$($Cryptanalysis$,2\mathrm{.}5$pts $4=10$ pts$)$ We use $$X$,$Y$,$ and $$Z$$ to denote Sender, Reciever, and

Eavesdropper, respectively. $$X$$ is sending a message to $$Y$$ using one of the following cryptosystems.

The plaintext of the message consists of the letter a repeated a few hundred times. $$Z$$ knows what

cryptosystem is being used, but not the key, and intercepts only the ciphertext. For systems $($a$),($b$),$

$($c$)$ and $($d$),$ state how $$Z$$ will recognize that the plaintext is one repeated letter and decide whether

or not $$Z$$ can deduce the letter and key. $($Note: for system $($c$),$ the solution very much depends on

the fact that the repeated letter is a$,$ rather than b$,$ c$,...)$

$($a$)$ Shift cipher

$($b$)$ Affine cipher

$($c$)$ Hill cipher $($with a $22$ matrix$)$

$($d$)$ Vigen$`$ere cipher