MACs vs$.$ PRFs: We learned in class that a PRF is a strictly $$stronger$$ primitive than a MAC. That is$,$ all PRFs are MACs, but not all MACs are PRFs$.$ Show that this is true in two parts:

a$.$ Show that if a function F is a secure PRF$,$ then F is also a MAC. To show this, show that any attacker who can win the MAC security game $($existential unforgeability$)$ against F can also win the PRF security. To do so$,$ assume you have access to an algorithm A$1$ which wins the MAC security game and describe an algorithm A$2$ which uses A$1$ as a subroutine and wins the PRF security game.

b$.$ Show a toy example of a MAC that isn$$t a PRF$.$ One way to do this is to take a function F which is a PRF and define a tweaked version F$$ that is still a MAC but not a PRF$.$