Objectives: Understanding PRF and IND$-$CPA definitions, proving schemes insecure.

Notation for problems below:

$$xy refers to the XOR of x and y

$$x $||$ y refers to concatenation of x and y

$$ refers the bitwise complement of x

Problem $2\mathrm{.}1$

$(10$ points$)$

Let F: $\{0,1\}$x$\{0,1\}\{0,1\}"$ be a secure PRF$.$ Prove that the function family F$$: $\{0,1\}$x$\{0,1\}2$m $\{0,1\}2$n specified for all $21,22$ in $\{0,1\}$m and all K in $\{0,1\}*$ below is not a secure PRF:

F$1($K$,$ X$1||22)=$ F$($K$,21)||$ F$($K$,2122)$

Problem $2\mathrm{.}2$

$(10$ points$)$

Let E: $\{0,1\}$x$\{0,1\}\{0,1\}$ be a blockcipher known to be a secure PRF$.$ Prove that the function family F$$: $\{0,1\}*\{0,1\}2$m $\{0,1\}$m specified for all $21,22$ in $\{0,1\}$m and all K in $\{0,1\}*$ below is not a secure PRF:

F$2($K$,$ X$1||22)=$ E$($K$,$ E$($K$,\backslash$pi $))$ X$1$