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This cannot be used to compromise $P_0$. Suppose that, instead of generating $C_0$ and $C_1$ as $R_0\oplus R_1$ and $R_2\oplus R_1$, the attacker instead picked $C_0$ and $C_1$ at random from the set of bit strings the length of $C_2$. This situation is indistinguishable from yours; in both cases, $R_0$ and $R_2$ are independent and follow a uniform ...

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