Alice and Bob wish to use the RSA cryptosystem for communication. Alice announced the

public modulus and encryption key pair to be $(143,11).$

$(1)$ Suppose that Bob wish to send a message $2,$ what is the encrypted message?

$[7$ marks$]$

$(2)$ Derive the private key. $[8$ marks$]$

$(3)$ Assume Bob sent an encrypted message $4,$ what$$s the original unencrypted message?

$[5$ marks$]$