RSA Cipher Challenge - Cryptography Tutorial

1) Discover the working of the RSA Cipher. 

2) Understand how to perform MOD exponentiation.

3) Research the Internet to find 
out how about the inventors of RSA. What is RSA named after? When was RSA invented.

You are now leaving the zone of easily breakable, historical ciphers. You are about to enter modern FBI and NSA security-level encryption. Here, the inconceivable becomes reality: Two keys are involved: a publicly known encoding key e and a different secret decoding key d known only to the recipient. The keys are related such that the decoding key d is used to decode the e-encoded message. However, they are not the same keys, and one can not be derived from the other. 

Figure 1 (source: www.PGPi.com):   Public-Key Encryption Scheme. The encoding key is used to encode the plain text, the decoding key is used to decode the cipher text. Knowing the encoding key does not yield the decoding key

Mathematicians call such miraculous 2-key ciphers "asymmetric ciphers". By contrast, all previously studied ciphers such as the Caesar, the One Time Pad, the Linear Cipher or the Vigenere Cipher are "symmetric" Ciphers since the knowledge of the encoding key quickly produces the decoding key.  

Mathematically, the difference between asymmetric and symmetric ciphers does not seem dramatic at first sight: The previously studied ciphers are based on MOD-addition and MOD-multiplication to encode plain letters. The RSA Cipher, however, is based on MOD-exponentiation. Read here to learn how to perform MOD-exponentiation. Below, you can perform the RSA encryption and decryption. Your challenge is to try to understand how it works by answering the following 3 questions:   

Exercise 1: 
En- and decode "safe", verify the computations mod 33 below. The modulus (commonly huge numbers consisting of at least 100-digit numbers) is crucial for the security of the RSA Cipher. We will explore this on the next page. 

Exercise 2:  
Using the same exponents (=keys) as in exercise 1, en- and decode "cat" and "cryptography" using paper and pencil and the provided calculator for the MOD exponentiations. Afterwards, verify your answers below. Check below.

Exercise 3:  
What number(s) must the sender have knowledge of in order to encode the message properly? How about the recipient?   Encoding key (e,n) = (3,33), decoding key (7,33).