Alice and Bob decide to use RSA public key cryptosystem to share secret message. Alice publishes her RSA public key: modulus N = 55 and exponent e = 7. . . If Bob wants to send Alice the message m = 27. What ciphertext does Bob send to Alice? If Alice receives another ciphertext c = 2 from Bob, what is the corresponding plaintext? Remark: this problem tests your understanding of the RSA public key cryptosystem. 1. For n=pq, where p and q are distinct odd primes, define (n)= (p-1)(q-1) gcd(p-1,q-1) Suppose that we modify the RSA cryptosystem by requiring that ed=1 mod (n). a. Prove that encryption and decryption are still inverse operations in this modified cryptosystem. b. If p=37, q=79, and e-7, compute d in this modified cryptosystem, as well as in the original RSA cryptosystem.