# Question 1: (12 Marks) If exists, calculate multiplicative inverse of 7, 12, 22, 23, 66, 93, and 129 in Z164. If does not exists explain why? Note that gcd(164, X) =1, otherwise no multiplicative inverse possible. Formulas: x * d mod 164=1 d= (1+k *164)/e, k= 1… upto x, should be an integer number Question 2: (12 Marks) If exists, find the determinant and the multiplicative inverse of the residue matrix M1 and M2 over Z26

Question 1: (12 Marks)
If exists, calculate multiplicative inverse of 7, 12, 22, 23, 66, 93, and 129 in Z164. If does not exists explain
why? Note that gcd(164, X) =1, otherwise no multiplicative inverse possible.
Formulas:
x * d mod 164=1
d= (1+k *164)/e, k= 1… upto x, should be an integer number
Question 2: (12 Marks)
If exists, find the determinant and the multiplicative inverse of the residue matrix M1 and M2
over Z26

7 3 9
5 23 25
21 6 22
1 M

9 5 7
25 21 22
23 6 3
2 M
Question 3: (12 Marks)
If we want to use above matrices (M1 and/ or M2) of Question 5 as a key for constructing a Hill Cipher
cryptosystem, then which one between M1 and M2 you recommend to use as a key, and why?
Using your recommended key decrypt the following ciphertext.
Ciphertext: TJFKBSXXW
Question 4: (20 Marks)
Using Feistel Block Cipher Encryption technique with two rounds, encrypt the following plaintext .
Plaintext: be (01100010 01100101)
K1 : 10101011
K2 : 11001101
F is defined as follows:
F(K, R) = K [ 4-bit left circular shift of R]
Question 5: (20 Marks)
Ahmed is using RSA crypto-system with the following setup:
 p = 11 and q = 3
 n = pq = 11 × 3 = 33.
 Ф(n) = (p − 1)(q − 1) = 10 × 2 = 20.
 Ahmed publish his Public Key:
(n, e) = (33, 3).
A. Calculate Ahmed’s private key.
B. Charlie wants to send the message M = 13 to Ahmed. Using Ahmed’s public and private keys, calculate the ciphertext C, and the value for Message R, when Alice recovers the message.
C. Dixit wants to set up his own public and private keys. He chooses p = 23 and q = 19 with e = 283. Find his private and public keys.
Question 6: (12 Marks)
In a RSA cryptanalysis, you intercept the ciphertext C = 10 sent to a user whose public key is (e = 7, n = 35). What is the plaintext M?
Question 7: (12 Marks)
In a Deffie-Hellman key exchange setup, for simplicity, consider the large prime P = 53 and the primitive root of P is a = 5. A sender generates his random secret XA =12 and the receiver generates his random secret YB = 18. Calculate the session key.

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