# Problem 1.(10 marks) Modular Arithmetic is the basis of many cryptosystems. Integer ring Zm also used modular arithmetic. Answer the given questions for Z13 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

Problem 1.(10 marks) Modular Arithmetic is the basis of many cryptosystems. Integer ring Zm also used modular arithmetic. Answer the given questions for Z13 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

a (1 marks) Calculate value of x when x=17 mode 13.b (1 marks) Calculate value of x when x=15 × 17 mode 13.

c (1 marks) Calculate value of x when x=72 mode 13.d (2 marks) Calculate value of x when 7x=8 mode 13. 1

e (2 marks) How many elements is in Z13 that has Greatest Common Divisor (GCD) 1 with 13? f (3 marks) Calculate inverse of 7 in Z13 and calculate s and t so that GCD(13, 5) = 13s + 7t.

Problem 2.(10 marks) Old cipher can be a good illustration of basic cryptography. This question regarding the Substitution, Shift and Affine cipher.

a  (2 marks) Decrypt the given message encrypted by the Substation cipher and calculate key K. Fubswrjudskb fdq eh d fkdoohqjlqj vxemhfw wr ohduqc exw lw fdq dovr eh yhub

lqwhuhvwlqj dqg uhzduglqje

b  (4 marks) Perform an attack against the given ciphertext using letter frequency count. Recover

the plaintext for the given ciphertext.

Aol Jhlzhy Jpwoly aljoupxbl pz vul vm aol lhysplza huk zptwslza tlaovkz vm lu- jyfwapvu aljoupxbl. Pa’z zptwsf h afwl vm zbizapabapvu jpwoly, p.l., lhjo slaaly vm h npclu alea pz ylwshjlk if h slaaly dpao h mpelk ubtily vm wvzpapvuz kvdu aol hswohila.

c  (4 marks) Encrypt the plaintexts found in question 1 and 2 by Affine chipher using given keys: (a) K =[3,13] for question 1.

(b) K =[7,13] for question 2.

Problem 3.(10 marks) Random number generation is a process often called random number gener- ator (RNG) by which a sequence of numbers is generated which cannot be reasonably predicted better than by random chance. Answer following questions regarding random number.

a (3 marks) What are the main properties of true random number. Give few examples about the sources of true random number.

1This problem is called discrete logarithm and a hard problem

1

b  (3 marks) Pseudo random generator needs a true random number as seed to generate pseudo random number. What are the limitations of pseudo random number.

c  (4 marks) Write a short note on a generator of cryptographic secure pseudo random number.

Problem 4.(10 marks) A block cipher is a method of encrypting data in blocks to produce ciphertext using a cryptographic key and algorithm. DES, Triple DES and AES are three well known and widely acceptable block cipher algorithms. Answer the following questions regarding all three block ciphers:

a (3 marks) DES structure is based on Feistel network. Draw the structure of Feistel network. b (2 marks) What are the main limitation of DES algorithm.

c (2 marks) How triple DES solve the limitations of DES algorithm. d (3 marks) What are the main components of AES algorithm.

Problem 5.(10 marks) A Mode of Operation describes how to repeatedly apply a cipher’s single- block operation to securely transform amounts of data larger than a block. Two well known mode of operation are ECB and CBC mode.

a (2 marks) What are the main limitations of ECB mode operation. b (2 marks) What are the two main ideas of CBC mode operation?

c (2 marks) Draw a block diagram of CBC model operation.d (4 marks) Write a short note on (i) Quantum computer, and (ii) Post-quantum cryptography.

Bonus Question.(10 marks) In cryptography, we used different terminologies and they have their meaning. Write a short not for the given terms in your own words (not more than five sentences)

a (1 marks) Brute-force Attackb (1 marks) Unconditional Security

c (1 marks) Unpredictabilityd (1 marks) Confusion Operation

e (1 marks) Diffusion Operation f (1 marks) Integer Ring

g (1 marks) Prime Fieldh (1 marks) S-Box Operation

i (1 marks) Block Cipherj (1 marks) Modes of Operation 