Pseudorandom generators $[16$ Marks$].$

For each function below, state whether it is also a secure Pseduorandom Generator. If the function is a secure

Pseduorandom Generator, give a proof. If not, describe a successful distinguisher and explicitly compute its

advantage.

$($a$)(8$ marks$)$ Let $G$:$\{0,1{\}}^n\{0,1{\}}^{3n}$ be a secure length$-$tripling Pseudorandom Generator, for security

parameter $n.$ Define $G^{\text{'}}$ as follows. $G^{\text{'}}(s)$:$=G(s)o+G(0^n).$ Here $sin\{0,1{\}}^n,0^n$ denotes the all$-$zeros bit

string of length $n$ and $o+$ denotes the bitwise XOR.

$($b$)(8$ marks$)$ Let $G$:$\{0,1{\}}^n\{0,1{\}}^{3n}$ be a secure length$-$tripling Pseudorandom Generator, for security

parameter $n.$ Define $G^{\text{'}}$ as follows. $G^{\text{'}}(s)$:$=G(s)o+(0^{2n}||s)$ where $sin\{0,1{\}}^n,0^{2n}$ denotes the all$-$zeros bit

string of length $2n$ and $o+$ denotes the bitwise XOR.