$10.$ The DES block cipher uses a $56-$bit key to encrypt a plaintext message.

$($a$)$ If an operation consists of randomly choosing a possible DES key and testing if it is correct, how many operations are required to guarantee that the cipher will be broken? $(3$ points$)$

$($b$)$ If each operation took $228$ seconds $($about $3\mathrm{.}7$ nanoseconds$)$ approximately how many days would it take to perform this attack? $(4$ points$)$

$($c$)$ If this attack were performed in parallel on $128$ processors, approximately how many days would it take for one of the attacks to succeed? $(3$ points$)$