15C: Caesar's JVTIVK JRCRU IVTZGV

Exercises 15A, 15B, and 15C can be completed in any order.

Cryptography is the art and science of hiding the meaning of information, in a way that only some people can see it. In this lesson we will introduce one of the simplest cryptographic methods, the Caesar cipher (also known as a shift cipher), and you will to write a program to break it. You will design every aspect of your solution (unlike lesson 15A, which we broke into sub-parts).

The Caesar cipher works by replacing each letter of the alphabet by another letter. To be precise, whenever you want to encode some text you need to pick a shift value S, which is a number between 0 and 25. Then, you replace each letter in the text with the letter which is S positions later in the alphabet, circling around to the start after you reach Z at the end of the alphabet.

Example

Suppose we want to encode the secret message

JOIN ME AT EIGHT BY THE ZOO

using the shift value S=2. The encryption rule says each letter is replaced by the one which occurs 2 positions later in the alphabet. For example, since the alphabet is ABCDEFGHIJKL..., the first letter J will be replaced by the letter L. Continuing, the O is replaced by Q, the I is replaced by K, et cetera. To encode the letter Y, we have to circle back to the start: after Y comes Z, then A, so Y is replaced by A. Likewise Z is replaced by B. So the encoded version of our secret message is:

LQKP OG CV GKIJV DA VJG BQQ

If some spy saw this message, it would not be obvious to them what your message was about.

Short Answer Exercise: Spy Coder

What is the result of encrypting SPY CODER with the shift value S=5? (Use uppercase.) You can write a program in the console to compute this, if you like — it might be useful later.
Your answer:

Correct!

Decoding

Once your friend gets the message, if they know the secret shift value S then it is easy for them to decipher the message: each letter is replaced by the one appearing S places earlier in the alphabet. For example, they would look at L, step back two positions in the alphabet and find J, which they now know is the first letter of your secret message. Again, they have to treat the alphabet as cyclic, assuming that Z comes before A.

Short Answer Exercise: Spy Decoder

If the encrypted message is HUD, and the shift value is S=6, what was the original message? (Use uppercase.)
Your answer:

Correct!

Working for the Bad Guys

You have been hired by a spy to help decipher a message encrypted using a shift cipher. Sadly, the spy does not know the value of S. You will use statistics to write a program that has a good chance to automatically find the right value of S.

Our method will be to use letter frequency analysis. In English, some letters are very common (like E) and some are very rare (like J). Here is a table of frequencies, based on counting the occurrences in a large body of text.

For example, the frequency of L is .0402=4.02%, which means that in average English text, 4.02% of the letters are L.

Call the goodness of a letter its value in the chart above. Then for our statistical method, define the goodness of a sentence to equal the sum of the goodness of each of its letters. For example the goodness of the string GOOD is

goodness("GOOD") = .0202 + .0751 + .0751 + .0425 = .2129

Now, the idea behind frequency analysis is that strings with higher goodness are more likely to represent English text. For example, if the spy sees the encoded message "JRRG", this might either represent the original text "GOOD" with shift S=3, or "IQQF" with S=1. But the goodness of "IQQF" is

goodness("IQQF") = .0697 + .0009 + .0009 + .0223 = .0938

and since .0938 < .2129, your program should conclude that GOOD is more likely to be the correct message than IQQF.

Measuring strings by goodness is not perfect. Say the secret message was JAZZ and the secret shift value was S=10, so the encoded text was TKJJ. When you input TKJJ to your solver it will try S=10, giving JAZZ as a possibility with goodness .0846, but the best is S=5, giving OFEE with goodness of .3514 as the program's best guess for the secret message. Your program should go ahead and output the non-English word OFEE. (Frequency analysis of this type works better if the secret message is longer.)

Coding Exercise: Auto-Decryption

Write a program which does the following. First, it should read one line of input, which is the encoded message, and will consist of capital letters and spaces. Your program must try decoding the message with all 26 possible values of the shift S; out of these 26 possible original messages, print the one which has the highest goodness.

For your convenience, we will pre-define the variable letterGoodness for you, a list of length 26 which equals the values in the frequency table above,

letterGoodness = [.0817,.0149,.0278,.0425,.1270,.0223,.0202,...

Click here for some general advice.

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print(letterGoodness) #testing... it's predefined # delete this comment and enter your code here

Enter testing statements like print(myfunction("test argument")) below.

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The solution above is called a brute force solution since you just tried all 26 possibilities. You could have even done this by hand if you needed to. Modern cryptographic protocols, on the other hand, are designed so that brute force solutions fail, even if you use very fast computers to try all possibilities.

If you wanted to make this auto-decoding system more accurate, you could pay attention to other English statistics, such as "what letters are likely to occur in pairs next to each other? what letters are likely to start a word?" This is also useful for more general substitution ciphers, where letters replace letters but not in a cyclic way. For such ciphers brute force does not work since there are 26*25*...*1 possible substitution ciphers, which equals approximately 4 * 10²⁶ (four hundred septillion).

You have completed this lesson! Feel free to send your friends an encrypted message to celebrate.