Principle behind RSA Private Key / Public Key

 Step 01:

Select two large random prime numbers. Let’s take some small numbers p=11 and q=5 to understand the concept

Calculate RSA modulus by multiplying together

n = p.q
  = 11.5
  = 55


 Step 02:

Calculate the totient of RSA modules

@(n) = (p-1)(q-1)   
     = 10.4
     = 40

Step 03:

Select a number that is relatively prime to the totient 1 < e < @(n)

3, 7 , 9 , 11 , 13 ,….

e = 7

Step 04:

Find the modular inverse of e with respect to @(n)

call d it will be a part of private key

e*d mod @(n) = 1

7* d mod 40 = 1

solve for d with extended Euclidean algorithm

part 01: Euclidean algorithm

40x + 7y = 1
40 =  5(7) + 5
7 =  1(5) + 2
5 =  2(2) + 1

part 02: Back substitution

1 = 5 -2(2)
1 = 5 -2(7 -5)
1= 3(5) -2(7)
1= 3(40 - 5.7) -2(7)
1= 3(40) - 17(7)

Since the number in front of 7 is negative

let's take d = 40 - 17 = 23

Private key (23, 55)

Public key (7, 55)