Suppose Alice and Bob are using a cryptosystem with a $128-$bit key, so there are $2^128$ possible keys. Eve is trying a brute$-$force attack on the system.

$($a$)$ Suppose it takes $1$ day for Eve to try $2^64$ possible keys. At this rate, how long will it take for Eve to try all $2^128$ keys? $($HINT: The answer is not $2$ days.$)$

$($b$)$ Suppose Alice waits $10$ years and then buys a computer that is $100$ times faster than the one she now owns $($so it takes only $1/100$ of a day, which is $864$ seconds, to try $2^64$ keys$).$ Will she finish trying all $2^128$ keys before or after what she does in part $($a$)?($NOTE: This is a case where Aesop$$s Fable about the Tortoise and the Hare has a different ending.$)$Suppose Alice and Bob are using a cryptosystem with a $128-$bit key, so there are

$2$

$128$

possible keys. Eve is trying a brute$-$force attack on the system.

Suppose it takes $1$ day for Eve to try

$2$

$64$

possible keys. At this rate, how long will it take for Eve to try all

$2$