SURYA GROUP OF INSTITUTIONS SCHOOL OF

ENGINEERING & TECHNOLOGY

Vikiravandi – Villupuram
Department of Computer Science & Engineering (SC)

Subject Code / Subject: CB3491 CRPTOGRAPHY AND CYBER SECURITY

Year / Semester : II/ IV Unit Number : 3

1 What is public key cryptography?

Public key cryptography (or asymmetric cryptography) is an encryption scheme that
uses two mathematically related, but not identical keys – a public key and a private key.
Each key performs a unique function. The public key is used to encrypt and the private
key is used to decrypt.
2 What is the difference between symmetric key cryptography and public key
cryptography?
Symmetric Key Cryptography Public Key Cryptography
Involves only one key (a secret key) to encrypt Uses a pair of keys – a public key and a
and decrypt the information private key
Speed of encryption (decryption is very fast) Slow

Only (public key) either one of the keys

Both parties should know the key
is known by the two parties
E.g.: DES, AES E.g.: RSA, ECC
3 Consider the RSA encryption method with p  11 and q  17 as the two primes. Find n
and (n) .
1) p  11 and q  17 , so n  pq 1711 187
2) (n)  ( p 1)(q 1) 1610 160

4 Define primitive root

A primitive root of a prime number p is one whose power modulo 0 p generate all the
integers from 1 to p-1. That is if a is a primitive root of the prime number p, then the
numbers
a mod p, a2 mod p,……, ap-1 mod p
are distinct and consists of the integers from 1 through p-1 in some presentation.
5 State Fermat‟s theorem
Fermat‟s theorem states the following: If p is prime and a is a positive integer not
divisible by p, then
ap-1  1(mod p)
6 Check whether
1) 2 is a primitive root of mod 5 &
2) 4 is a primitive root of mod 5

21 mod 5  2 mod 5  2
22 mod 5  4 mod 5  4
1)
23 mod 5  8 mod 5  3
24 mod 5  16 mod 5  1

So, 2 is primitive root of mod 5

41 mod 5  4 mod 5  4
42 mod 5  16 mod 5  1
2)
43 mod 5  64 mod 5  4
44 mod 5  256 mod 5  0
So, 4 is not a primitive root of mod 5
7 Name any 2 methods for testing prime numbers.

 Miller – Rabin test

 Fermat primality test
 Solovay – Strassen primality test
 Frobenius primality test
8 Define Euler‟s totient function.
Euler‟s totient function, written (n), and defined as the number if positive integers less
than n and relatively prime to n.
By convention (1)  1
9 State Euler‟s theorem.
Euler‟s theorem states that for energy a and n that are relatively prime:
a(n)  1(mod n)

10 Determine (37) and (35)

To determine (37) : Because 37 is prime, all the positive integers from 1 through 36 are
relatively prime to 37. So, (37)  36 .
To determine (35) :List all the positive integers less than 35 that are relatively prime to
it. (i.e) 1,2,3,4,6,8,9,11,12,13,16,17,18,19,22,23,24,26,27,29,31,32,33,34. There are 24
numbers on the list. So, (35)  24
11 State alternative form of Fermat‟s theorem with example.
The alternative form of Fermat‟s theorem: If p is prime and a is a positive integer than
ap  a(mod p)
12 List the 6 ingredients of public key encryption.
 Plaintext
 Encryption algorithm
 Public key
 Private key
 Cipher text
 Decryption algorithm
13 Perform encryption for the plaintext M=88 using the RSA algorithm.
P=17, q=11 and public component e=7

i. p=17, q=11
ii. Calculate n=p*q = 17*11 =187
iii. Calculate (n) = (p-1)(q-1) = 16*10=160
iv. Select e=7
v. Determine d such that de  1(mod 60). The correct value of d is 23
Public key (7,187) and private key (23,187)
Encryption: 887 mod 187 = 11
14 Perform encryption and decryption using the RSA algorithm for the following.
P=7, q=11, e=17 and M=8
i. p=7, q=11
ii. Calculate n=p*q = 7*11 =77
iii. Calculate (n) = (p-1)(q-1) = 6*10=60
iv. Select e=17
v. Determine d such that de  1(mod 60). The correct value of d is 53
Public key (17,77) and private key (53,77)
Encryption: 817 mod 77 = 56
Decryption: 5653 mod 77 = 8
15 List the 5 possible approaches to attacking the RSA algorithm
 Brute force
 Mathematical attacks
 Timing attacks
 Hardware fault-based attack
 Chosen ciphertext attacks
16 Define discrete logarithm
For any integers b and a primitive r not a of prime number p, we can find a unique
exponent I such that
b  a i (mod p) where 0 ≤ I ≤ (p-1)
The exponent I is referred to as the discrete logarithm of b for the base a, mod p.
17 What is the principal attraction of ECC, compared to RSA? (Dec 2021)
The principal attraction of ECC, compared to RSA, is that it appears to offer equal
security for a far smaller key size, thereby reducing processing overhead. But the
confidence level of ECC is not yet as high as than in RSA. (i.e. ECC is fundamentally
more difficult to explain than either RSA or Diffie-Hellman
18 What is an ellipse curve?
Elliptic curve is a plane algebraic curve defined by an equation of the form y 2=x3+ax+b
which is non-singular. Formally, an elliptic curve is a smooth, projective, algebraic curve
of genius arc, on which there is a specified point O.
19 Give the significance of key control
Hierarchies of Key Distributor Center (KDC) requires for large networks. A single KDC
may be responsible for a small number of users since it shares the master keys of all the
entities attached to it. If two entities in different domains want to communicate, local
KDCs communicate through a global KDC.
20 Why is asymmetric cryptography bad for huge data? Specify the reasons (May 18)
 Asymmetric cryptography takes more time
 Key management is difficult
 Slower encryption speed due to long keys
21 Give the applications of the public key crypto system
 To provide confidentiality (a message that a sender encrypts using the recipients
public key can be decrypted only by the recipient‟s private key
 Digital signature (used for sender authentication)
 Further applications built on this include: digital cash, password authenticated
key agreement, time-stamping services, non-repudiation protocol, etc.

PART-B
1 State Chinese Remainder Theorem and find X for the given set of congruent equations
using CRT
X = 2 (mod 3)
X = 3 (mod 5)
X = 2 (mod 7)
2 State and prove Fermat‟s theorem.
3 Explain RSA algorithm, perform encryption and decryption to the system with
p=7, q=11, e=17, M=8
4 Users Alice and Bob use the Diffie-Hellman key exchange technique with a common
prime q=83 and a primitive root α=5.
i. If Alice has a private key XA=6, what is Alice‟s public key YA?

ii. If Bob has a private key XB=10, what is Bob‟s public key YB?
iii. What is the shared secret key?
5 State Chinese Remainder Theorem and find X for the given set of congruent equations
using CRT (Dec 2020)
X=1 (mod 5) X=2 (mod 7) X=3 (mod 9) X=4 (mod 11)
6 Explain Diffie-Hellman key exchange algorithm in detail
7 Perform encryption and decryption using RSA algorithm for p=17, q=11, e=7 and u=88
8 Why ECC is better than RSA? However, why is it not widely used? Defend it.
9 State and prove Chinese remainder theorem. What are the last two digits of 4919?
10 (ii) With a neat sketch explain the Elliptic curve cryptography with an example (8)
(ii) Alice and Bob use the Diffie – Hellman key exchange technique with a common
prime number 11 and a primitive root of 2. If Alice and Bob choose distinct secret
integers as 9 and 3, respectively, then compute the shared secret key. (5) (Dec 2020)
11 Describe RSA algorithm & Perform encryption and decryption using RSA algorithm for
the following: p=7, q=11, e=7, M=9
12 Explain briefly about Diffie-Hellman key exchange algorithm with its merits and
demerits.
13 Explain public key cryptography and when it is preferred?
14 Explain the working of RSA and chose an application of your choice for RSA and explain
how encryption and decryption is carried out?
15 Prove Fermat‟s theorem and Euler‟s theorem
16 Demonstrate encryption and decryption for the RSA algorithm:
Parameters – p=3, q=11, e=7, d=?, M=5