Codebreakers Victory (5 page)

Bertrand's first meeting with Schmidt was arranged by a wily go-between named Rodolphe Lemoine, code-named Rex. At the meeting, Schmidt supplied documents that included the instruction manual for the use of the Enigma and the directions for setting its keys. Bertrand's enthusiasm for his find, however, was dampened by France's leading cryptanalyst of the time, who found the material of little value without an indication of the wiring of Enigma's rotors and the actual keys in use for a given period. Bertrand received the same lukewarm response from British intelligence. Not one to give up, he sent photocopies of the two booklets to Warsaw by diplomatic courier while he himself flew there to meet with the Poles.

There, his reception was much warmer. The Poles immediately recognized the value of the booklets Bertrand had received from Asché. As recalled by Rejewski, "Asché's documents were like manna from heaven, and all doors were immediately opened."

Of the three young Polish recruits joining the Cipher Bureau, Rejewski was the one in whom those in charge had the most confidence. Previously he had spent a year studying advanced mathematics at a German university to prepare himself for a career as an insurance actuary. After four years in which they themselves had made no headway in solving the Enigma, the bureau elders handed Rejewski the task. In addition to the manuals received from Schmidt, all he had to work with were some scraps of paper left over from earlier attacks against the Enigma, an outmoded machine and stacks of intercepts.

Studying these meager resources, Rejewski saw that two enormously complex tasks awaited him. One was to determine the internal wiring of the Germans' military Enigma so that replicas of it could be built. The second was to find a way to match what the German operators did in setting their keys for the day.

He went at the wiring problem first. An obvious beginning, Rejewski recognized, was to reduce the number of unknowns. He concentrated on the procedures the manual set forth for German code clerks to follow in operating their Enigmas. Clerks in all units were to arrange the three rotors on their shaft in the same sequence for a given time period—at that early date, for a quarter year. The clerks also followed daily directions for turning the rotors to their assigned letters as that day's "ground setting" common to all the operators. The instructions then told the clerk the order for coupling the plugboard cables. With all the machines set uniformly, it was left to the individual operator to choose, at random, three letters—the message key—indicating the starting positions for a specific transmission. After using that day's ground setting to encipher his message key twice, the operator was ready to change his rotors to the three letters of his choice and begin encoding the message. The procedure was complicated, but it had the virtue of giving each message its own key and then concealing that key in the enciphered first sequence of six letters—the "indicator."

Rejewski's attention focused on the penultimate step of the instructions: the requirement that the operator tap in the three letters of his message key
a second time.
As an example, he might type in
LTBLTB,
and the glow lamps would light up, say,
XMYRVO
as the indicator to be sent over the air. The Germans' apparent intention in doubling the encipherment of the message key was to guard against garbles in transmission because of radio interference or operator error. The repeat of the three letters gave the receiving operator a second chance to set
his
rotors correctly in order to convert the scrambled code groups back into German.

Those six letters at the head of each message, Rejewski saw, represented a major flaw in the German system. For in each grouping of letters, the plaintext behind the first letter was the same as for the fourth, the second as for the fifth, and the third as for the sixth.

These relationships held for the entire day's transmissions. Every time an operator began the indicator with
L,
enciphered into
X,
his fourth letter was also
L,
this time enciphered, say, into
R.
By stacking a series of indicators one beneath the other, Rejewski saw ways to begin sequences of interconnected letters. If, for example, the indicators were

 

XMYRVO

RTLGAS

 

he would know that the first and fourth letters
Xand R
were the same in the plaintext but then, also, that in the next series beginning with an
R,
the plaintext letter underlying
G
was the same as
R
and
X.
He would link together
XRG
as the first links in a chain. If an additional indicator was
GOVXPW,
the appearance of another
X,
now in the fourth position, would close the loop and complete the chain. Rejewski could assume that all the cipher letters in that chain were the same plaintext letter. Some chains extended for many cipher letters, some for only a few, but in the end, by stacking the first six letters of sixty to eighty intercepts, he would fashion chains covering the whole alphabet. He would repeat the process for the second and fifth linkages and the fourth and sixth. He called the three sets of chains the "characteristics" of that setting.

From the commercial Enigma, Rejewski would have learned that the three rotors worked on different cycles. The right-hand one turned a notch every time a typewriter key was pressed. It was the "fast" rotor. Only after it had gone through an entire twenty-six-letter cycle did it trigger a move in the middle rotor, which in turn had to edge forward twenty-six times before activating the third rotor. Rejewski's quick mind seized upon the fact that while the fast rotor was going through its cycle, the other two rotors and the reflector remained fixed as a single unknown factor that could be disregarded while he solved the wiring of the fast rotor.

He also applied mathematical theory to determine that his chains of letters, his characteristics, were entirely the product of the rotors. The plug board could change the individual letters within a chain but could not alter the number of the chains or their lengths. At least for this part of the crypt-analysis, encipherment by the plugboard could be ruled out.

His characteristics told him the alphabetic substitutions performed for a given day in the six consecutive positions of the indicators. By applying numbers to his characteristics, he used them to set up six complex equations that, if he could solve them, would disclose the fast rotor's wiring sequence. When he tackled the equations, however, he was overwhelmed by too many unknowns. His Cipher Bureau superiors came to his aid. Hoping that Rejewski could solve the Enigma on his own and make the Polish program independent of external help, they had deliberately withheld much of the information they had received from Schmidt. Now they relented and, in early December 1932, gave him a copy of the daily keys for the past months of September and October.

The keys removed one of Rejewski's unknowns—the plugboard connections—and simplified his work on the rest of his equations. Yet they still resisted solution—until it occurred to him that the linkage between the typewriter keyboard and the rotor entry could be the problem. Studying the top row of letters on the typewriter keyboard—
QWERTZUIO
—he had thought the far-left typewriter keyboard
Q
connected with the first, or
A,
position on the input to the fast rotor, that the next letter,
W,
connected with
B,
the
E
with C and so on through the alphabet. This was, after all, the order of pairings in the commercial Enigma. But what if, in their military Enigmas, the Germans had decided on a different order of pairings?

If the connections were randomized, he knew, they would present an almost infinite number of variations, an all but insuperable stumbling block. But what if this was a point at which the Germans' fondness for orderliness, or a desire to make life less complicated for Enigma operators, had prevailed? What if instead of randomizing, the German planners had taken the simplistic course of arranging the connections in straight alphabetical order, keyboard letter
A
to the
A
contact on the entry rotor,
BtoB
and so on through the alphabet?

It turned out that was exactly what they had done. When Rejewski adjusted his equations, he later recalled, "the very first trial yielded a positive result. From my pencil, as by magic, began to issue numbers designating the wiring of drum N"—his "drum" was that rightmost rotor.

The wiring of one rotor had been converted into a known quantity. What about the other two rotors and the reflector? By a stroke of good fortune, Schmidt's two monthly key tables spanned two different calendar quarters. This meant that in the second quarter another of the three rotors was shifted into the right-hand slot. Rejewski could apply to it the same formula he had used on the first rotor. After that, he wrote, "finding the wiring in the third drum, and especially in the reflecting drum, now presented no great difficulties."

Now, with Rozycki and Zygalski assigned to help him, Rejewski was ready to turn over to the Bureau's collaborative electrical firm the details for building a working model of the Germans' military Enigma. He and his coworkers could also use Schmidt's information to begin learning how to decipher German messages.

In his reminiscences, Rejewski describes the scene that unfolded during the last days of 1932. While others were celebrating Christmas, the three young analysts, sleepy, unshaven, exhausted, but very content, placed on their superior's desk the first completely decrypted German army messages enciphered by the Enigma.

 

 

Mastering the Daily Key Settings

 

The Poles now faced a second, equally formidable challenge. They had learned how to reconstruct the Enigma machine, but they still lacked the know-how, without relying on the uncertain largesse from the German turncoat, to determine the daily settings. The messages they had decrypted at Christmastime were from the past, using outdated keys. The test now was to find out how to unravel the current keys and decrypt ongoing traffic.

They had to deal with four keying elements: the order of the three rotors on their shaft; the three letters—one for each rotor—which were that period's "ground setting" common to all operators; the three-letter key the individual operator chose for enciphering a specific message; and the connections of the plugboard cables.

They began their attack on the key settings with the discovery that also eased the way for later analysts: German code clerks misused their machines. Whether from boredom, laziness or overconfidence in the security of their Enigmas, they took shortcuts the Poles were able to exploit.

Faced with the necessity of choosing three-letter combinations as their message keys for every message they sent, as an example, clerks frequently chose not to make random selections but to use repeated letters such as
AAA
or
ZZZ.
The letter chains Rejewski derived from indicators tipped him off when the letters of the message key included a repeat. The discovery greatly reduced the number of trials the Poles had to run through in their search for the message key.

When German intelligence officers woke up to the shortcuts the code clerks were taking and issued orders prohibiting repeated letters in key settings, Rejewski was not discouraged. He saw that he could use his letter chains the other way around and rule out any combinations that included a repeat. By the process of elimination extended over a sufficient number of messages, he arrived at the point where only the correct settings were left.

These methods, however, identified only the three rotors' top letters that showed through the windows on the Enigma's lid. They did not reveal which rotor was where. Yet there were only six possible arrangements of the rotors. Rejewski and his colleagues invented an ingenious but laborious process by which they could identify the fast rotor. They used sheets of paper with six slots cut into them marked with the letters from one of Rejewski's chains. They then slid these grilles over tables of the cipher alphabets generated by each rotor, searching for pairings of the letters. Six pairings told them which was the right-hand rotor as well as its starting position. Repeating the process enabled the Poles also to identify the middle rotor, with the remaining rotor obviously going into the third slot.

Use of these techniques was tedious and time-consuming. Rejewski realized his letter chains offered a better way. They differed with each day's setting of the Enigma. Because of the reciprocal nature of the Enigma's wiring connections, the number of chains for each rotor was limited to thirteen (twenty-six letters divided by two). One day's setting might have thirteen chains for each rotor; the next day's could be thirteen, thirteen, twelve, one, and so on through all the various possibilities. He realized that these distinctive patterns were the fingerprints for identifying each day-key. If he and his colleagues could catalog all the variations in the chains and the number of links in each chain, they would have a ready reference with which to determine the ground setting for that day's transmissions. It was a daunting task since there were six variants for the order of rotors multiplied by 17,756 different letter placements on the three rotors (26 x 26 x 26) for a total of 105,456 entries that had to be tracked down and recorded. Still, that total was far less than the billions of permutations the Germans thought they had built into their machine.

Even with all this, the Poles still did not know where the alphabet rings had been set on the rotors. This step had to be taken in order to know where the notch on each rotor was located to trigger the turnover of its neighbor. Rejewski and his colleagues observed that a great many messages began with
AN
(German for "To") followed by an
X
used as a word separator. They ran a message through all positions of the rotors in search of
ANX.
Finding a message with this beginning left "only" 26 x 26, or 676, ring positions to be tested for the other two rotors. That was a lot easier and faster than having to run through all 17,756 possible positions.