$(5)($RSA public key cryptosystem$)$

Bob chooses the secret primes $p=5$ and $q=11$ and the encryption expo$-$

nent $e=3.$ Bob publishes $N=55$ and $e=3.$

$($a$)$ Alice chooses the plaintext $m=7.$ Compute the ciphertext $c$ that

Alice sends to Bob.

$($b$)$ Compute Bob's decryption exponent $d.$

$($c$)$ Bob receivers the ciphertext $c=24$ from Alice. Compute the plaintext

$m$ that Bob obtains by deciphering $c?$

$($d$)$ Explain why the RSA public key cryptosystem is a secure system.