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4

The question errs precisely at the point when it writes: $$\text{and so }H(x')=y.$$ Problem is: that holds for $x'$ matching some garbage $z'$, including $z'=z$, and perhaps other garbage $z'$, but we do not know which garbage (or $z$ unless we know $x$), and I see no reason that it holds for $x'$ obtained from any garbage $z'$ except for trivial $H$; and ...

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While trying to reverse the 'hash' that was mentioned in one of the comments, I discovered the real problem. In short, the problem of finding the valid garbage is bigger than the problem of finding the input. That is, the procedure is correct, but it makes the problem more difficult than the original. The detailed explanation Suppose you are trying to ...

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Your basic problem is, that your circuit would require infinite many output bits (to be precise, the garbage is infinite). The input for each gate is 3 bit, and you end up with 3 bit. You can duplicate bits in a circuit (by using it as input to multiple gates), but you can NEVER throw away bits or reduce the circuits information. This means, that the output ...

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