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2

Suppose $F \colon \lbrace 0,1 \rbrace^n \to \lbrace 0,1 \rbrace^{\ell(n)}$ is a secure PRG. Create $G \colon \lbrace 0,1 \rbrace^{n+1} \to \lbrace 0,1 \rbrace^{\ell(n)}$ as follows: $$G(b ||s) := F(s), \quad b \in \lbrace 0,1 \rbrace, s \in \lbrace 0,1 \rbrace^n.$$ $G$ is as secure as $F$, and clearly $G(0||s) = G(1 || s)$. To prove the "$G$ is as ...

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Adding the source and length actually collects more entropy. Consider the case of using system counters as entropy sources. If at some timestamp CounterX=123, and CounterY=123, we actually retain at least one bit more entropy by stirring in a source identifier ("CounterX=123CounterY=123") instead of simply "123123". The source of the value is additional ...

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You can't contaminate the pool with low entropy data. Hash functions "extract" up to the output size or the hash - slightly less, because of collisions. The source number makes the output different for different sorces even if the input is the same. The length is needed because else we could not differate two small inputs from one large input with special ...

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