Public Key Cryptography - Understanding the Process and Generating Private and Public Key

Running head: PUBLIC KEY CRYPTOGRAPHY
PUBLIC KEY CRYPTOGRAPHY
Name of the Student
Name of the University
Author Note

PUBLIC KEY CRYPTOGRAPHY1
Introduction:
In this assignment the objective is understand the process of public key cryptography and
then to generate private and public key using a certain algorithm from the given Student ID.
The ideas of public and private keys and their relationship are presented in this assignment.
Furthermore, as an example private and public keys are generated by prime number
calculation method using Wolframalpha online software. The concepts of cryptography are
presented by answering the questions in part 1 and part 2.
1. i) Public key cryptography alternatively known as asymmetric cryptography is a particular
encryption scheme that basically uses two related but not same keys known as public key and
private key. This is different from the symmetric key algorithm where the same key is used
for encryption and decryption. In public key cryptography the public key is used for
encryption and private key is used for decryption.
A private key is an extremely large number generated by using some algorithm like wallet
program or any other algorithm. The private key is not shared and only kept to the generator
itself. The public key is generated from the private key and shared to all other persons in the
network. The public key is basically the (x,y) co-ordinates in the elliptical curve after
multiplication of private key number of times. At first a publicly known point is assumed as
initial point G in the curve and then the co-ordinate of the point G is multiplied with the co-
ordinate itself by private key number of time. This reflects the point G on the elliptical curve
by x axis and this process continues until multiplication count reaches private key number of
times. This process is known as pin-ball effect. Hence, after completion of the pin-ball effect
the public key is generated.

PUBLIC KEY CRYPTOGRAPHY2
ii) Now, the RSA algorithm is used here for generation of public and private key. Here, we
have assumed that p and q number are small for ease of calculation but practically p and q are
very large prime numbers.
Let, p = 7 and q = 13.
Hence, pq = 91.
Now, a number e =5 is chosen as e is co-prime to (p-1)(q-1) = 6*12 = 72
Hence, the public key is formed by the pair of numbers (n, e) = (91, 5) an this is made
available to anyone in the network.
Now, inputting the values of p,q and e in the extended Euclidean algorithm outputs the
number d = 29.
Hence, the private key obtained from public key and the extended Euclidean algorithm is (91,
29).
iii) Now, the given student ID is 1464130.
As the bit length of the number is 7 so the number of bits that the number can be represented
with is 7.
iv) Let in the student ID the a particular number of binary bits is 30.
Hence, the number of prime numbers available in that range is found using Wolframalpha as
given below.

PUBLIC KEY CRYPTOGRAPHY3
Hence, the number of prime numbers less than 30 is 10. The 10 primes are
2,3,5,7,11,13,17,19,23,29.
v) Now, by using the nextprime function in wolframalpha the two prime numbers are
generated in the following process. The nextprime of 1464130 is found and checked whether
nextprime(1464130) – 1 is divisible by 4 or not and then next prime number is found and the
process is repeated until (prime number – 1) is divisible by 4.
In this way the found prime number is of the form (4x1 + 1), where x1 is any integer.