In any other context where I encountered the concept of soundness it was very simple: if the input does not belong to the language then the protocol fails or fails with great probability.

But in the paper Witness Indistinguishable and Witness Hiding Protocols there is an interaction between a prover $P$ (with a set of valid witnesses $w(x)$), a verifier $V$ and an input $x$ and the definition of soundness is: $\exists M\ \forall P'\ \forall x\ \forall w'$, $$ P[V_{P'(x, w')}(x) accepts] < P[M(x, w'; P') \in w(x)] + negligible $$ Where the probability is over the random tapes of $V$, $P'$ and $M$. $M$ uses $P'$ as a blackbox.

I just can not grasp this, can someone explain it to me?


The soundness definition that you are familiar with is the normal soundness definition of proof systems. The soundness definition in this paper is a definition of soundness for "proofs of knowledge", i.e., the goal of the prover is not only to convince the verifier that the statement is true but also that it knows a witness. This can be a much stronger requirement.

Now, there are two main aspects in which this differs to the notion you're familiar with. The first aspect is related to this question: How can we formalize that the prover "knows" a witness? The short answer in informal words is: if there's a witness in the prover, then there must be some way to get it out of the prover. This extraction process is the job of the knowledge extractor $M$ in the definition you found. The knowledge extractor $M$ is given black-box access to $P$ in order to extract the witness. I could expand on this concept but I believe there are a lot of very good resources with different levels of rigor that elaborate specifically on this point [1,2,3,4] because this is often a source of confusion for newcomers.

The second difference is that this definition relates the accepting probability of $P$ with the probability that $M$ extracts. In other words, whenever $P$ convinces the verifier, then $M$ should be able to extract a witness (with a negligible error). An alternative (but flawed) definition which is closer to the definition you're aware of would require that if $P$ convinces with non-negligible probability, then $M$ can extract successfully.* Variants of this definition have indeed been used early but they're too weak for what we typically need, as Bellare and Goldreich [4] explain in an entire paper that I very much recommend for a deeper understanding. In Appendix A, they explain why the definition you found is better than the flawed definition but still not optimal.

A final remark is that the paper you've found is already pretty old. These were the first days of zero-knowledge when the community was still in the process of finding good definitions for zero-knowledge proof systems, which can be nicely seen from the paper by Bellare and Goldreich [4]. The definitions in newer works may be clearer.

*Note that the condition "if the statement is in the language" would obviously be too weak: just because the statement is in the language does not mean that some $P$ (which could be just a dummy machine outputting only zeros) knows a witness.


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