# In the reduction proof below, where OWF exists only if is a PRG. I am not able to understand the highlighted part  I am able to understand how G(x) id generated. But then what is the use of variable z. Also if the probability is >1/2 + e then the distinguisher wins! Then how is this still a OWF

This is an example of proof by contradiction. If you have ever seen a proof that the square root of 2 is irrational, that probably used a similar logical argument. Our goal is to prove that $$\mathcal A$$ does not exist, but we do that by assuming it does exist and then deducing that something goes wrong. The place to pay attention is bullet 4 where we assume $$\mathcal A$$ exists; mathematicians and computer scientists could make life easier at this point by adding "SPOILER ALERT: It doesn't!". The next three bullets proceed from this assumption and at the end of bullet 7 the wheels fall off because, as you realised, we have constructed a case where the distinguisher wins and because we know that $$G$$ is a PRG this cannot happen. We go back to bullet 4 and realise that we must have gone wrong by assuming the existence of $$\mathcal A$$. We conclude that no such $$\mathcal A$$ can exist.
The variable $$z$$ is meant to represent a response to a challenge from the adversary. After that the proof shows how you can distinguish the response if you can lay your hands on $$\mathcal A$$ (SPOILER ALERT: You can't).
• You are correct that the proof sketch formulates the argument as a proof by contradiction. This is very annoying, confusing and completely unnecessary. You can formulate it in a much nicer way as simply upper bounding the success probability of any ppt $\mathcal{A}$. In my opinion that's how it should be done. Mar 21 at 12:02