Problem $4$ In this problem you will illustrate RSA with an example similar to $3\mathrm{.}3\mathrm{.}3.$ You are given

$p=31$ and $q=53,$ so $n=pq=1643.$ You are also given $e=97.$ So to recap, the

public key is $(n,e)=(1643,97).$

$($a$)$ Compute $(1643).$

$($b$)$ Solve $97d-=1(mod(1643))$ for $d.$

$($c$)$ Find the encryption function $E$: $??$ defined by $E(x)=?($fill in the $?).$

$($d$)$ Find the decryption function $D$: $??$ defined by $D(x)=?($fill in the $?).$

$($e$)$ Encrypt the letter $B$ if it's encoded as the number $2.$

$($f$)$ What letter is encrypted as $1555,$ assuming letters are encoded by their place in

the alphabet as in part $($e$)?$ In other words, decrypt $1555.$