Discrete Math - Day 33

Today we took our first look at cryptography. This was a shortened day and this lesson fit in perfectly.

Class started with a simple example to illustrate the ideas of plaintext, ciphers, and ciphertext. The class was asked how they might make the text "secret." After brief discussions in their groups most were unsure how they would proceed. A couple of students suggested shifting letters.

This idea is exactly what Julius Caesar used. The traditional Caesarean cipher shifts letters by three characters. A student asked how a translation would occur as they were not clear on the process. I used the plaintext "zoo time" to illustrate how the cipher would convert this to the ciphertext "crr wlph."

I asked students to convert the plaintext "You too Brutus" to ciphertext. This allowed students to get a better sense of the process of ciphering. The result was "Brx wrr Euxwxv." Students seemed comfortable with the idea, so we were ready to move on to the next challenge.

Students picked their own value to shift and then used their cipher on the plaintext "I conquer." They swapped messages with a classmate and tried to determine the value used to shift letters. Even though the students knew what the plaintext message was, there were still some challenges in determining the value used in the shift.

Next, I had students create a brief plaintext message of 10 characters or less. They then used their own shift cipher. Students swapped ciphertext and tried to determine the plaintext message. Students found this extremely challenging. Most students made little or no headway in determining the plaintext. A couple of students wondered if they could look at the most frequently used characters.
I displayed the relative frequency of occurrence for different letters in English. I told students they could try assuming the most frequently used letter was e and see if this led to a translation. If not, assume the most frequently used letter was a t and try again. They could continue on in this manner to see if they could de-cipher the ciphertext.

This process still had little success. I pointed out that computers can easily be programmed to de-cipher ciphertext using shift ciphers. I used an Caesarean shift applet and a student's ciphertext that was school appropriate to show how easily a computer could de-cipher ciphertext.

Shift ciphers are not secure. A more sophisticated and secure method for creating ciphers is needed. As it turns out, prime numbers play a key role in the development of secure ciphers.

Class concluded with students summarizing their thoughts about ciphers and questions they have about secure ciphers.
Visit the class summary for a student's perspective and to view the lesson slides.