Please solve c and d only

RSA and digital signatures. Recall that in the RSA public-key cryptosystem, each user has a public key P = (N, e) and a secret key d. In a digital signature scheme, there are two algorithms, sign and verify. The sign procedure takes a message and a secret key, then outputs a signa- ture o. The verify procedure takes a public key (N, e), a signature o, and a message M, then returns "true" if o could have been created by sign (when called with message M and the secret key corresponding to the public key (N,e)); "false" otherwise. (a) Why would we want digital signatures? (b) An RSA signature consists of sign(M, d) = M (mod N), where d is a secret key and N is part of the public key. Show that anyone who knows the public key (N,e) can perform verify((N, e), Md, M), i.e., they can check that a signature really was created by the pri- vate key. Give an implementation and prove its correctness. (c) Generate your own RSA modulus N = pq, public key e, and private key d (you don't need to use a computer). Pick p and , so you have a 4-digit modulus and work by hand. Now sign your name using the private exponent of this RSA modulus. To do this you will need to specify some one-to-one mapping from strings to integers in (0, N - 1). Specify any mapping you like. Give the mapping from your name to numbers m. m2, ... me, then sign the first number by giving the value m (mod N), and finally show that (m) = m (mod N). (d) Alice wants to write a message that looks like it was digitally signed by Bob. She notices that Bob's public RSA key is (17,391). To what exponent should she raise her message