# RSA Algorithm mathematical explanation

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• RSA algorithm is used majorly for key exchange for symmetric encryption

• The RSA cryptosystem is the most widely-used public key cryptography algorithm in the world.

• It can be used to encrypt a message without the need to exchange a secret key separately.

• The RSA algorithm can be used for both public key encryption and digital signatures.

• Its security is based on the difficulty of factoring large integers.

1. See, public key has 2 parts ‘e’ and ‘n’.

2. No we need ‘n’.

1. n=P * Q.

2. Here, P and Q and 2 Prime numbers.

3. Lets assume p=53, and Q =59

4. n=P * Q

5. n=53 * 59=3127

3. Now we need ‘e’.

1. ‘e’ is such that it should be and integer and,

2. 1 < e < Ф(n)

3. Here Ф(n)=(P-1)*(Q-1)

4. So, Ф(n)=(53–1)*(59–1)

5. Ф(n)= 3016

6. Let’s take any value of e in the range of 1 and 3016

7. let’s assume e=3

4. Now our public Key is (n,e)=(n=3127, e=3)

5. Now, we need to work for private key ‘d’.

1. d=[(K* Ф(n))+1]/e

2. for any integer k=2,

→ d =[(2*3016)+1]/3

→ d=6033/3=2011

3. So your private key is d==2011.

6. Now Let’ Encrypt any Message.

→Suppose A=1 and Z=26.

→So HI = 89.

7.Now to encrypt our message we need public key(n=3127, e=3)

8. Encrypted message c = 89^e mod n =1394.

7. Now, Decrypting:

→Our encrypted message is 1394.

→Decrypted Data = c^d mod n=89.