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The Theorem that Won the War: Part 1 – Enigma on Paper

Jeff Suzuki (Brooklyn College)


In 1923, the firm of German engineer Arthur Scherbius (1878–1929) began selling a machine that could encrypt messages automatically. Marketed as Enigma, it was soon adopted (and adapted) by the German military, and by the outbreak of World War II, it was used for almost all important German military communications.

To understand the mathematics behind Enigma, it helps to have an Enigma machine. Unfortunately, most of the remaining machines are in museums, whose curators don’t take too kindly to handling their exhibits, so we’ll have to make do with a mockup. Online Enigma emulators are available, but to understand the mathematical details, we’ll set up a simple “paper” Enigma to analyze key points. We will do this in two parts, pausing to take part in some specific activities along the way.

Enigma machine in use during the Battle of France.
Figure 3.
Enigma machine in use in a radio tank
during the Battle of France in May 1940.
Wikimedia Commons. CC BY-SA 3.0 de.



Display of Enigma machines at National Cryptology Museum, US.
Figure 4. A display of Enigma machines and paraphernalia exhibited at the U.S. National Cryptologic Museum.
Left side of display: Enigma machines from 1923 to 1939. Right side of display: Enigma machines from 1939 to 1945.
Photographs by Robert Malmgren. Creative Commons Attribution-Share Alike 3.0 Unported.


Jeff Suzuki (Brooklyn College), "The Theorem that Won the War: Part 1 – Enigma on Paper," Convergence (October 2023)