1. Polyalphabetic Substitution Ciphers

2. First Steps Towards Complexity  If one alphabet is good, then two alphabets must be better!  By doubling the number of cipher alphabets, the frequency of each coded letter is (approximately) halved.

3. Practice Encrypting/Decrypting  Choose 2-3 sentences from The Code Book as your plaintext message.  Then, using the cipher key below, encrypt the message by alternating between the first and second cipher alphabet.  Exchange ciphers and decipher your partner’s message.

4. Decryption with Multiple Alphabets  Does frequency analysis fail us?  Some of our tricks don’t help us out anymore:  Double letters will be encrypted in two different ways.  This is true throughout—an e will be encrypted as both a K and an F in different places in the ciphertext.

5. Frequency Analysis—Not All is Lost!  Assuming our plaintext message is in English, we can predict that “e” will be the most common letter in the text. Although there are two options that will encrypt the letter “e”, these two letters “K” and “F” should appear most frequently in the ciphertext.  Given a long enough message, we can still use frequency analysis to decrypt it.

6. So…Add More Alphabets  This seems like the obvious solution, right?  It’s much harder to break  Ergo, it’s more secure So, why didn’t Vigenère’s 26-alphabet cipher catch on?

7. Working By Hand  Vigenère’s Cipher is a huge leap forward in terms of security (it took another couple centuries to break), but it’s not easy to encrypt and decrypt messages quickly when you are working by hand.  Instead, the more commonly used cipher of the time was Homophonic Substitution Cipher.

8. Homophonic Substitution Cipher  In this cipher each plaintext letter has some number of symbols that represent it in the ciphertext; the number of different symbols for each letter is based on the proportions of the letter frequency.  For example, the letter “a” appears ~8% of the time in English, so there will be 8 different symbols that can substitute for the letter “a”.  Each letter in the ciphertext should appear ~1% of the time.

9. What Type of Cipher is This?  Is the Homophonic Substitution Cipher a:  Mono-alphabetic substitution cipher?  Poly-alphabetic substitution cipher?  Something different?

10. One-to-One Functions  Definition of a Function  For every input (x-value), there is exactly one output (y-value).  Vertical Line Test  If BOTH the original function and its inverse are functions, we say this function is “one-to-one.”

11. Some Functions From Math Which of these are “one-to-one” functions?  Linear  Exponential  Quadratic  Logarithms  Inverse Power  Trigonometric

12. Homophonic Substitution Cipher as a Function  The encryption of a message using a Homophonic Substitution Cipher is NOT a function (more than one output for each input)  The decryption of a message using a Homophonic Substitution Cipher IS a function (only one output for each input)  Therefore this function is not “one-to-one”