# How Cut-and-Choose can leak information?

I'm currently reading the book "A Pragmatic Introduction to Secure Multi-Party Computation". On pages 103-104 I came across the following related to the Cut-and-Choose technique:

If $$P_2$$ sees inconsistent outputs, then it is obvious to $$P_2$$ that $$P_1$$ is misbehaving. It is tempting to suggest that $$P_2$$ should abort in this case. However, to do so would be insecure! Suppose $$P_1$$’s incorrect circuits are designed to be selectively incorrect in a way that depends on $$P_2$$’s input. For example, suppose an incorrect circuit gives the wrong answer if the first bit of $$P_2$$’s input is 1. Then, $$P_2$$ will only see disagreeing outputs if its first bit of input is 1. If $$P_2$$ aborts in this case, the abort will then leak the first bit of its input. So we are in a situation where $$P_2$$ knows for certain that $$P_1$$ is cheating, but must continue as if there was no problem to avoid leaking information about its input.

To add some context to the above. They are referring to the naive Cut-and-Choose technique where $$P_1$$ is the garbled circuit generator and $$P_2$$ is the evaluator. So $$P_1$$ generates a bunch of GC for the preagreed function $$f$$ and sends them to $$P_2$$. Then $$P_2$$ chooses which of them to open based on a random coin and tells $$P_1$$ to open them, e.g. send all the corresponding wire labels (keys) for $$P_2$$ to be able to reconstruct the Look-up Table and check if it is equal to the pre-agreed one.

How can $$P_1$$ create an incorrect circuit that depends on the $$P_2$$ input so it can reveal information about it?

$$P_1$$ garbles the circuit, so he can manipulate it (e.g. swap AND and OR gates) since $$P_2$$ only receives permuted encrypted truth tables.
In particular, $$P_1$$ could add an "if-else" gate s.t. the correct output is XORed with hardcoded random noise iff the first bit of $$P_2$$ is one.
If $$P_2$$ chooses this manipulated circuit and therfore aborts, $$P_1$$ learns that $$P_2$$'s first bit is zero.