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1

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||\...


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There are many possible constructions but the simplest is probably to initalize a counter with a random seed and hash the counter to produce more random bits and then increment the counter.


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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

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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