RSA Principle 2

Generate cipher code

Step 1 Choose two coprime number randomly

53 and 271

The bigger these number is the more difficulty to break it.

Step 2 First number multiplied by second number

$$n = 53 \times 271 = 14363 = 11100000011011$$

totle 14bits
In daily application use 1024 bits[128bytes] usually.
In some important application use 2048 bits[256bytes] usually.

Step 3 Calculate n’s Euler function

$$\phi(n) = (p-1)(q-1) = 52 \times 270 = 14040$$

Step 4 Choose a number e

$$1 < e < \varphi(n)$$

and the number e has coprime relation with

$$\varphi(n)$$

Choose between (1, 14040), suppose 97 as e;

Step 5 Calculate mod inverse element

Calculate the mod inverse element d for e with

$$\varphi(n)$$ $$ed \equiv 1 \quad (mod \quad \phi(n))$$

->

$$ed -1 = k\phi(n)$$

->

$$ex + \phi(n)y = 1$$

->

$$97x + 14040y = 1$$

Based on Extended Euclidean algorithm
x = -4487, y = 31 -> (-4487, 31) is a solution pair
positive d
(-4487, 31) -> (9553, -66)

Step 6 Packaging

Pack n and e into public key,n and d into private key
n = 14040 e = 97 d = 9953 and k = -66

So , (14363, 97) is public key, (14363, 9553) is private key.