$(Q5)$

$[5$ marks$]$

Suppose that Bob adopts the RSA cryptosystem with primes $p=73$ and $q=79.$ He chooses the $($public key$)$ encryption exponent to be $e=1211.$ Write each answer as an integer in $\{1\mathrm{.}2,$dots,$m-1\}$ if vou are working modulo $m.$

$($a$)$ Show that Boh;s choice of encryption exponent is legitimate, and find his corresponding $($privite key$)$ decryption exponent $d.$

$($b$)$ Supposs that Alice encrypts the phantext mesage $M^{\text{'}}=231$ using the RSA cryptosystem with Bob's public key $($n$,$e$)$ Whit is the resilting ciphertext that would be sent to Bob?

$($a$)$ Go thirohg the decryption process that would ued to be done at Bob's end using his private key $(p,q,d)$ with decrytion $($xponent that whis deternined in item $($a$).$