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\paragraph{Problem 3 (18 points)}
Let $F:=\{ F^n: \{0,1\}^{3n} \to \{0,1\}^{2n} \}_{n\in\mathbb{N}}$ be a family of one-way functions (OWFs).
Let $G:=\{ G^n: \{0,1\}^n \to \{0,1\}^{2n} \}_{n\in\mathbb{N}}$ be a family of PRGs.
Let $H:=\{ H^n: \{0,1\}^{n} \to \{0,1\}^{2n} \}_{n\in\mathbb{N}}$ be defined by: for any $x\in\{0,1\}^n$
\[ H^n(x) := F^{n}( x, G^n(x) ) . \]
Assuming one-way functions exist, show that there exists a family of OWFs $F$ and a family of PRGs $G$ such that $H$ is NOT a OWF family. |