Show mathematically why C1 - C2 (mod 26) is equal to P1 - P2 (mod 26). Explain in your own words why this is a catastrophic failure for the spies. What information has been leaked to Eve, and how could she use it to potentially recover the original messages if she were to correctly guess just one of them?

Your friend receives the ciphertext numbers. Show the step-by-step process they would use to decrypt it (by subtracting the key and taking the result modulo 26) to recover the original message "HI".

You want to send a secret two-character message, "HI", to your friend. You have both agreed on a secret numerical key, which you will use only once.

• Plaintext Message: HI
• Secret Key (as numbers): [3, 15]

Show the step-by-step process of encrypting and decrypting this message using modular arithmetic (mod 26).

(d)  Suppose that in the case of intercepting BRKKLK we also know that the plaintext is the name of a country. What is the plaintext?

(e)  What important lesson does the BRKKLK example illustrate?

(b)  If a Simple Substitution Cipher is used (with an unknown key) and we intercept the ciphertext OXAO then which of the following four-letter words could be the plaintext: JOHN, SKID, SPAS, LOOT, PLOP, OSLO? 

(c)  If a Simple Substitution Cipher is used (with an unknown key) and we intercept the ciphertext BRKKLK then what do we know about the plaintext? 

The Simple Substitution Cipher is a major improvement on the Caesar Cipher. (a)  In what different ways is it an improvement?

29⁷ mod 32

2⁵ mod 25