Is this zero knowledge protocol for honest verifiers?

Assume the zero knowledge protocol where the prover knows a $$x$$ such that: $$g^x = h \pmod{p}$$.

• The prover chooses a random $$t \in \mathbb{Z}^*_m$$ and sends $$y = g^t \pmod{p}$$
• The verifier sends random $$c \in \mathbb{Z}^*_m$$ and sends it
• The prover calculates $$s = t + c + x$$ and sends it
• The verifier accepts if and only if $$g^s = yg^ch \pmod{p}$$

Is this for honest verifiers?

Attempt:

The real transcript is $$(t, g^t \pmod{p}, c, t +c+x)$$, but I cannot find a simulated transcript $$(t, \cdot, c, \cdot)$$ that has the the same distribution as the real one. Can you give me some hints?

It's not a proof of knowledge, as someone without knowledge of $$x$$ can complete this protocol successfully with an honest verifier.
Hint: what happens if the ignorant prover sends $$y = h^{-1}$$?
• Then the verifier will choose a $c$ and the prover will send back $s=c$ so that $g^s = h^{-1} g^c h = yg^ch$ and the prover will pass the challenge without knowing $x$ as you said. Can you elaborate on what does that show for the protocol? Does it mean that it's not a zero knowledge protocol or that it's not a zero knowledge protocol for honest verifiers? Jan 23, 2021 at 17:03
• @Paris: it's not a zero knowledge protocol for proof of knowledge of $x$, as you can 'prove' it without knowing $x$ Jan 23, 2021 at 17:08