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Codebreakers: Arne Beurling and the Swedish Crypto Program During World War II

American Mathematical Society
Number of Pages: 

I have met many gifted persons in my day, but only one genius. I need not mention his name.

He was a demi-god, so I was very proud.


These are but two of many similar quotes concerning the cryptographer/mathematician Arne Beurling. (The first quote is from Ake Lundqvist, one of Beurling's fellow cryptographers; the second quote is from Gunnar Blom, a conscript in the cryptography department, on how he valued any praise received from Beurling.) This book, through the use of interviews and historical documents, gives an insider's view of the history of Swedish cryptography, with special attention paid to Arne Beurling. The (original Swedish version of this) book was written by Bengt Beckman, a notable name in Swedish cryptography circles, and the English translation was provided by Kjell-Ove Widman, Director of Sweden's Mittag-Leffler Institute.

Let's start with a brief description Codebreakers. The first part of this book, which turns out to be most of the book, covers the history of cryptography in Sweden. The reader is given an example of a simple replacement cipher, a description of the world's first known ciphering machine (the Gripenstierna device), and the beginning of the Swedish crypto industry (the Damm brothers' AB Cryptograph). Karl Wilhelm Hagelin, who invests in the Damms' company, goes on to devise ciphering machines to compete with ciphering devices on the market from other countries (including the German Enigma machine). He develops many of his machines based on advances in the area of cryptanalysis—especially advances made by the mathematically and linguistically gifted Yves Gylden. We also learn of some the Swedish efforts in Radio signal interception in World War I and the 1930s, which leads to the creation of the Forsvarsstaben—the Defense Staff Headquarters.

With the beginning of World War II, the book takes off. At the outbreak of WWII, Arne Beurling was assigned to study Russian transmissions. To assist him in this task, he has Ake Lundqvist (mentioned above). Arne's section succeeds in deciphering 10,400 of the Russian Baltic Navy's telegrams. The Russians used a four-digit code (which was broken by Gosta Wollbeck and Olle Sydow) and an “additively super-enciphered five-digit code” (which was broken by Beurling and Lundqvist). Soon, the cryptographers begin handling traffic from other countries—most notably Germany. Beurling's greatest accomplishment was probably his breaking of the algorithm for the German Geheimschreiber (or G-Schreiber); a feat which took him all of two weeks. Having done this, he designed the apps, an apparatus which could emulate the G-Schreiber, allowing the Swedes to decrypt German messages. This process of decrypting—and the roles of several of the “31-groups”—is described. Another of Beurling's accomplishments, which pleased him to no end, what the breaking of a Czech double transposition code; a great accomplishment, considering Beurling did not know Czech. (Incidentally, despite great successes such as these, Beurling was not able to break the German Enigma machine.)

In 1940, the Swedish government created the FRA (the Radio Bureau of the Defense Services), a permanent signal intelligence agency. The agency was quite successful in deciphering German messages, ending up with well over 1000 miles of paper tape as a result of their work. Some of their success, naturally, was due to the quality work of many gifted people, but some of it was also due to the ineptitude of some of the German forces (making for some humorous stories). However, the agency wasn't perfect: they had to deal with several leaks from within their ranks. But ultimately, they were of great help to Sweden, as well as the Finns (in the Stella Polaris operation) and the Norwegians (with their resistance movement). In fact, several members of the FRA were awarded the Norwegian King Haakon VII Freedom Medal, an award that had to be kept secret at the time, as it would have exposed the Swedish violation of their neutrality policy. There is also a brief mention of the Swedes' Crypto Systems, with some discussion concerning its efficacy.

The second part of the book, covering the last thirty or so pages, focuses more on the person of Arne Beurling. We learn of his personality and his study and mathematics. For example, how his doctoral degree was delayed a year (part of the delay involved the mathematics of Lars Ahlfors, part of the delay involved alligator hunting); how, as a professor, he considered students' clumsy solutions to be a personal insult. And on the more personal level, we learn of his affect on women, his reputation as a womanizer, and his explosive temper (having a fist fight with Yves Gylden, threatening to shoot Andre Weil). Perhaps most interesting, from a mathematical point of view, was his rather unique possessive view of his mathematics. Much of what he wrote was not published. Being something of an insecure perfectionist, he wouldn't reveal his results until he thought they were just right. In fact, his great friendship with Lars Ahlfors was ended because Ahlfors gave a lecture in which he mentioned work that he had done with Beurling. Ahlfors made sure to state how much of a role Beurling had played in the work. However, Beurling was offended, because he felt that the material was unfinished and therefore should not have been mentioned. (The friendship was eventually renewed, albeit many years later; in fact, just two years before Beurling's death in 1986.) After the substance of the first part of the book, the second part almost acts as the dessert to this meal.

So, who is the book written for? The writing is geared towards a general audience. If you are interested in the history of cryptography, or perhaps even in history in general, then you'll find much to enjoy in this book. If you are interested in the mathematics of cryptography, then this book, while appealing, may not satisfy you. Codebreakers does deal with the mechanics of cryptography (letter shifts, digraphs, additive super-position, the notion of depth), but not much with the actual mathematics. Outside of those interested in cryptography, mathematics, or history, there probably isn't much of an audience. The book presents the history of Swedish cryptography well; but where it truly shines is in the stories of the people. Not just stories about Beurling, but of many of the other characters as well. Stories about the cryptographers dealing with military bureaucracy, their working conditions, the occasional public gaffe, and their methods for getting the job done in less-than-ideal circumstances, make for some fun reading.

Donald L. Vestal is Assistant Professor of Mathematics at Missouri Western State College. His interests include number theory, combinatorics, reading, and listening to the music of Rush. He can be reached at

Date Received: 
Friday, June 6, 2003
Include In BLL Rating: 
Bengt Beckman
Publication Date: 
Donald L. Vestal

Part 1

  • An 18th century cipher
  • The world's first ciphering machine
  • Damm, Hagelin, and Gyldén
  • Radio signal interception and cryptanalysis before 1939
  • War
  • Enter Arne Beurling
  • The Russian Baltic Navy
  • Mysterious signals
  • Teleprinters
  • Beurling's Analysis
  • The G-Schreiber and the apps
  • Continued cryptanalysis
  • Exit Gyldén--but Beurling comes back
  • The double transposition
  • Operation Barbarossa
  • The work place
  • Contents
  • The birth of the FRA
  • Brilliant results--despite everything
  • Downturn and leakage
  • The Red Army and the Arctic Sea
  • The doubly enciphered Russian code
  • Stella Polaris
  • Gradual loss of German traffic
  • Borelius pays a visit to the Germans
  • Information--but of what value?
  • Norway
  • The last years of the war
  • The Swedes' own crypto systems
  • Arne Beurling 1943-1945

Part 2

  • Arne Beurling
  • Through the eyes of a woman
  • A magical friendship
  • Sources
  • Index of names
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Wednesday, October 17, 2007