# Breaking Enigma the first time

The 10th Annual Pi Mu Epsilon Lecture was given by Dr. David Saltman on Oct. 7, 2016.

Saltman is the director of the Center for Communications Research, a federally funded research and development center in Princeton, New Jersey. Formerly, he was the Mildred Caldell and Baine Perkins Kerr Centennial Professor of Mathematics at the University of Texas in Austin. David Saltman is well known for his remarkable research contributions in algebra and number theory.

During his stay at Nebraska, Saltman met with students to offer his perspective on academic vs. non-academic employment, gave a colloquium talk for faculty and graduates, and presented the 10th annual Pi Mu Epsilon address “Breaking Enigma the First Time.”

“Enigma” was the name given to the cipher machines that the German Armed Forces used to safeguard their communications during World War II. The lecture featured a real Enigma unit that the audience had a chance to try out. Saltman described the fundamentals of its construction wherein any position of rotors defines a permutation of the alphabet by mapping each letter to another one. Every keystroke rotates the rotors and thus yields a new permutation for each next character. The encoded message can be read on any other Enigma machine by entering the resulting combination of letters, provided the two Enigma units had the same starting rotor configurations.

The story of how the Allies broke the Enigma codes (and kept it a secret from the Nazis) is a compelling and important one. The course of the war might have been far different without the vital intelligence gained by reading Enigma messages. However, the breaking of Enigma started with the lesser known work of three Polish mathematicians and cryptanalysts Marian Rejewski, Jerzy Rozycki and Henryk Zygalski. The first key weakness of the machine they exploited was the fact that the starting rotor position had to be transmitted in the clear, after which an operator would choose a secret key that would be sent over twice (to avoid errors). Though the rotor structure was not fully known, the beginning of such a message had relations between the 1st and 4th, the 2nd and 5th, 3rd and 6th, etc., characters. This information along with other clever insights could be used to successfully decode many Enigma messages.

The Germans kept updating the machine and changed communication procedures. Rajevsky and his team in the Polish Cipher Bureau were quick to respond by developing more sophisticated decoding methods. Yet in 1939 it became apparent that their resources were insufficient to keep pace with the evolution of Enigma encryption. The Polish General Staff and government decided to initiate their Western allies into the secrets of Enigma decryption. The Polish methods were then revealed to French and British intelligence.

The Enigma study during WWII was led by the British and the genius of Allan Turing – a mathematician, logician, theoretical biologist and regarded as the ‘father’ of theoretical computer science. Turing led the section responsible for German naval cryptanalysis. He devised a number of techniques for speeding the breaking of Enigma ciphers including an electromechanical machine that could find the rotor settings for the Enigma.

In light of these latter successes, the pioneering efforts of the Polish cryptanalysts are less well known. One may even wonder if their work was truly indispensable. Yet, it is hard to forget that one of the most difficult research questions is often not computing the solution, but proving that one exists to begin with. Rajewski and his team provided not merely an impetus and a knowledge base for further study of Enigma, their monumental achievement was to solve the greatest mystery of this cipher device: They proved that Enigma could be broken.

– Daniel Toundykov