All zero-knowledge protocols I have read so far for discrete log works like this:
1) Prover generate a random number $r$, creates a commitment $t=g^r$ and sends $t$ to the verifier.
2) Verifier generates a random challenge $c$ and sends it to the prover.
3) Prover creates a response $s=r+x*c$ and sends it to the verifier
My question is: is the security affected in any way if, in step 3, the prover responds with $s=r+x+c$?
I don't understand why multiplication $x*c$ is used instead of sum $x+c$.