Seizing the Enigma (13 page)

The instructions showed Rejewski that the clerk plugged in the six plugboard connections according to the key list. He inserted the rotors in the order given in the key list for that quarter of the year. He set each rotor’s alphabet ring so that its spring-driven stud fit into the hole at a letter given in the key list for that day. Next he looked up that day’s basic setting in the table of daily keys. He turned his rotors until the three letters given as the basic setting—say, PDX—appeared in the windows of the cover. He enciphered MGKMGK. Suppose that the six letters that lit up were OFLWZZ. This was his indicator, which he placed at the head of his message. He then turned his rotors until MGK appeared in the rotor windows. Only then did he begin enciphering the actual text of the message. This complicated procedure gave each message its own key and concealed that key in its transmission to the decipherer.

Now another cipher clerk in the same net that day might have chosen MIH as his message key. Since he would have the same plugboard connections, rotor order, alphabet ring positions, and rotor starting position as the first clerk, his two M’s would be enciphered into the same letters as the two M’s in MGKMGK, namely, O and W, even though the other letters differed. This relationship led Rejewski to build chains from the first and fourth letters of each indicator. If, for example, on a single day, two indicators were RTMGNU and GWAIZZ, Rejewski could string RG and GI together to make RGI. This constituted a chain—or at least the first links in one. Other indicators provided other links. Eventually each chain closed upon itself, returning to its first letter. Rejewski rapidly found that no single chain included all 26 letters, but that if he had enough indicators (usually around 60) all 26 would be included in other chains. The maximum
of 26 was reached in only three ways, or cycles: two chains of 13; six chains of 10, 10, 2, 2, 1, and 1 letters each; and six chains of 9, 9, 3, 3, 1, and 1 letters each.

At this point, Rejewski’s analysis branched into a path that differed fundamentally from all methods hitherto used in cryptanalytic attacks. In the past, cryptanalysts had depended upon statistics. Which letter was the most frequent? Which of several possible plaintexts was the most likely? Even the only known previous solution of a rotor machine, the dazzling 1924 success of American William F. Friedman in reconstructing the wiring of Edward Hebern’s five-rotor machine, used a probabilistic and lower-algebraic approach. But Rejewski, for the first time in the history of cryptanalysis, utilized a higher-algebraic attack. He applied one of the first theorems taught in the theory of groups. In simplified form, the theorem states that if
P
and
Q
are permutations, then the permutation
PQP
⁻¹
(read
P, Q, P
inverse) has the same cycle structure as the permutation
Q
. In the Enigma encipherment
P
could represent the plugboard input;
P
⁻¹
, the plugboard output; and
Q
the total rotor encipherment. Group theory thus told Rejewski that his cycles depended only on the rotor setting and not on the plugboard encipherment. It told him, in other words, that the plugboard, in which the Germans placed great trust as enhancing the machine’s security, could be ignored in at least part of the cryptanalysis.

The cycles Rejewski had discovered were produced by the substitutions generated by the six steps of the rightmost, or fast, rotor (the one that turned at the encipherment of each letter of the six letters of the key). Rejewski used the cycles to set up six huge equations that, if solved, would disclose the wiring of the fast rotor. The unknown terms of the equations were not simple ones like 3
x
, but arrays of 26 elements. These elements consisted of Rejewski’s quantification of the machine encipherment. If a rotor input contact was at the 12th position and the wire inside connected it to the output contact at the 20th position, the encipherment for that input position would be given the
numerical value of 8. But for Rejewski, all 26 values, representing all 26 connections, were unknown.

Each of Rejewski’s six equations had four complex terms. Three terms were unknown: the array of numbers representing the wiring of the fast rotor (which moved each time a letter was enciphered); the array of numbers representing the combined wiring of the middle and left rotors (which were assumed to be stationary, as they were in 21 cases out of 26) plus the reflector; and the connections of the six letter-pairs that were enciphered in the plugboard. (The plugboard could be ignored in the cycles but not in the eventual recovery of plaintext.) Rejewski assumed that he knew the fourth term, but in fact it was unknown. It represented the connections of the typewriter keys to the input plate that fed the current to the rightmost rotor. On the basis of the commercial Enigma that BS-4 had bought, Rejewski thought that these connections ran in keyboard order, from key Q to the first, or A, position on the input plate, from key W to input plate position B, from E to C, and so on. Finally, Rejewski introduced a permutation that would correct for the movement of the fast rotor as successive letters were enciphered.

He then tackled the equations. But the number of their unknowns overwhelmed him. It became clear to Ciȩżki, who visited him in his solitary office every day, that Rejewski was not going to succeed by himself. He would have to be given some of the material that Schmidt had supplied. Ciȩżki and Langer had thus far withheld this material, perhaps to make Polish cryptanalysis less dependent on gifts from France, which was just then cooling toward Poland’s insistence on retaining the Corridor and on being superior in armed forces to Germany. On December 9, 1932, some six or eight weeks after Rejewski had started work, Ciȩżki gave him a copy of the daily keys for September and October 1932, which Schmidt had given
REX
in August and which Bertrand had brought to Warsaw in September.

The keys at once transformed one of the unknowns—the plugboard connections—into a known and simplified the rest of the
equations. But they remained complex, and Rejewski continued to wrestle with them for several weeks. Then one day it struck him that his assumption for the wiring from the typewriter keys to the input plate could be wrong. Perhaps the wire from key Q ran to position Q rather than position A. He adjusted his equations. “The very first trial yielded a positive result. From my pencil, as if by magic, began to issue numbers designating the wiring in rotor N [the rightmost, or fast, rotor],” he wrote.

The twenty-seven-year-old cryptanalyst had uncovered part of the secret heart of the Enigma: the wiring of one rotor. This enabled him to lay the first Enigma solutions on Ciȩżki’s desk at the end of December, as Christmas and New Year’s lifted people’s spirits in the Polish capital.

But these solutions comprised only a selected few, and further work was needed to complete the reconstruction of the machine. Here the Poles had a stroke of luck. The Germans changed the order of the rotors in the machine every three months, or quarter of a year. Fortunately, the keys that Schmidt had supplied straddled two different quarters: the third, for the September keys, and the fourth, for the October ones. This meant that in October the rotor in the right-hand position was different from the one in that position in September. Using the same technique as before, Rejewski determined the wiring on this rotor. After this, “finding the wiring in the third rotor, and especially in the reflecting rotor, now presented no great difficulties.” Cleaning up the work—eliminating ambiguities to obtain completely correct information on wiring and rotor stepping—was greatly eased by a sample encipherment in one of the manuals that Schmidt had provided.

The solution was Rejewski’s own stunning achievement, one that elevates him to the pantheon of the greatest cryptanalysts of all time. Much of the solution was due to his brilliance. Yet mathematics—even with Rejewski’s extraordinary ability—had not sufficed. Pure analysis alone had not achieved a solution. The machine was too
complex. Rejewski needed help from outside information, as he acknowledged:

To this day, it is not known whether equation 3 [of the set of six equations with the arrays of 26 unknowns] is solvable. Admittedly, another approach to the reconstruction of the rotor wirings was found, in theory at any rate. But that approach is imperfect and laborious…. It requires the possession of messages from two days of identical or very similar settings of the rotors; therefore, finding the wiring of the rotors would depend on luck. In addition, it requires so many trials that it is not clear whether the director of the Cipher Bureau would have had enough patience to employ several workers for a long period without certain attainment of success, or whether he would have once more discontinued work on the Enigma. Hence the conclusion is that the intelligence material furnished to us should be regarded as having been decisive to the solution of the machine.

Britain and France also had these documents. Why had they not solved the Enigma?

They lacked mathematical cryptanalysts. Their cipher establishments, like generals still fighting the last war, saw no need to change the linguistic orientation that had brought them their successes of 1914–1918 and that was continuing to solve many diplomatic codes in the 1920s. France, for example, was breaking the codes of some ten countries. The cipher bureaus had no guarantee that an inexperienced mathematical cryptanalyst would succeed where experienced linguistic cryptanalysts had failed: Dillwyn Knox, a leading light of Britain’s agency, had not broken the German Enigma despite great efforts. Though this agency had once considered training university mathematicians as reserve cryptanalysts, it had rejected the idea for fear that their indiscretions might reveal its codebreaking efforts. But Britain, at least, seemed justified in not expending more of its resources on the Enigma. The Admiralty maintained that Japan was Britain’s chief threat, not Germany, where even that far-right exponent
of revanche, Adolf Hitler, had written in
Mein Kampf
that to win England as an ally he would offer a “renunciation of a German war fleet.”

In the end, what France and Britain lacked—and not only in cryptology—were vision and will. Poland had both. It was the great merit of Pokorny and Ciȩżki to have seen, before their counterparts in Europe’s other cipher bureaus, the value of cryptanalysts with a strong mathematics background. And the great need to know what Germany was planning drove those cryptanalysts to extraordinary efforts. So Poland did what no other country had done—and what the Germans believed impossible.

But the work of Rejewski, that modern magus, was far from finished. It had, in fact, just begun.

R
EJEWSKI’S GREAT SOLUTION HAD REPRODUCED THE
E
NIGMA AND
solved a few German messages. But it did not enable the Poles to read messages regularly. Indeed, the very concept of the Enigma, the reason for the machine’s adoption, was that even if the enemy had a machine, he would not be able to obtain useful information from messages enciphered with it. So many keys were available, the thinking ran, that no cryptanalyst, or even team, would be able to find the right one before the messages had lost all military value. And the time it would take to solve an intercept was measured not in hours, not in days, but in years, in millennia.

Arthur Scherbius had enlarged the possible number of keys by proposing a machine with seven or even ten rotors. The German army had chosen instead to make the rotors changeable, to add alphabet rings, and to attach a plugboard. Their permutations raised the number of keys available to the astronomical figure of 10½ quadrillion. If 1,000 cryptanalysts, each with a captured or copied Enigma, each tested four keys a minute, all day, every day, the team would take 1.8 billion years to try them all. Since on average the codebreakers would reach plaintext halfway through, the typical solution would take them “only” 900 million years. For the Germans, this sufficed.

Rejewski thus faced a new and daunting task. Using Schmidt’s keys, he had reconstructed the Enigma. Now, using the reconstruction, he had to find each message’s key. And to decipher the German
messages quickly enough to make current use of them, he obviously had to find a method other than exhaustive search.

For there were no flaws in the theory of the machine. It offered all the defenses that its inventor and proponents said it did; the Germans’ reasoning was impeccable, and their confidence was, in theory, not misplaced. But it was in practice. The ways in which men used the machine undermined its defenses. The army cryptographers’ requirement that message keys be duplicated and then enciphered created relationships that vastly reduced the number of trials cryptanalysts would have to make to find the right key. Cipher clerks made up message keys, such as ZZZ, that were so easy to guess that the number of trials was reduced even more, and signal officers drafted stereotyped plaintexts, also easy to guess, that cut the number still further. The failure was not, as a later generation would say, in the hardware: it was a software problem.