From the lecture notes I have learned the following:
- $P \rightarrow V$: $T=g^t$ ($t$, randomly chosen from $Z_q$)
- $P \leftarrow V$: $c$ (randomly chosen from $Z_q$)
- $P \rightarrow V$: $s=xc+t \mod q$
- $V$ accepts if $g^s=A^cT$, (where $A=g^x$)
Question: Can $P$ instead of sending $g^s$ send directly $A^cT$?
In other words, $P$ has all it need to generate $A^cT$ which is the same as $g^s$. Can someone explain what I am seeing wrong?