape of the PPT algorithm that calculated $\pi$ can be used as the input to a PPT extractor to output $w$ such that $(\phi,w) \in R$.
If $w$ does not exist, then there can be no such extractor. Thus any zero-knowledge proof of knowledge is also a zero-knowledge proof. However, not every zero-knowledge proof is a zero-knowledge proof of knowledge. In particular, Groth-Sahai proofs do not have knowledge extraction.