In a Non-Interactive $Zero-Knowledge-Proof$, the challenge is chosen by the Prover.
I am trying to find a Non-Interactive Zero-Knowledge-Proof based on the following problem:
DISCRETE LOGARITHM
Input: Prime number $p$, generator $g$ of $Z^{*}_{p}$ , and $y\in Z^{∗}_{p}$ .
Question: find $x \in \lbrace1, . . . , p − 1\rbrace$ with $y ≡ g^{x}\;mod\;p$?
It seems, what you are looking for is a Non-Interactive Zero Knowledge Proof of Knowledge (NIZKPok) of a discrete logarithm.
The Schnorr identification scheme is an (interactive) ZKPoK and can be transformed in to a non interactive one (in the random oracle model) using the Fiat-Shamir transformation.
It works in any group, is perfectly sound and computationally zero knowledge under the discrete logarithm assumption for that group in the random oracle model.
http://publikationen.stub.uni-frankfurt.de/files/4280/schnorr.pdf
