How to write a Zero-Knowledge Proof of Knowledge of input to a one-way function?

n-l-i

10/13/23, 10:48 AM

I'm having a bit of difficulty understanding how to construct Zero-Knowledge proofs. So given a one-way function $f$ and a secret message $x$ so that $f(x)=y$, $f$ and $y$ being public, how could one construct a simple Zero-Knowledge Proof of Knowledge algorithm proving that one knows $x$?

I think I understand how this could be done if the verifier also knows $x$, because then they should be able to send random bits to each other and both run them through $f$. For example: the prover generates a random message $r$ and shares it with the verifier, they both calculate $f(x+r)=y_r$, the prover generates a random number $n$ and asks if the $n$:th bit of $y_r$ is a one or a zero. And then repeat all of that until satisfied.

I have no idea, however, how one would start to construct a proof for a verifier without knowledge of $x$.

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