Consider the Meet$-$in$-$the$-$Middle attack of Double DES $($the weakened, exportable version$),$ with $64-$bit data blocks and two $40-$bit keys. If a pair of keys $($K$1,$ K$2)$ can satisfy EK$1($P$1)=$ DK$2($C$1),$ i$.$e$.$ lead to the same output in the middle stage, we call it a collision. The corresponding pair of keys is referred to as a colliding key$-$pair, which may or may not be the pair of authentic keys. $($a$)$ For a random pair of $40-$bit keys, what is the probability that they will produce a collision at the middle stage? $($b$)$ If one enumerates all possible key$-$pairs $(40-$bit per key$),$ on average, how many colliding key$-$pairs do we expect to find if we only check each key$-$pair against one given pair of plaintext and ciphertext, say $($P$1,$ C$1)?($c$)$ For a random pair of $40-$bit keys, what is the probability that it will lead to a collision at the middle stage for $2$ different pairs of plaintext and ciphertext, i$.$e$.($P$1,$ C$1)$ and $($P$2,$ C$2)?$ That is$,$ the probability that $($K$1,$ K$2)$ will satisfy: EK$1($P$1)=$ DK$2($C$1)$ and EK$1($P$2)=$ DK$2($C$2).($d$)$ To follow up on part $($c$),$ if we enumerate over all possible key$-$pairs, what is the expected number of key$-$pairs that can satisfy EK$1($P$1)=$ DK$2($C$1)$ and EK$1($P$2)=$ DK$2($C$2)$ simultaneously?