Variant of Schnorr Protocol (Difference pair of response and verification) When I am trying to learn deeper to Schnorr Protocol. I found that for deference there is more than one response and verify pair. But I am not sure am I right.

We will use Schnorr Protocol to prove the knowledge of $x$ in $y=g^x$. Here is the details of Schnorr Protocol, first prover send $t=g^r$ where $r$ is a random number to verifier. Verifier send challenge $c$ to prover, next step prover will calculate response and let verifier calculate is it correct by a function. Here is the question.

For $y=g^x$, $t=g^r$ and $c$, we have 3 equation below.


$g^x g^{rc}=yt^c$ $\rightarrow$  $g^{x+rc}=yt^c$
$g^{xc} g^r=y^ct$ $\rightarrow$  $g^{xc+r}=y^ct$
$(g^x g^r)^c=y^ct^c$ $\rightarrow$  $g^{c(x+r)}=(yt)^c$


Can I say I can choose either one of that pair to finish the protocol?


response $s=x+rc$, verify $g^s=yt^c$
response $s=xc+r$, verify $g^s=y^ct$
response $s=c(x+r)$, verify $g^s=(yt)^c$


Am I correct?
response $s=c(x+r)$, verify $g^s=(yt)^c$
  


This one doesn't work; a lying prover can chose $t = y^{-1}g^n$, for an arbitrary $n$.  Then, when the challenger responds with a $c$, the lying prover can respond with $s = nc$, satisfying the relationship.

The other two are good; the second is the standard Schnorr, and the first is standard Schnorr proof of the inverse relationship, that is, if you're proving the knowledge of $y^{x'} = g$ (which is equivalent to knowledge of $g^x = y$)