# The Theorem that Won the War: Notes on Implementation

Author(s):
Jeff Suzuki (Brooklyn College)

This activity is divided into three parts.

Part 1 (Enigma on Paper) requires no specialized background. After some introduction to the nature of the Enigma rotors, Activity 1.1 could be completed during class. In the next class, setting up a sample Enigma encryption could be done as a class activity, then students should be able to finish Activity 1.2 in the remaining time. The last question (the prohibition on self-encryption) will probably require some guidance, but the biggest challenge is likely to be getting students to understand the directional nature of the “wiring”.

Part 2 (The Algebra of Enigma) is a little more technical, but can be explored with any student with an interest in mathematics. After introducing the details of the Enigma system, Activity 2.1 could be finished at the end of a class period (it’s mostly setup for utilizing cycle notation). Activity 2.2 would make for a good full class period activity. Activities 2.3 and 2.4 would be good homework assignments. In a first course in abstract algebra, the activities in Part 2 would form a good set of homework problems (after the working details of the Enigma system are presented), as they introduce several key concepts such as: permutation groups; cycle notation; conjugates; and cycle decomposition.

Part 3 (Breaking Enigma) is suitable for later in a first course on abstract algebra. Activity 3.1 should probably be done in class, as several of the proofs rely on some careful reasoning; more importantly, there are several key observations that need to be made about the products that students working at home might easily miss. Part of the next class period could be used to prove Rejewski’s Theorem and the Enigma Theorem, at which point Activity 3.2 and 3.3 become reasonable homework assignments.

Figure 2. First page of the 1980 article in which Rejewski first published his theorem.
The European Digital Mathematics Library.

Jeff Suzuki (Brooklyn College), "The Theorem that Won the War: Notes on Implementation," Convergence (October 2023)