Definitions

# Marian Rejewski

Marian Adam Rejewski (16 August 1905 – 13 February 1980) was a Polish mathematician and cryptologist who, in 1932, solved the Enigma machine, the main cipher device used by Germany. The success of Rejewski and his colleagues jump-started British reading of Enigma in World War II, and the intelligence so gained, code-named "Ultra", contributed, perhaps decisively, to the defeat of Nazi Germany.(Note 1)

While studying mathematics at Poznań University, Rejewski had attended a secret cryptology course conducted by the Polish General Staff's Cipher Bureau, which he joined full-time in 1932. The Bureau had achieved little success reading Enigma and in late 1932 set Rejewski to work on the problem. After only a few weeks, he deduced the secret internal wiring of the Enigma. Rejewski and two mathematician colleagues then developed an assortment of techniques for the regular decryption of Enigma messages. Rejewski's contributions included devising the cryptologic "card catalog", derived using his "cyclometer", and the "bomba".

Five weeks before the German invasion of Poland in 1939, Rejewski and his colleagues presented their results on Enigma decryption to French and British intelligence representatives. Shortly after the outbreak of war, the Polish cryptologists were evacuated to France, where they continued their work in collaboration with the British and French. They were again compelled to evacuate after the fall of France in June 1940, but within months returned to work undercover in Vichy France. After the country was fully occupied by Germany in November 1942, Rejewski and fellow mathematician Henryk Zygalski fled, via Spain, Portugal and Gibraltar, to Britain. There they worked at a Polish Army unit, solving low-level German ciphers. In 1946 Rejewski returned to his family in Poland and worked as an accountant, remaining silent about his cryptologic work until 1967.

## Education and early work

Marian Rejewski was born 16 August 1905, in Bromberg, now Bydgoszcz.(Note 2) His parents were Józef, a cigar merchant, and Matylda, née Thoms. He attended a German-speaking Königliches Gymnasium zu Bromberg (Royal Grammar School in Bromberg) and completed high school with his matura in 1923. Rejewski then studied mathematics at Poznań University, graduating on 1 March 1929.

In early 1929, shortly before he graduated, Rejewski began attending a secret cryptology course organized for selected German-speaking mathematics students by the Polish General Staff's Cipher Bureau (Biuro Szyfrów). Rejewski and fellow students Henryk Zygalski and Jerzy Różycki were among the few who could keep up with the course while balancing the demands of their normal studies.

Rejewski graduated with a master's degree in mathematics on 1 March 1929; his thesis was titled, "Theory of double periodic functions of the second and third kind and its applications." A few weeks later, he began the first year of a two-year actuarial statistics course at Göttingen, Germany. He would not complete the course, for, while home for the summer in 1930, he accepted the offer of a mathematics teaching assistantship at Poznań University. At the same time, he also began working part-time for the Cipher Bureau, which by then had concluded the cryptology course and set up an outpost at Poznań to decrypt intercepted German radio messages. Rejewski worked some twelve hours a week near the Mathematics Institute in an underground vault referred to puckishly as the "Black Chamber".

In the summer of 1932, the Poznań branch of the Cipher Bureau was disbanded. On 1 September 1932, as a civilian employee, Rejewski joined the Cipher Bureau at the General Staff building (the Saxon Palace) in Warsaw, as did Zygalski and Różycki.

Their first assignment was to work out a four-letter code used by the Kriegsmarine (German Navy). Progress on solving this system was initially slow, but sped up considerably after a test coded-message exchange was received—a six-group signal, followed by a four-group response. The cryptologists guessed correctly that the first signal was the question, "When was Frederick the Great born?" followed by the response, "1712.

## Enigma machine

In October 1932, while work on the Naval code was still underway, Rejewski was set to work, alone and in secret, on the output of the new standard German cipher machine, the Enigma I, which was coming into widespread use. While the Cipher Bureau had, by later report, succeeded in solving an earlier, plugboard-less Enigma,(Note 3) it had had no success with the Enigma I.

The Enigma machine was an electromechanical device, equipped with a 26-letter keyboard and a set of 26 lamps, corresponding to the letters of the alphabet. Inside was a set of wired drums ("rotors" and a "reflector") that scrambled the input. The machine also featured a plugboard to swap pairs of letters. To encipher a letter, the operator pushed the relevant key and noted down which of the lamps lit. Each key press caused one or more rotors to advance, and thus the encipherment varied from one key press to the next. In order for two operators to communicate, both Enigma machines had to be set up in exactly the same way. The large number of possibilities for setting the rotors and the plugboard combined to form an astronomical number of configurations, each of which would produce a different cipher. The settings were changed daily, with the consequence that the machine had to be "broken" anew each day if the messages were to be read continually.

To decrypt Enigma messages, three pieces of information were needed:

1. A general understanding of how Enigma functioned
2. The wiring of the rotors
3. The daily settings: the sequence and orientations of the rotors (of which there were three initially), and the plug connections on the plugboard

Rejewski had only the first at his disposal, based on information already acquired by the Cipher Bureau.

## Solving Enigma's wiring

First, Rejewski tackled the problem of finding the wiring of the rotors. To do this, he pioneered the use of pure mathematics in cryptanalysis. Previous methods had largely exploited linguistic patterns and the statistics of natural-language texts — letter-frequency analysis. Rejewski, however, applied techniques from group theorytheorems about permutations — in his attack on Enigma. These mathematical techniques, combined with material supplied by French military intelligence, enabled him to reconstruct the internal wirings of the machine's rotors and nonrotating reflector. "The solution", historian David Kahn writes, "was Rejewski's own stunning achievement, one that elevates him to the pantheon of the greatest cryptanalysts of all time". Rejewski used a mathematical theorem that one mathematics professor has since described as "the theorem that won World War II".

Rejewski studied the first six letters of all the Enigma messages intercepted on a single day. For security, each message sent on Enigma was encrypted using a different starting position of the three rotors, chosen by the operator. This was termed the message setting, and was three letters long. To convey this to the receiving operator, a sending operator began each message by sending the message setting in a disguised form — a six-letter indicator. The indicator was formed using the Enigma with the rotors set to a common global setting for that day, termed the ground setting, shared by all operators. Unfortunately for the Germans, the particular way the indicator was constructed introduced a fundamental weakness into the system.

For example, suppose the operator chose the message setting KYG for a message. The operator would first set the Enigma's rotors to the ground setting, which might be GBL on that particular day, and then encrypt the message setting on the Enigma twice; that is, the operator would enter KYGKYG (which might come out to something like QZKBLX). The operator would then reposition the rotors at KYG, and encrypt the actual message. A receiving operator could reverse the process to recover first the message setting, then the message itself. The repetition of the message setting was apparently meant as an error check to detect garbles, but it had the unforeseen effect of greatly weakening the cipher. Due to the indicator's repetition of the message setting, Rejewski knew that, in the plaintext of the indicator, the first and fourth letters were the same, the second and fifth were the same, and the third and sixth were the same. These relations could be exploited to break into the cipher.

Rejewski studied these related pairs of letters. For example, if there were four messages that had the following indicators on the same day: BJGTDN, LIFBAB, ETULZR, TFREII, then by looking at the first and fourth letters of each set, he knew that certain pairs of letters were related. B was related to T, L was related to B, E was related to L, and T was related to E: (B,T), (L,B), (E,L), and (T,E). If he had enough different messages to work with, he could build entire sequences of relationships: the letter B was related to T, which was related to E, which was related to L, which was related to B (see diagram). This was a "cycle of 4", since it took four jumps until it got back to the start letter. Another cycle on the same day might be A$rightarrow$F$rightarrow$W$rightarrow$A, or a "cycle of 3". If there were enough messages on a given day, all the letters of the alphabet might be covered by a number of different cycles of various sizes. The cycles would be consistent for one day, and then would change to a different set of cycles the next day. Similar analysis could be done on the 2nd and 5th letters, and the 3rd and 6th, identifying the cycles in each case and the number of steps in each cycle.

Using the data thus gained, combined with Enigma operators' tendency to choose predictable letter combinations as indicators (such as girlfriends' initials or a pattern of keys that they saw on the Enigma keyboard), Rejewski was able to deduce six permutations corresponding to the encipherment at six consecutive positions of the Enigma machine. These permutations could be described by six equations with various unknowns, representing the wiring within the entry drum, rotors, reflector, and plugboard.

### Help from France

At this point Rejewski ran into difficulty: the large number of unknowns made the equations complex. He would later comment in 1980 that it was still not known whether such a set of six equations was solvable without further data. But he was assisted by cryptographic documents that a section of the French intelligence organization (the Deuxième Bureau), under future General Gustave Bertrand, had obtained and passed on to the Polish Cipher Bureau. The documents had been procured from a spy in the German cipher office, Hans-Thilo Schmidt, and included the Enigma settings for the months of September and October 1932. On December 9 or 10,(Note 4) 1932, the documents were given to Rejewski, who used their information to eliminate the effect of the plugboard from the equations. With the reduced number of unknowns, solving the equations became a tractable problem.

Another obstacle had to be overcome, however. The military Enigma had been modified from the commercial Enigma, of which Rejewski had had an actual example to study. In the commercial machine, the keys were connected to the entry drum in German keyboard order ("QWERTZU…"). However, in the military Enigma, the connections had instead been wired in alphabetical order: "ABCDEF…" This new wiring sequence foiled British codebreakers working on Enigma, who dismissed the "ABCDEF…" wiring as too obvious. Rejewski, perhaps guided by an intuition about a German fondness for order, simply guessed that the wiring was the normal alphabetic ordering. He later recalled that, after he had made this assumption, "from my pencil, as by magic, began to issue numbers designating the connections in rotor N. Thus the connections in one rotor, the right-hand rotor, were finally known".

The settings provided by French Intelligence covered two months which straddled a changeover period for the rotor ordering. A different rotor happened to be in the right-hand position for the second month, and so the wirings of two rotors could be recovered by the same method.(Note 5) This simplified the analysis, and by the end of the year, the wirings of all three rotors and the reflector had been recovered. An example message in an Enigma instruction manual provided a sequence of plaintext and corresponding ciphertext enciphered at a given setting; this helped Rejewski eliminate remaining ambiguity from the wiring.

There has been speculation as to whether the rotor wirings could have been solved without the documents supplied by French Intelligence. Rejewski recalled in 1980 that another way had been found that could have been used to achieve this, but that the method was "imperfect and tedious" and relied on chance. In 2005, mathematician John Lawrence published a paper arguing that it would have taken four years for this method to have had a reasonable likelihood of success. Rejewski wrote that "the conclusion is that the intelligence material furnished to us should be regarded as having been decisive to solution of the machine".

## Solving daily settings

After Rejewski had determined the wiring in the remaining rotors, he was joined in early 1933 by Różycki and Zygalski in devising methods and equipment to break Enigma ciphers routinely.(Note 6) Rejewski later recalled:
Now we had the machine, but we didn't have the keys and we couldn't very well require Bertrand to keep on supplying us with the keys every month ... The situation had reversed itself: before, we'd had the keys but we hadn't had the machine — we solved the machine; now we had the machine but we didn't have the keys. We had to work out methods to find the daily keys.

### Early methods

A number of methods and devices had to be invented in response to continual improvements in German operating procedure and to the Enigma machine itself. The earliest method for reconstructing daily keys was the "grill", based on the fact that the plugboard's connections exchanged only six pairs of letters, leaving fourteen letters unchanged. Next was Różycki's "clock" method, which sometimes made it possible to determine which rotor was at the right-hand side of the Enigma machine on a given day.

After 1 October 1936, German procedure changed, increasing the number of plugboard connections. As a result, the grill method became considerably less effective. However, a method using a card catalog had been devised around 1934 or 1935, and was independent of the number of plugboard connections. The catalog was constructed using Rejewski's "cyclometer", a special-purpose device for creating a catalog of permutations. Once the catalog was complete, the permutation could be looked up in the catalog, yielding the Enigma rotor settings for that day.

The cyclometer comprised two sets of Enigma rotors, and was used to determine the length and number of cycles of the permutations that could be generated by the Enigma machine. Even with the cyclometer, preparing the catalog was a long and difficult task. Each position of the Enigma machine (there were 17,576 positions) had to be examined for each possible sequence of rotors (there were 6 possible sequences); therefore, the catalog comprised 105,456 entries. The preparation of the catalog took over a year, but when it was ready about 1935, it made obtaining daily keys a matter of 12–20 minutes. However, on November 1 or 2, 1937, the Germans replaced the reflector in their Enigma machines, which meant that the entire catalog had to be recalculated from scratch. Nonetheless, by January 1938 the Cipher Bureau's German section was reading a remarkable 75% of Enigma intercepts, and according to Rejewski, with only a minimal increase in personnel, this could easily have been increased to 90%.

### Bomba and sheets

In 1937 Rejewski, along with the German section of the Cipher Bureau, transferred to a secret facility near Pyry in the Kabaty Woods south of Warsaw.

On 15 September 1938, the Germans put into effect new rules for enciphering message keys (a new "indicator procedure"), rendering obsolete the cryptologic techniques then used by the Poles.(Note 7) The Polish cryptologists rapidly responded with new techniques.

One was Rejewski's bomba, an electrically powered aggregate of six Enigmas, which enabled the daily keys to be solved in about two hours. Six bombas were built and ready for use by mid-November 1938. The bomba exploited the fact that the plugboard connections did not affect all the letters; therefore, when another change to German operating procedure occurred on 1 January 1939, increasing the number of plugboard connections, the usefulness of the machines was greatly reduced. The British bombe, the main tool that would be used to break Enigma messages during World War II, would be named after, and likely inspired by, the Polish bomba, although the cryptanalytic methods embodied by the two machines were very different.

Around the same time, a manual method was invented by Zygalski, that of "perforated sheets" ("Zygalski sheets"), which was independent of the number of plugboard connections.

However, application of both the bomba and Zygalski's sheets was complicated by yet another change to the Enigma machine on 15 December 1938. The Germans had supplied Enigma operators with an additional two rotors to supplement the original three, and this increased the complexity of decryption tenfold. Building ten times as many bombs was beyond the Biuro's ability—that many bombs would have cost fifteen times their entire annual equipment budget.

The following month, things became even worse when the number of plugboard cables increased from six to ten. Instead of twelve letters being swapped before entering the rotor scrambler, there were now twenty swapped letters, reducing the effectiveness of the bomba and increasing the number of possible plugboard settings by more than a thousandfold.

### Allies informed

As it became clear that war was imminent and Polish resources were insufficient to keep pace with the evolution of Enigma encryption (i.e., due to the Poles' difficulty in producing in time the required 60 series of "Zygalski sheets"), the Polish General Staff and government decided to let their Western allies in on the secret. The Polish methods were revealed to British and French intelligence representatives in a meeting at Pyry on 26 July 1939.

The Poles' gift of Enigma decryption to their Western allies, a month before the outbreak of World War II, came not a moment too soon. Knowledge that the cipher was crackable was a morale boost to Allied cryptanalysts. The British were able to manufacture at least two complete sets of perforated sheets—they sent one to PC Bruno, outside Paris, in mid-December 1939—and began reading Enigma within months of the outbreak of war.

Without the Polish assistance, British code-breakers would, at the very least, have been considerably delayed in reading Enigma. Author Hugh Sebag-Montefiore concludes that substantial breaks into German Army and Air Force Enigma ciphers by the British would have occurred only after November 1941 at the earliest, after an Enigma machine and key lists had been captured, and similarly Naval Enigma only after late 1942. Former Bletchley Park cryptologist Gordon Welchman goes further, writing that the Army and Air Force Enigma section, Hut 6, "would never have gotten off the ground if we had not learned from the Poles, in the nick of time, the details both of the German military ... Enigma machine, and of the operating procedures that were in use".

Intelligence gained from solving high-level German ciphers—intelligence codenamed "Ultra" by the British and Americans—came chiefly from Enigma decrypts. While the exact contribution of Ultra intelligence to Allied victory is disputed, Kozaczuk and Straszak note that "it is widely believed that Ultra saved the world at least two years of war and possibly prevented Hitler from winning. The English historian Sir Harry Hinsley, who worked at Bletchley Park, similarly assessed it as having "shortened the war by not less than two years and probably by four years". The availability of Ultra was, at the least, due largely to the earlier Polish breaking of Enigma.

## In France and Britain

### PC Bruno

In September 1939 after the outbreak of World War II, Rejewski and his fellow Cipher Bureau workers were evacuated from Poland to Romania. Rejewski, together with Zygalski and Różycki, managed to avoid being interned in a refugee camp and made their way to Bucharest, where they contacted the British embassy. Having been told by the British to "come back in a few days", they next tried the French embassy, introducing themselves as "friends of Bolek" (Bertrand's code name). The French army officer on duty called Paris for instructions and immediately had the three mathematicians evacuated to France. They arrived in Paris by the end of September.

On 20 October they resumed their work on German ciphers at a joint French-Polish-Spanish radio intelligence unit stationed at Château de Vignolles, forty kilometers northeast of Paris, code-named "PC Bruno". Enigma keys were being broken again by December 1939 or January 1940. The staff at PC Bruno collaborated by teletype with their opposite numbers at Bletchley Park in England. For communications security the allied Polish, French and British cryptological agencies used the Enigma machine itself closing Bruno's Enigma-encrypted messages to Britain with an ironic "Heil Hitler! On 24 June 1940, Bruno was disbanded after Germany's victory in the Battle of France, and Rejewski and his colleagues were evacuated to Algeria.

During September 1940 they returned to work in secret in unoccupied southern (Vichy) France. Rejewski's cover was as Pierre Ranaud, a lycée professor from Nantes. A radio intelligence station was set up at the Château des Fouzes near Uzès, code-named "Cadix". Cadix began operations on 1 October. Rejewski and his colleagues solved German telegraph ciphers, and also the Swiss version of the Enigma machine (which had no plugboard). Rejewski may have had little or no involvement in working on German Enigma at Cadix.(Note 8)

In early July 1941, Rejewski and Zygalski were asked to try solving messages enciphered on the secret Polish Lacida cipher machine, which was used for secure communications between Cadix and the Polish General Staff in London. Lacida was a rotor machine based on the same cryptographic principle as Enigma, yet had never been subjected to rigorous security analysis. The two cryptologists created consternation by breaking the first message within a couple of hours; further messages were solved in a similar way.

On 9 January 1942, Różycki, the youngest of the three mathematicians, died in the sinking of a French passenger ship as he was returning from a stint in Algeria to Cadix in southern France.

By summer 1942 work at Cadix was becoming dangerous, and plans for evacuation were drawn up. Vichy France itself was liable to be occupied by German troops, and Cadix's radio transmissions were increasingly at risk of detection by the Funkabwehr, a German unit tasked with locating enemy radio transmitters. Indeed, on 6 November a pickup truck equipped with a circular antenna arrived at the gate of the chateau where the cryptologists were operating. The visitors, however, did not enter, and merely investigated — and terrorized — nearby farms. Nonetheless, the order to evacuate Cadix was given, and this was done by 9 November. The Germans occupied the chateau only three days later.

### Escaping France

The Poles were split into twos and threes. Rejewski and Zygalski were sent to Nice on 11 November, which was in a zone occupied by the Italians. They had to flee again after coming under suspicion, constantly moving or staying in hiding, to Cannes, Antibes, Nice again, Marseilles, Toulouse, Narbonne, Perpignan and Ax-les-Thermes, close to the Spanish border.

The plan was to smuggle themselves over the Pyrenees across into Spain. Accompanied by a local guide, Rejewski and Zygalski began their trek through the Pyrenees on 29 January 1943. They avoided German and Vichy patrols, but near midnight and near the border, their guide pulled out a pistol and demanded they hand over the rest of their money. Despite being robbed, they succeeded in reaching the Spanish side of the border, only to be arrested by Spanish security police within hours. The Poles were sent first to a prison in Séo de Urgel until 24 March, then moved to a prison at Lerida. The pair were eventually released on 4 May, after the intervention of the Polish Red Cross, and sent to Madrid. Leaving Madrid on 21 July, they made it to Portugal; from there, aboard HMS Scottish, to Gibraltar; and thence, aboard an old Dakota, to Britain, arriving 3 August 1943.

### Britain

On 16 August Rejewski and Zygalski were inducted as privates into the Polish Army and employed at cracking German SS and SD hand ciphers at Boxmoor. The SS and SD ciphers were largely based on the Doppelkassettenverfahren system (a double Playfair scheme). On 10 October 1943, Rejewski was commissioned a second lieutenant, and on 1 January 1945, he was promoted to lieutenant.

Enigma decryption, however, had become an exclusively British and American domain; the two mathematicians who, with their late colleague, had laid the foundations for Allied Enigma decryption were now excluded from the opportunity of making further contributions to their métier. British code-breaker Alan Stripp suggests that by that time, at Bletchley Park, "very few even knew about the Polish contribution" because of the strict secrecy and the observance of the "need-to-know" principle. Stripp comments further that "setting them to work on the Doppelkassetten system was like using racehorses to pull wagons".

## Back in Poland

On 21 November 1946, Rejewski, having been discharged from the Polish Army in Britain, returned to Poland to be reunited with his wife, Irena Maria Rejewska (née Lewandowska, whom Rejewski had married on 20 June 1934) and their two children, Andrzej (born 1936) and Janina (born 1939, who would follow in her father's footsteps to become a mathematician). One option now open to Rejewski was to resume teaching mathematics at a university in Poznań or Szczecin, as suggested by his old Poznań University professor, Zdzisław Krygowski. Taking a university post, however, would have entailed yet another separation from his family and his elderly in-laws, with whom the Rejewskis were now living in Bydgoszcz. A grievous blow to Rejewski, too, soon after his return, was the death in summer 1947 of his 11-year-old son Andrzej from poliomyelitis. Instead, Rejewski took a position in Bydgoszcz as a supervisor in the sales section of a cable manufacturing plant, Kabel Polski (Polish Cable).

The Polish Security Service repeatedly investigated him between 1949 and 1958 but never found out about his history of success with Enigma; in 1950 they demanded that he be fired from his employment. Thereafter, he worked briefly at the State Surveying Company as a supervisor, then at the Union of Surveyors of Poland. Between 1951 and 1954 he was employed at the Branch Union of Timber and Varied Manufactures Cooperative. From 1954 until his retirement in February 1967, he worked as a bookkeeper at the Provincial Union of Labour Cooperatives.

Until this point, Rejewski had remained silent about his prewar and wartime work. Shortly after his retirement, he wrote a memoir of his work on Enigma and deposited it at the Military Historical Institute. In 1969 he moved back with his family to Warsaw. When the role of the Poles in the Enigma story emerged in 1973, Rejewski published a number of papers on his cryptologic work and contributed generously to articles, books and television programs the world over on the subject. He maintained a lively correspondence with his wartime French host, General Gustave Bertrand, author of the first book published on Enigma (1973), which Rejewski at Bertrand's suggestion began translating into Polish. A few years before his death, Rejewski broke enciphered correspondence of Józef Piłsudski and his fellow Polish Socialist conspirators from 1904. On 12 August 1978, he received the Officer's Cross of the Order of Polonia Restituta. Rejewski, who had been suffering for some time from heart disease, died of a heart attack at his home on 13 February 1980, aged 74, and was buried with military honors at Warsaw's Powązki Cemetery.

The story of Rejewski and his co-workers has been celebrated both in Poland and abroad. He was decorated with a number of Polish medals, both before and after World War II, and after the Enigma story had become public knowledge. In 2000, Rejewski and his colleagues Zygalski and Różycki were posthumously awarded the Grand Cross of the Order of Polonia Restituta. On 4 July 2005, Rejewski's daughter received the War Medal 1939–1945 on his behalf, presented by the British Chief of the Defence Staff.

In 1979 Rejewski and his colleagues became the heroes of Sekret Enigmy ("The Enigma Secret"), a Polish-produced movie thriller documenting, with a few embellishments, the Polish solution of Enigma. Shortly afterwards a Polish TV series was produced with a similar theme, Tajemnice Enigmy ("The Secrets of Enigma"). In 1983, a Polish postage stamp marked the 50th anniversary of the German military Enigma's first solution. Memorials to the trio have been unveiled at Bletchley Park and the Polish Embassy in the UK, and at Uzès in France. In Rejewski's home city of Bydgoszcz, a street and school have been named for him, a plaque placed on the building where he had lived, and a sculpture commissioned (pictured above, left). In 2005 a postcard (below, center) was issued on the 100th anniversary of Rejewski's birth.

## Notes

1. The exact extent of the contribution of Ultra to Allied victory is debated. The typical view is that Ultra shortened the war; Supreme Allied Commander Dwight D. Eisenhower called Ultra "decisive" to Allied victory. For a fuller discussion, see Ultra's strategic consequences.
2. Bydgoszcz (called "Bromberg" in German) was then part of the Prussian Province of Posen. Bydgoszcz — which had been seized by Prussia in the 1772 First Partition of the Polish-Lithuanian Commonwealth — returned to Poland in 1919 after the Greater Poland Uprising.
3. An early Naval Enigma model (the "O Bar" machine) had been solved before 1931 by the Polish Cipher Bureau, but it did not have the plugboard of the later standard Enigma. Mahon cites, as his source for "most of the information I have collected about prewar days", Alan Turing, who had received it from the "Polish cryptographers", who Mahon says had done "nearly all the early work on German Naval Enigma [and] handed over the details of their very considerable achievements just before the outbreak of war."
4. Some writers, after Bloch (1987), argue that Rejewski is more likely to have received these documents in mid-November 1932, rather than 9/10 December.
5. Lawrence (2004) shows how Rejewski could have adapted his method to solve for the second rotor, even if the settings lists had not straddled the quarterly changeover period.
6. More Enigma settings were provided to the Polish Cipher Bureau by French Intelligence, but these were never passed on to Rejewski and his colleagues. A possible explanation for this is that the Poles wished to remain independent of French assistance for reading Enigma, and without outside help the cryptologists were forced to develop their own self-sufficient techniques.
7. The Navy had already changed its Enigma indicator procedure on 1 May 1937. The SD net, which lagged behind the other services, changed procedure only on 1 July 1939.
8. Rejewski later wrote that at Cadix they did not work on Enigma. Other sources indicate that they had, and Rejewski conceded that this was likely the case. Rejewski's correspondent concluded that "Rejewski either had forgotten or had not known that, e.g., Zygalski and Różycki had read Enigma after the fall of France".

## Bibliography

• Gustave Bertrand, Enigma ou la plus grande énigme de la guerre 1939–1945 (Enigma: the Greatest Enigma of the War of 1939–1945), Paris, Librairie Plon, 1973.
• Gilbert Bloch, "Enigma before Ultra: Polish Work and the French Contribution", translated by C.A. Deavours, Cryptologia, July 1987, pp. 142–155.
• Zbigniew Brzezinski, "The Unknown Victors". pp.15–18, in Jan Stanislaw Ciechanowski, ed. Marian Rejewski 1905–1980, Living with the Enigma secret. 1st ed. Bydgoszcz: Bydgoszcz City Council, 2005, ISBN 83-7208-117-4.
• Stephen Budiansky, Battle of Wits: the Complete Story of Codebreaking in World War II, New York, The Free Press, 2000.
• Chris Christensen, "Polish Mathematicians Finding Patterns in Enigma Messages", Mathematics Magazine, 80 (4), October 2007.
• James Gannon, Stealing Secrets, Telling Lies: How Spies and Codebreakers Helped Shape the Twentieth Century, Washington, D.C., Brassey's, 2001, ISBN 1-57488-367-4, pp. 27-58 and passim.
• I. J. Good and Cipher A. Deavours, afterword to: Marian Rejewski, "How Polish Mathematicians Deciphered the Enigma", Annals of the History of Computing, 3 (3), July 1981. (This paper of Rejewski's appears as Appendix D in Kozaczuk, 1984.)
• F.H. Hinsley and Alan Stripp, eds., Codebreakers: The Inside Story of Bletchley Park, Oxford University Press, 1993, ISBN 0-19-820327-6.
• Stanisław Jakóbczyk and Janusz Stokłosa, editors, Złamanie szyfru Enigma. Poznański pomnik polskich kryptologów (The Breaking of the Enigma Cipher: the Poznań Monument to the Polish Cryptologists), Poznań, Wydawnictwo Poznańskiego Towarzystwa Przyjaciół Nauk, 2007, ISBN 978-83-7063-527-5. This 140-page book was published in connection with the 2007 dedication, before the Poznań Castle, of a three-sided bronze monument, each side bearing the name of one of the three Polish mathematician-cryptologists who attended the cryptology course there and subsequently collaborated on breaking the Enigma cipher. The volume recounts the history of the cipher's breaking before and during World War II and the importance of this achievement in the prosecution of the war, provides brief biographies of a number of Interbellum Poznań mathematicians, and includes photographs of documents and of a growing number of Enigma-decryption-related memorials to be found in various Polish locales.
• David Kahn, The Codebreakers: The Comprehensive History of Secret Communication from Ancient Times to the Internet, 2nd edition, New York, Scribner, 1996, ISBN 0684831309.
• David Kahn, Seizing the Enigma: the Race to Break the German U-Boat Codes, 1939-1943, Boston, Houghton Mifflin, 1991, ISBN 0-395-42739-8.
• Władysław Kozaczuk, Enigma: How the German Machine Cipher Was Broken, and How It Was Read by the Allies in World War Two, edited and translated by Christopher Kasparek, Frederick, MD, University Publications of America, 1984, ISBN 0-89093-547-5. (The standard reference on the Polish part in the Enigma-decryption epic. This English-language book is substantially revised from Kozaczuk's 1979 Polish-language W kręgu Enigmy, and greatly augmented with documentation, including many additional substantive chapter notes and papers by, and interviews with, Marian Rejewski.)
• Władysław Kozaczuk, "A New Challenge for an Old Enigma-Buster", Cryptologia, 14 (3), July 1990.
• Jerzy Kubiatowski, "Rejewski, Marian Adam", Polski słownik biograficzny (Polish Biographical Dictionary), vol. XXXI/1, Wrocław, Wydawnictwo Polskiej Akademii Nauk (Polish Academy of Sciences), 1988, pp. 54–56.
• John Lawrence, "A Study of Rejewski's Equations", Cryptologia, 29 (3), July 2005, pp. 233–247.
• John Lawrence, "The Versatility of Rejewski's Method: Solving for the Wiring of the Second Rotor", Cryptologia, 28 (2), April 2004, pp. 149–152.
• John Lawrence, "Factoring for the Plugboard — Was Rejewski's Proposed Solution for Breaking the Enigma Feasible?", Cryptologia, 29 (4), October 2005.
• A.P. Mahon, "The History of Hut Eight: 1939–1945", June 1945, 117 pp., PRO HW 25/2,
• A. Ray Miller, "The Cryptographic Mathematics of Enigma", 2001,
• Wojciech Polak, "Marian Rejewski in the Sights of the Security Services," in Jan Stanisław Ciechanowski, ed., Marian Rejewski, 1905–1980: Living with the Enigma Secret, Bydgoszcz: Bydgoszcz City Council, 2005, ISBN 83-7208-117-4, pp. 75–88.
• Marian Rejewski, "An Application of the Theory of Permutations in Breaking the Enigma Cipher", Applicationes Mathematicae, 16 (4), 1980, pp. 543–559 (PDF)
• Marian Rejewski, interview in: Richard Woytak, Werble historii (History's Drumroll), edited by and with introduction by Stanisław Krasucki, illustrated with 36 photographs, Bydgoszcz, Poland, Związek Powstańców Warszawskich w Bydgoszczy (Association of Warsaw Insurgents in Bydgoszcz), 1999, ISBN 83-902357-8-1.
• Hugh Sebag-Montefiore, Enigma: the Battle for the Code, London, Weidenfeld and Nicolson, 2000.
• Simon Singh, The Code Book: the Evolution of Secrecy from Mary Queen of Scots to Quantum Cryptography, Doubleday, 1999, pp. 149–160, ISBN 0-385-49531-5.
• Alan Stripp, "A British Cryptanalyst Salutes the Polish Cryptanalysts", Appendix E in: Władysław Kozaczuk and Jerzy Straszak, Enigma: How the Poles Broke the Nazi Code, New York, Hippocrene Books, 2004, ISBN 0-7818-0941-X, pp. 123-25.
• Gordon Welchman, The Hut Six Story: Breaking the Enigma Codes, New York, McGraw-Hill, 1982.
• Gordon Welchman, "From Polish Bomba to British Bombe: the Birth of Ultra", Intelligence and National Security, 1 (1), January 1986.
• Fred B. Wrixon, Codes, Ciphers, & Other Cryptic & Clandestine Communication: Making and Breaking Secret Messages from Hieroglyphics to the Internet, 1998, Black Dog & Leventhal Publishers, ISBN 1-57912-040-7, pp. 83–85.