Caesar encryption by alphabet shifts. We start by choosing a fixed integer, such as 7.

​

To encipher a message, each letter is replaced by the letter 7 places down the alphabet. To encrypt

​

T,U,V, W, X, Y, or Z, we return to the beginning of the alphabet. For instance, the message “HELLO”

​

gets transformed to “OLSSV''. So in this exemple :

 

H = 7 7 + 7 = 14 = O

E = 4 4 + 7 = 11 = L

L = 11 11 + 7 = 18 = S

L = 11 11 + 7 = 18 = S

O = 14 14 + 7 = 21 = V

 

The mathematical concept here is addition in a cyclic group. Schematic wheels provide a vivid way to

 

illustrate this simple but important idea.

 

Perhaps the children can figure out a way to break the cipher on their own; otherwise,

 

frequency analysis can be suggested to them. That is, we first ask them to guess which letter appears

 

most frequently in English (most will say either `e' or `a'). Then we have each student choose

 

a paragraph in a book, count the a's, e's, etc., and in this way decide whose conjecture is correct.

 

The students might want to investigate how long a text one needs in order to have

 

reasonable confidence that `e' occurs the most often. What strategies could one use for a short text to

 

which the assumption about the frequency of `e' does not apply? What are the second and third

 

most frequently occurring letters?

 

​