Elliptic curve encryption or decryption scheme

Question

Question 1:

a)  If m = p•q•r where p, q, and r are prime numbers, what is Φ (m)?

b) Therefore, compute Φ (440).

Question 2: Explain the given terms as used in the cryptography:

a) Avalanche effect
b) Stream cipher
c) Primitive root
d) Confusion
e) Steganography

Question 3: Consider an elliptic curve encryption or decryption scheme. The cryptosystem parameters are E₅(1,6)  and G = (2,4). B's secret key is n_B = 2.

a) Determine B's public key P_B.

b) A wishes to encrypt the message P_m = (0, 2) and selects the random value k = 2. Find out the ciphertext C_m.

Question 4:

a) Bob has public RSA key (n = 77, e = 7). Show that Bob’s private key is (d = 43).

b) Alice wants to send the message m = 13 to Bob. She encrypts the message by using Bob’s public key. Find out the value of the ciphertext which Alice sends to Bob?

c) David has as well sent an encrypted message to Bob. The ciphertext value that Bob receives from David is 17. Showing all your working, use Bob’s key to decrypt this ciphertext and recover the value of the David’s message.

Question 5: Consider cryptographic hash functions.

a) What is meant by collision resistance for hash function?

b) What common usage do hash functions have in connection with the digital signatures?

c) What is the difference between a hash function and a MAC (message authentication code)?

d) Why is it proposed to use a hash function with 2n bits of output when it is used as the component in a system designed for n bits of security?