Yet Enigma was to be broken and not long after it had been put into use. Those who achieved the solution were cryptanalysts of the Polish army which, as the defender of the Versailles state most resented by post-war Germany, took a keen and necessary interest in German military encrypted transmissions. What is extraordinary, positively intellectually heroic, about the Polish effort is that it was done initially by the exercise of pure mathematics. As Peter Calvocoressi, an initiate of the British cryptanalytic centre at Bletchley Park, has succinctly put it, “in order to break [a machine] cipher, two things are needed: mathematical theory and mechanical aids.”15 The Poles eventually designed a whole array of mechanical aids—some of which they passed to the British, some of which the British replicated independently, besides inventing others themselves—but their original attack, which allowed them to understand the logic of Enigma, was a work of pure mathematical reasoning. As it was done without any modern computing machinery, but simply by pencil and paper, it must be regarded as one of the most remarkable mathematical exercises known to history.
To do the work the Polish army recruited in the late 1920s a number of young civilian mathematicians from university mathematics faculties, including Henryk Zygalski, Jerzy Rozycki and Marian Rejewski. Marian Rejewski was to prove the most creative; like the others, he came from western, formerly German Poland, and spoke German fluently. In 1932, soon after the German army had adopted, on 1 June, the Enigma machine as its principal encryption instrument, and his own return from postgraduate study at Göttingen, he began to work on intercepted German encrypts in the Polish general staff building in Warsaw. The Poles had already learnt how to break German super-enciphered codes. From 1928 onwards, however, they had been defeated by strange messages which were clearly enciphered and probably, they concluded, the product of a machine system. The young cryptanalysts were set to learn its secrets.
What the Poles were intercepting were five-letter groups which betrayed no frequency. In technical terms, the message was itself the key, a continuous one which did not repeat unless at very long mathematical intervals (once in many millions of times, as we have seen). Yet it must, as Rejewski knew, obey a mathematical rule. He set out to construct the cipher’s mathematical basis.
The messages he was given were, we now know, produced in the following manner. After setting up his machine by printed instruction, which prescribed the disc (or rotor) order, the position of the rim and the plugging, the operator chose his own preliminary rotor setting and typed in a three-letter group, which he then repeated; this instructed the recipient how to set up his own machine for that particular transmission (and was to reveal clues to decipherment that were to be of great use, particularly to Bletchley Park). He then typed in the message with his left hand, writing down with his right hand the letters as they appeared illuminated one by one on the lamp board. Next, he passed what he had written to a radio operator, who transmitted it to the receiving station; it was this process which denied Enigma the status of an on-line system, though it would have been easy to achieve had it been linked directly to a transmitter. At the receiving end, the recipient typed in the letters he received and took down those illuminated on his lamp board, which disclosed the decrypted meaning.
Rejewski got only the encrypt. Quite quickly, however, he recognised that the first three letters were separate from the body of the message, and that the second three letters were an encryption of the first three. These two three-letter groups provided, in short, a key to the very much larger key which was the message itself. If the two preliminary three-letter groups could be broken, two results would follow: first, the electro-mechanics of Enigma itself could be reconstructed, in part at least; second, some intercepted messages could be decrypted.
Rejewski devised a set of equations which would allow him to allot real alphabetical values to the first six encrypted letters. He was able to deduce that, in the groups, say, ABC followed by DEF, D would be an encryption of A (via electromechanical permutation), E would be an encryption of B and F would be an encryption of C. He decided to designate the permutations produced by the first (fixed) disc as S, those produced by the rotors as L, M, N and that produced by the reflector as R. As a result he wrote three equations, the first of which he expressed as:
AD = SPNP-1 MLRL-1M-1PN-1p-3NP MLRL-1 M-1 p N-1 p- S-1
The other two were equally complex and, he writes, “the first part of our task [was], essentially, to solve this set of equations in which the left sides, and on the right side only the permutation P and its powers are known, while the permutations S, L, M, N, R are unknown. In this form, the set is certainly insoluble.”16
“Therefore,” Rejewski goes on, “we seek to simplify it. The first step is purely formal and consists in replacing the repeated product MLRL-1 ? M-1 . . . with the single letter Q. We have thereby temporarily reduced the number of unknowns to three, namely S, N, Q.”
Non-mathematicians will be unable to follow Rejewski’s subsequent pages of equations. They conclude, however, as follows: “the method described above for [recovering] N could be applied by turns to each rotor, and thus the complete inner structure of the Enigma machine could be reconstructed.”17
That was the Polish triumph: the penetration of the Enigma secret by pure mathematical reasoning. During the thirties, the Poles also managed to keep abreast of successive German refinements of Enigma, both electromechanical and procedural, and they succeeded in manufacturing duplicates of the Enigma machine. As its transmissions became more difficult to break, they also devised an electromechanical device (the “bombe,” apparently so-called after its ticking, which was thought to resemble that of an infernal machine) which tested solutions of encrypts faster than was possible by paper methods. Meanwhile they shared their knowledge with the French cryptanalytic service, France being Poland’s principal ally. The French themselves, through a financially corrupt German informant, known as Asché (French pronunciation of HE, the initials of his cover name), were acquiring documents which revealed many of Enigma’s operating secrets; Asché, the brother of a general, appears eventually to have been unmasked and to have been shot for treason in 1943.18 The Poles and the French certainly worked together closely on German ciphers throughout the thirties: latterly the French were also co-operating with the British Government Code and Cipher School (GCCS) located at Bletchley Park. During the period 24–25 July 1939, just before Germany’s invasion of Poland, French and British officials visited Warsaw, where the Poles passed them each a reconstructed model of the Enigma machine.