Cryptography and Matrices

  Cryptography or the ability to code and distribute data or messages so only the party receiving the information is able to read it.  One of the most famous cryptography machines is the Enigma machine from World War II. What is interesting is that Enigma is not a machine, it is the name of a manufacturer who produced a series of machines before and during World War II.  In fact, the Enigma machines inspired other companies to produce their own cipher machines.

This situation provides math teachers with a perfect activity to introduce students of all ages to cipher machines and matrices as used in real life.  Students relate better when given an example of when they might use it.  Yes, I know they won't be working for the government but they could share messages so no one else would understand them.

Now the process involved in encrypting a machine is rather complex.  It involves several steps described as follows.
1.  Write out the message.

2. Create an alpha- numeric substitution so each letter has a numerical substitution.  Save 0 for spaces and use 1 to 26 for the letters but make sure its not sequential i.e. A=4, B = 5, C = 6 and so on.  Mix them up so its more like A=4, B = 9, C = 5, D = 21, etc. 

3. Choose an invertible square n x n matrix to help encode the message.  Then take your message and translate it into numerical form using the substitution chosen in step 2.

4.  Convert the message into an m x n matrix.  n representing the number of columns comes from step 2 and m represents the number of rows which changes depending on how many letters are in the message.

5. Now you multiply the matrix from step 3 and the one from step 4 to create the encoded message you are trying to keep secret.  You take the resulting matrix, write the letters out in a string and send it.

6.  The person receiving the message then takes the numerical values, places them in the same size matrix as step 4 and multiplies by the inverse of the n x n matrix from step 3 to get the m x n matrix in step 4. 

7.  Using the original substitution key, you translate the numerical values from the m x n matrix to find the original message.

If that gets a bit confusing, check this example to see how it works.  Engage NY has a nice little lesson on this topic that is well organized and set up to run stations so students rotate from one to another as they work through the activities.  There is lots of detailed examples of the math in using matrix along with the answers.

You do not have to wait until students reach advanced class to use this activity.  There are now matrix calculators available for both online and mobile devices so no more having to sit there with fingers as you multiply each term by each term and then spend hours trying to figure out where you messed up.  These apps allow you to integrate activities like this sooner so you can give students a taste of real life.

This topic is really appropriate for a cross curricular unit on World War II because you have this for the math component, social studies can discuss the enigma machine and other ways they communicated or the various intelligence agencies,  Language arts can focus on reading a fictional tale and science could build their own enigma machine.

Let me know what you think, I'd love to hear.