The Theorem that Won the War: Part 2 – The Algebra of Enigma

In an invited address at a 1941 meeting of the American Mathematical Society, the American mathematician Abraham Adrian Albert (1905–1972) famously noted that

It would not be an exaggeration to state that abstract cryptography is identical with abstract mathematics.

In fact, around the same time that Hardy reveled in the “uselessness” of higher mathematics, Albert and Olive Clio Hazlett (1890–1974), both of whom published pioneering work in abstract algebra, began working with the U.S. Army on the problem of breaking coded military communications.

  
Figure 13.  American mathematicians Abraham Adrian Albert and Olive Clio Hazlett.
Left: Convergence Portrait Gallery; Right: Wikimedia Commons, Creative Commons Attribution-Share Alike 4.0.

At this point, we'll thus need to introduce some abstract algebra. If you haven't taken abstract algebra yet, don't worry: we'll introduce the important ideas as we go along.

We will again do this in several parts, pausing to look at some specific activities along the way.

• Part 2.1: Compositions
  • Activities for Part 2.1
• Part 2.2: Conjugates
  • Activities for Part 2.2
• Part 2.3: Cycle Notation
  • Activities for Part 2.3
• Part 2.4: The Message Key
  • Activities for Part 2.4