Q$1.$ Consider RSA Cryptosystem that can be used to ensure non$-$repudiation $($refers to a situation where a statement's author cannot successfully dispute its authorship or the validity of an associated contract$).$ Assume that Bob has received a signed message S $=165$ and a public key $(21,187)$ from Alice. Consider that the message $($and not the hash of the message$)$ is signed.

a$)$ What is the range of distinct signed$-$messages? Assume that there is countably infinite number of plaintext messages of any length available at sender$.$

b$)$ Perform factorization attack to find private key.

c$)$ What is the message $M$ that was signed?

d$)$ Construct a message $M^{\text{'}}M,$ that will produce a same signature as $S.$

e$)$ In part d$)$ do you think non$-$repudiation is assured? If so then why, and if not then why not?