The Other Enigma
On one of the greatest achievements in the history of cryptanalysis
It took some time, but Alan Turing is now a household name in the United Kingdom. He is probably best known for two things - first, his central role in breaking the German Enigma encryption machine, thereby saving the lives of countless British soldiers and mariners during WWII, and second, the atrocious cruelty with which he was treated by the same government that had used his mathematical insights to defeat Hitler's Germany. He is the subject of a (rubbish) film, and at least one (very good) biography. He also has secured a sort of secular immortality as the face of the latest £50 note, which now carries both his likeness and the image of an array from his famous paper on the Entscheidungsproblem.
I wanted to use this post to share another story from the Second World War which is less well known, at least here in the UK, but is arguably even more astounding than Turing's (even if it was ultimately far less impactful). It takes place in Sweden, one of a handful of European countries that maintained its neutrality between 1939 and 1945. Despite its official posture as a neutral state, Sweden found itself entangled in the war in different ways, exporting iron ore to Germany for use by the Wehrmacht, while also allowing its intelligence services to share strategic information with the Allies. The reason Sweden had any intelligence to share was because of the efforts of one man, Arne Beurling, a professor of mathematics who had been invited by the Swedish Defence Cryptography Department to evaluate encrypted German signals traffic intercepted from Swedish-leased telegraphy lines in Norway.
The signals running through those cables had been encrypted using a more advanced system than the Enigma machine that was later cracked by Turing et al., the so-called Der Geheimschreiber' or 'G-writer', a mechanical teleprinter manufactured by Siemens, which combined the "overlaying" encryption process of older systems with a "transposition" that introduced a systematic permutation into the order of pulses:
The Geheimschreiber's ten code wheels had the periods 47, 53, 59, 61, 64, 65, 67, 69, 71 and 73. In the first models all the wheels moved one step for each enciphered character. Since the wheel periods were relatively prime, that is they had no common factor, the total period of the machine — the number of steps the machine must make to return to its starting position — was equal to the product of all the individual wheel periods, that is 893 622 318 929 520 960 steps. This number also indicates the number of possible wheel starting positions. The “transposition circuit", that is the insertion of the “transposition relays" between the rows, could be varied. Eight basic patterns were possible, each with 2 612 736 000 variations. The combinations of connections and wheel adjustments were, before the creation of the computer, considered to be extremely large numbers. In addition there were the number of ways of connecting the code wheels to the relays. This may have given the Germans the impression that the Geheimschreiber was a very secure cipher machine. It was probably considered to be more secure than the Enigma machine which was intended for tactical use. The Enigma had, for example, a period of 17576.
Lars Ulfving, 'The Geheimschreiber Secret' (1992)
Beurling was first shown the intercepted teleprinter tapes, which had been printed out and stitched together on long strips of paper, sometime in the summer of 1940. He was given 24 hours of traffic. Two weeks later he had reconstructed the complete encryption mechanism. He had no access to any machines (unlike Turing), no captured plain text or 'cribs' (unlike Turing), and had no prior knowledge about the principles used by Siemens to construct the machine (unlike Turing). His only tools were a pencil and some paper. When asked how he did it, he would respond 'a magician does not reveal his tricks', which became something of a motto. This sort of thing would not be acceptable where I work.
It is a wonderful story, which is told at length in Bengt Beckman's book Codebreakers. Beurling would eventually go on to Princeton, where he was given Einstein's old office at the Institute for Advanced Study and did important work in pure and applied analysis. Here is his former student, the Fields Medallist Lennart Carleson, with an interesting account of Beurling's unique way of thinking about mathematical problems:
When we do mathematics most of us use only the intellectual 50% of the brain and keep the emotional side idle. For Arne Beurling the world of mathematics was somehow integrated with the real world and with life itself and his brain was 100% active. This seemed to give him access to information that was hidden for the rest of us and this can only be described as related to Mysticism […] This emotional relation to mathematics explains why he judged mathematical results in terms of beauty rather than, or even opposed to, complexity. He wrote his papers as pieces of art and was anxious to hide as well as possible how he found his results.
Lennart Carleson, 'A Mathematical Genius' (2007)
Elsewhere Carleson describes him as one of the few mathematicians that Ibsen could 'learn from'.
To crowbar in some philosophy, this strange combination of total rigour and profound aesthetic feeling would seem to give the lie to an idea which is becoming increasingly popular, that emotions only impede rational thought, which becomes defective under their influence. To quote one American commentator, 'facts don't care about your feelings'. This is a vulgarised version of the more ancient stoic idea that the virtuous man is also the one who is free from emotion (apathes). Emotion is understood in this tradition as 'an impulse which is excessive and disobedient to the dictates of reason, or a movement of soul which is irrational and contrary to nature', according to Stobaeus. Clearly this is not what Carleson meant; 'emotion', as it is used in the passage quoted above, suggests more a sort of elan which when harmonised with reason opens the mind to beauty. And there is a tradition as old as the stoic one which holds that beauty and truth are two aspects of the same fundamental reality.