1. Find all the square roots of x2 â‰¡ 53 (mod 77) by hand. (2 marks)

2. Convert the decimal-based numbers 79, 83 to bytes, and then multiply these two bytes in GF(28) with respect to the irreducible polynomial x8+x4+x3+x+1 by hand. (3 marks)

3. Alice has the RSA public key (n, e) = (11413, 251) and private key d = 1651. And Bob also has his own RSA public key (n’, e’) = (20413, 2221) and private key d’ = 6661. Alice wants to send the message 1314 to Bob with both authentication and non-repudiation. Use Maple, calculate what is the ciphertext sent by Alice. And Verify that Bob is able to recover the original plaintext 1314. (3 marks)

4. Use the Fermat difference of squares method to factor the RSA modulus n = 1052651. (2 marks)

5. Alice and Bob want to establish a common key pair using the Diffie-Hellman key exchange. Alice is in Australia while Bob is in Brazil. Carl, a friend of Alice, has been tracking the email received and sent by Alice and decides that he wants to listen in on conversations Alice intends to have with Bob.

Carl, therefore, sets up a man-in-the-middle attack as follows. Bob and Alice agree by email on a prime p = 877 and a primitive root (generator) a = 453. Carl sees these agreed numbers. Then Alice chooses secret x = 20 while Bob chooses secret y = 14. Carl chooses secret z = 19. Alice computes her 45320 mod p and sends it to Bob; however, Carl intercepts this email. Similarly, Carl intercepts Bob’s e-mail containing 45314 mod p. Using Maple, determine what common key Carl sets up with Alice, and the key Carl sets up with Bob. (3 marks)

6. In a (2, 5) Shamir secret sharing scheme with modulus 19, two of the shares are (0, 11) and (1, 8). Another share is (5, k), but the value of k is unreadable. Find the correct value of k. (2 marks)