# RSA Encryption

🎯**What is RSA encryption?**

👉RSA (Rivest-Shamir-Adleman) is an algorithm used for secure data transmission. It is an asymmetric encryption that is widely used for secure data transmission.  
👉In RSA, each person has a pair of keys: a public key and a private key.  
👉The public key can be shared with anyone, while the private key must be kept secret.  
👉When a message is sent using RSA, the sender encrypts the message using the recipient's public key.  
👉The recipient then decrypts the message using their private key.  
👉This ensures that only the intended recipient can read the message, as only they have the private key needed to decrypt it.

![](https://cdn.hashnode.com/res/hashnode/image/upload/v1672664604949/c29cdad8-ce93-420d-9fab-7b1747a9f4d7.jpeg align="center")

📌**A simple demonstration of how the RSA algorithm works**

🌲Suppose Swetha wants to send a secure message to Vijay.  
🌲Vijay generates a pair of keys using the RSA algorithm and sends his public key to Swetha.  
🌲Swetha uses Vijay's public key to encrypt her message and sends the encrypted message (ciphertext) to Vijay.  
🌲Vijay then uses his private key to decrypt the ciphertext and read the original message (plaintext).  
   
📌**Key generation:**

🌲Vijay selects two prime numbers, p, and q, and calculates n = p \**q and in this example, p = 5 and q = 11, so n = 5* \*11 = 55.  
🌲Vijay calculates a value called the totient, denoted as φ(n), which is the number of positive integers less than n that are relatively prime to n. In this case, φ(55) = (p - 1) *(q - 1) = 4* \*10 = 40.  
🌲The factors of 40 are 2\*2\*2\*5.  
🌲Vijay selects a public key, e, that is relatively prime to φ(n) and none of the factors of 2 and 5. In this example, let's say e =7.  
🌲Vijay calculates his private key, d, such that e *d ≡ 1 (mod φ(n)) or we can derive d= (1+ x* φ(n))/e while x can be 0,1,2, 3, etc.  
After some calculations using Excel, d is calculated as d = (1+4\*40)/7 = 23.

![](https://cdn.hashnode.com/res/hashnode/image/upload/v1672664730999/1b2313c8-cf7c-4620-ad8e-e9fc6e264d04.jpeg align="center")

🌲We now have n=55, e=7, d=23  
🌲Vijay's public key is the pair (e, n), and his private key is the pair (d, n). Vijay sends his public key to Swetha.  
   
📌**Encryption:**

🌲Swetha wants to send the message "HELLO" to Vijay.  
🌲She converts the message using a predetermined scheme (e.g., A=1, B=2, till Z=26.).  
🌲Swetha calculates the ciphertext, c, using the formula c ≡ m^e (mod n), where m is the plain text.  
🌲In this case, c is calculated as c = 8^7 (mod 55) 5^7 (mod 55) 12^7 (mod 55) 12^7 (mod 55) 15^7 (mod 55).  
🌲c= 2 25 23 23 5. Swetha sends the ciphertext to Vijay.  
   
📌**Decryption:**

🌲Vijay receives Swetha's ciphertext, 2 25 23 23 5.  
🌲Vijay calculates the plaintext, m, using the formula m = c^d (mod n).  
🌲In this case, m = 2^23 (mod 55) 25^23 (mod 55)23^23 (mod 55)23^23 (mod 55)5^23 (mod 55) = 8 5 12 12 15.  
🌲Vijay converts the number back to the message "HELLO" using the predetermined scheme.   
👉The link for the calculation is a [calculator](https://www.calculator.net/big-number-calculator.html?cx=128&cy=437&cp=20&co=mod).

🎯**Below are the screenshots of the Python code used to create the RSA public, private, encryption, and decryption algorithms.**

👉Please refer to the link to generate RSA keys using python [RSA Keygen](https://pycryptodome.readthedocs.io/en/latest/src/examples.html#generate-an-rsa-key)

🎯**The Public and Private key Generation**

![](https://cdn.hashnode.com/res/hashnode/image/upload/v1673121979707/02a585bd-cd78-4e03-8f24-533406e729c4.jpeg align="center")

🎯**Encryption Algorithm**

![](https://cdn.hashnode.com/res/hashnode/image/upload/v1673121946539/09cc06e0-169b-4dc0-a44e-7875893cfea3.jpeg align="center")

🎯**Decryption Algorithm**

![](https://cdn.hashnode.com/res/hashnode/image/upload/v1673122113282/626bc72e-67ff-4eaf-b538-4432d9bfd388.jpeg align="center")

🌲**Strengths**  
👉When compared to symmetric encryption, there is no need to exchange the keys ahead of time.  
👉"Non-repudiation" is achieved because data cannot be changed during communication.  
👉It's a one-way function, so knowing one prime key won't get you the other primes.  
   
🌲**Weakness**  
👉The difficulty of generating keys.  
👉The RSA algorithm is relatively slow when compared to symmetric algorithms.

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