Proof of knowledge outside of Zero Knowledge

Question

Are there interesting protocols for which proof of knowledge is required, but not zero knowldege?

Is Proof of Knowledge an interesting proof by itself?

Answer 1

If you don't have any zero knowledge requirement, then a proof of knowledge is achieved by just having the prover send the "witness" in the plain to the verifier. Thus, no, a proof of knowledge that has no privacy requirement is not interesting.

Note that it can be interesting to have a witness indistinguishable proof of knowledge, or a witness hiding proof of knowledge. These are not full blown zero knowledge but have meaningful privacy requirements.

The best place to read about witness indistinguishability and witness hiding is in Oded Goldreich's book on the Foundations of Cryptography (Volume 1). (Note the errata on strong witness hiding.) You can also look at Uri Feige's PhD Thesis, which is extremely clear and easy to read (but less general than the treatment in the Foundations of Cryptography).

The reason why one would want these not "full blown" zero knowledge notions is that it is possible to achieve them more efficiently in some sense. In addition, they can be used as important stepping stones to full zero knowledge.