$1.$ Let F be a pseudorandom function, and consider the following construction of a stream cipher accepting an n$-$bit initialization vector:

Init$($s$,$ IV$)$ outputs st $=($s$,$ IV$).$

Next$($s$,$ IV$)$ outputs y $$ Fs$($IV$)$ and st$=($s$,$ IV $+1).$

Show that this stream is not secure$.$

$2.$ Let F be a length preserving pseudorandom function. For the following constructions of a keyed function F$$ : $\{0,1\}^$n $*\{0,1\}^$n $-1->\{0,1\}^2$ n$,$ state whether F$$ is a pseudorandom function. If yes, prove it; if not, show an attack.

F$\text{'}($x$)$: $=$F$(0||$ x$)||$ F$(0||$ x$)$

F$\text{'}($x$)$: $=$F$(0||$ x$)||$ F$(1||$ x$)$

F$\text{'}($x$)$: $=$F$(0||$ x$)||$ F$($x $||0)$

F$\text{'}($x$)$: $=$F$(0||$ x$)||$ F$($x $||1)$

$3.$ Let G : K $->\{0,1\}^$ n be a PRG such that from the last n$/2$ bits of G$($k$)$ it is easy to compute the first n$/2.$ Is G a secure PRG$?$ Why?