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As far as I know, the "random oracle game" doesn't exist. What your are speaking (I think) is about pseudo-random functions. And when the challenger (not the oracle) is in the "RANDOM" mode, he could keep in memory the pair input/output, according to previous queries. Then it doesn't need to output "error", he could return the same output as in the previous ...

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The problem is in the last equation: $$Pr[\mathcal{A}^{\{U\}}(1^{\lambda})=1]=Pr[\mathcal{A'}^{\{U\}}(1^{\lambda})=1]$$ It does not hold because $\mathcal{A}^{\{U\}}$ is the result of $\mathcal{A}'$ using $U$ instead of $F_k$. This is actually equivalent to $F'$ and will be distinguished by $\mathcal{A}'$, so we get  Pr[\mathcal{A}^{\{U\}}(1^{\lambda})=...

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It is not necessary, that $G_1$ is a PRG. Let $G_2: \{0,1\}^{n-1} \rightarrow \{0,1\}^n$ be a PRG, define $G_1: \{0,1\}^n \rightarrow \{0,1\}^{n+1}$ as \begin{align*} G_1(s_1||\ldots||s_n) := 1||G_2(s_2||\ldots||s_n). \end{align*} and consider the distinguisher $\mathcal{D}_1$, which returns the first bit $w_1$, when given the $n+1$ bit string $w := w_1||\... -2 Pseudorandom permutation Confusingly, nothing really gets permuted here in the classic definition of a permutation. It's more of a cryptographic twist on the common notion of permutation. The actual input/output behaviour is like:- And each input is mapped to exactly one output value. The above diagram is simplistic in that the size of the inputs and ... 3 In term of CPA and KPA security, does$\boxplus$replacement add security over 3 rounds, with proof? No. There's nothing special about$\oplus$in the Feistel struture. In fact any group operation$\boxplus$over$\{0,1\}^n$will do just fine. The main point of these random functions is to "blind" the previous rounds' other halves and for that a group ... 3 All three are families of functions. For example,$f_k(x) = k \oplus x$, where$\oplus$is xor and$k$and$x$are 256-bit strings, is a family of functions; for any 256-bit string$k$, there is a function$f_k$which given another 256-bit string$x$returns the xor of$k$and$x\$. The input and output spaces need not be the same; we could imagine a family ...

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