Given the message x ∈ {0,1,. ...,N-1} and N = pq, where p and q are prime numbers, conventional RSA encrypts x with y = E(x) = x^e (mod N). The decryption is done by D(y) = y^d (mod N), where d is the inverse of e (mod (p − 1)(q – 1)). Alice is trying to send a message to Bob, and as usual, Eve is trying to decipher what the message is. One day, Bob gets lazy and tells Alice that he will now use N = p, where p is a 1024-bit prime number, as part of his public key. He tells Alice that it’s okay, since Eve will have to try out 2 1024 combinations to guess x. It is very likely that Eve will not find out the secret message in a reasonable amount of time! In this problem, we will see whether Bob is right or wrong. Assume that Eve has found out about this new setup and that she knows the public key.

Required:
a. Prove that the message x is recovered after it goes through your new encryption and decryption functions, E (x) and D(y).
b. Can Eve compute d in the decryption function? If so, by what algorithm and approximately how many iterations does it take for it to terminate?
c. Given part (b), how would Eve recover x and what algorithm would she use? Approximately how many iterations does it take to terminate?
d. Based on the previous parts, can Eve recover the original message in a reasonable amount of time? Explain.