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\paragraph{Problem 3 (18 points)}
Let $F:=\{ F^n:=\{ f^n_k: \{0,1\}^n \to \{0,1\}^n \}_{k\in K_n} \}_{n\in\mathbb{N}}$ be a family of PRFs.
Let $G:=\{ G^n: \{0,1\}^n \to \{0,1\}^{2n} \}_{n\in\mathbb{N}}$ be a family of PRGs.
Let $H:=\{ H^n:=\{ h^n_k: \{0,1\}^n \to \{0,1\}^{2n} \}_{k\in K_{2n}} \}_{n\in\mathbb{N}}$
be defined by: for any $x\in\{0,1\}^n$
\[ h^n_k(x) := f^{2n}_k(G^n(x)) \]
Prove: there exists a family of PRFs $F$ and a family of PRGs $G$ such that $H$ is NOT a PRF family.
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