WikiPortallus: Cryptanalysis of the Enigma.html

The cryptographic key for the Vigenère cipher consists of a word or phrase that is repeated many times, to cover the length of the message. The letters of the key indicate which line of the Vigenère Square is used to encipher each letter of the plaintext to produce the ciphertext. It was this repetition that allowed Babbage and Kasiski to achieve their break. Towards the end of the First World War, American scientists realised that a purely random key sequence, containing no repetitive pattern, would make a polyalphabetic substitution unbreakable.⁴ This led to the development of rotor cipher machines such as Scherbius' Enigma.

Repetition or guessable text in either the message or the key are the weaknesses that allow cryptanalysts to seek patterns which lead them into breaking a cipher. These techniques were used by those who broke Enigma ciphers.

Strengths of Enigma

Enigma was designed to defeat basic cryptanalytic techniques by continually changing the substitution alphabet by means of a scrambler. Like other rotor machines, it implemented a polyalphabetic substitution cipher with a long period. With single-notched rotors, the period of the machine was 17,576 (26 × 26 × 26).⁵ This long period prevented repetition in the enciphering sequence. It was achieved by each rotor of the scrambler having a variable starting position (any letter of the alphabet) and the sequence of the three rotors being changed (e.g. I, III, II, or II, I, III or II, III, I); later Enigma models added a variable alphabet ring like a tyre around each rotor, which specified which letter was opposite the notch that caused the next wheel to advance. Later still, the three rotors used were selected from a set of five, or, in the case of the German navy, eight.

Most of the military Enigmas also featured a plugboard (German: Steckerbrett) which exchanged letters. Even so, this complex combination ground key setting (German: Grundstellung) was easily distributed to all users in a network by means of "setting sheets" (codebook).⁶ These specified the letter positions of the three rotors, the rotors to be used and their order (German: Walzenlage), the ring positions (German: Ringstellung) and the plugboard connections (German: Steckerverbindungen ). Potentially this made the Enigma an excellent system.

The most common versions of the Enigma had the major operational convenience of being symmetrical (or self-reciprocal). This meant that decipherment worked in the same way as encipherment — when the ciphertext was typed in, the sequence of lamps lit yielded the plaintext. This did, of course, require that the deciphering machine's plugboard and scrambler rotors were set identically to those of the enciphering machine. These ground keys of plugboard and scrambler rotor settings were changed regularly (at first monthly or weekly, but soon daily, as specified on the settings sheets; and even, toward war's end in some networks, many times a day).

Security properties

The various Enigma models provided different levels of security. The presence of a plugboard substantially increased the security of the encipherment. In general, the unsteckered Enigma was used for commercial and diplomatic traffic and could be broken relatively easily using hand methods, while attacking versions with a plugboard was much more difficult. The British read unsteckered Enigma messages sent during the Spanish Civil War,⁷ and also read some Italian traffic enciphered early in World War II (see Ultra).

The Enigma machine did, however have two major weaknesses that proved helpful to cryptanalysts. First, a letter could never be encrypted to itself (with the exception of the early models A and B, which lacked a reflector). This was of great help in using cribs — short sections of plaintext that are known (or suspected) to be somewhere in the ciphertext. This property can be used to eliminate a crib in a particular position. For a possible location, if any letter in the crib matches a letter in the ciphertext at the same position, the location can be ruled out; this was termed a "crash" at the British Government Code and Cipher School at Bletchley Park.⁸ It was this feature that the British mathematician and logician Alan Turing exploited in designing the Bombe.

A second weakness was that the plugboard connections were reciprocal, so that if A was plugged to N, then N likewise became A. This property was used by Gordon Welchman at Bletchley Park in the "diagonal board" of the Bombe, which substantially reduced the number of rotor settings that it had to try.⁹ A third weakness for many Enigma models was that the rightmost rotor turned a constant number of places before the next rotor turned.

A number of the officially-specified procedures for using Enigma also provided avenues for attack. For example, for the machines where there was a choice of more rotors than there were slots for them, there was a rule that no rotor should be in the same slot in the scrambler as it had been for the immediately preceding configuration. Also, the plugboard setup rules forbade a letter being connected to an adjacent one on the alphabet. Once detected, these constraints reduced the number of alternatives that needed to be tried. The specified operating procedures and good cryptological practice were not, however, adhered to by all the Enigma operators.

It has been suggested by some of those working on its cryptanalysis at Bletchley Park that the Enigma should have been unbreakable in practice had its operating procedures been better thought out, and had its operators been less ill-disciplined. Post-war debriefings of German cryptographic specialists, conducted as part of project TICOM, tend to support this view — German cryptographers were well aware that Enigma was theoretically breakable, but felt that the resources required to mount a pure brute-force attack on the system would require too much effort to be worthwhile. Had they considered the potential consequences of widespread poor operator procedure, and acted accordingly, it is likely that breaking Enigma on a regular basis would have proved impractical. Up to the end of the war, various enhancements were made to the system, although the Germans considered it unbreakable for all practical purposes.

Polish breakthrough

In 1928, the German Army began using a 3-rotor Enigma with a 6-cable plugboard.¹⁰ British, French and American cryptanalysts had no success in their attempts to crack this version of Enigma. In Poland, however, the threat from Germany was much greater, and the Polish Cipher Bureau (Biuro Szyfrów) continued to work on it. In September 1932, a 27-year-old Polish mathematician, Marian Rejewski joined the Bureau. In December of that year the Poles received two documents from their French allies through a French military intelligence agent named Rex who had obtained them from an agent in Berlin (Hans Thilo-Schmidt, codenamed Asché by the French). These were entitled Gebrauchsanweisung für die Chiffriermaschine Enigma and Schlüsselanleitung für die Chiffriermaschine Enigma and provided instructions for using Enigma, and sufficient information to start to deduce the wirings of the three rotors and build an Enigma double (a functioning copy of a 3-rotor Enigma).¹¹ This enabled Rejewski to make one of the most important breakthroughs in cryptological history by using algebraic mathematical techniques to solve the Enigma wiring and rotor setting.¹² He worked out a method which enabled him to derive the rotor settings independently of the plugboard connections. The Poles were then able to decrypt a large portion of German Enigma traffic from December 1932 to December 1938.

At the time, the indicator, (rotor set-up) procedure for each message, was to send a 9-letter sequence. The machine was first set to its ground setting as detailed in the setting sheet. The first three of the nine letters were selected by the operator and sent to give the key for setting the rotors for the next six letters. This rotor setting, called the "indicator setting" at Bletchley Park, would be used to encrypt an operator-selected 3-letter key which would then be sent twice — the "indicator" in Bletchley Park terminology.¹³ For instance, if an operator picked QRS as their "message setting", the operator would set the machine to the day's ground settings, type a 3-letter key, re-set the rotors to that setting and then type QRSQRS.¹⁴ This might be encrypted as JXDRFT. The feature of Enigma that Rejewski exploited was that the rotor moved three positions between the two sets of QRS — knowing that J and R were originally the same letter, as were XF and DT, was vital information. Although the original letters were unknown, it was known that, while there were a huge number of rotor settings, there were only a small number of rotor settings that would change a letter from J to R, X to F and D to T, and so on. Rejewski called these patterns chains. The chain patterns were independent of the plugboard settings, which reduced the number of options to be tried from 10,000 trillion to 105,456.¹⁵

Finding the matching chains from the many possible rotor positions and settings, was a tremendous task. The Poles, particularly Rejewski's classmates Jerzy Różycki and Henryk Zygalski, developed a number of methods. The British had independently developed a similar technique when they succeeded in breaking the unsteckered Enigma. Initially Rejewski compiled an index of the chain patterns from different scrambler settings, but later he and his colleagues developed a machine to do this — the Cyclometer.¹⁶

Polish Bomba

Analysis of the 105,456 possible rotor settings represented a vast human effort, if done by hand. To help with this, Rejewski in about October 1938 invented an electro-mechanical device which was dubbed the Bomba kryptologiczna or "cryptologic bomb": the name possibly originating from the characteristic muffled noise it produced when operating; alternative names puckishly given the device by Polish Cipher Bureau personnel were "washing machine" and "mangle." In mid-November 1938 the Polish bomby¹⁷ were ready, and reconstruction of daily keys went on apace. Rejewski has written about the device: "The bomb method, invented in the fall of 1938, consisted largely in the automation and acceleration of the process of reconstructing daily keys. Each cryptologic bomb (six were built in Warsaw for the Cipher Bureau before September 1939) essentially constituted an electrically powered aggregate of six Enigmas. It took the place of about one hundred workers and shortened the time for obtaining a key to about two hours." ¹⁸

In December 1938 the German Army increased the complexity of its Enigma operating procedures. Initially only three rotors had been in use, and their sequence in the slots was changed periodically. Now two additional rotors were introduced; three of the five would be in use at any given time. This increased the number of possible rotor arrangements in the scrambler by a factor of ten, effectively "blacking out" Polish reading of Enigma messages. In January 1939 the number of plugboard cables was increased from six to ten.¹⁹ As Rejewski wrote in a 1979 critique of appendix 1, volume 1 (1979), of the official history of British Intelligence in the Second World War, "we quickly found the [wirings] within the [new rotors], but [their] introduction [...] raised the number of possible sequences of drums from 6 to 60 [...] and hence also raised tenfold the work of finding the keys. Thus the change was not qualitative but quantitative. We would have had to markedly increase the personnel to operate the bombs, to produce the perforated sheets ..."

World War II

Polish intelligence had been reading Enigma-generated messages since December 1932. Subsequent modifications in the machine and its operating procedures caused periodic "blackouts" requiring the Poles (and, after July 1939, also the British and French) to find new ways of breaking into the ciphers. On 27 April 1939, Germany withdrew from its non-aggression treaty with Poland. The Poles, realizing the pace and direction of changes in the European political situation, decided in mid-1939 to share their work. At a conference in Warsaw on 26 July 1939,²⁰ they revealed to the French and British that they had broken Enigma and they pledged to give them each a Polish-reconstructed Enigma, along with details of Enigma-solving techniques that they had developed, such as Zygalski's perforated sheets "Zygalski sheets" and the "cryptologic bomb" (Polish: bomba kryptologiczna). The two "Enigma doubles" were shipped to Paris, whence Gustave Bertrand brought one to London for the British, turning it over at Victoria Station, as he was to recall in his Enigma, to Stewart Menzies of Britain's Secret Intelligence Service. Until then, German military Enigma traffic had defeated the British, French and Americans, and they had faced the disturbing prospect that German communications would remain "black" to them for the duration of the coming war.

During the German invasion of Poland in September 1939, key Cipher Bureau personnel were evacuated southeastward and — after the Soviets invaded eastern Poland on 17 September — into Romania, on the way destroying their cryptological equipment and documentation. Eventually, crossing Yugoslavia and still-neutral Italy, they reached France. There, at PC Bruno outside Paris, they resumed their work on breaking German Enigma ciphers, continuing it into the subsequent Battle of France.

British codebreakers at Bletchley Park gained greatly from the knowledge that the Poles had broken Enigma, but had to remain alert to German cryptographic advances. The Government Code and Cipher School (GC&CS), before its move to Bletchley Park, had realised the value of recruiting mathematicians and logicians to work in codebreaking teams. In 1938, Alan Turing had started to work for GC&CS on a part-time basis.²¹ Gordon Welchman, another mathematician from Cambridge University, also received initial training in 1938.²² Both he and Turing then reported to Bletchley Park on 4 September 1939, the day after Britain declared war on Germany.

Alan Turing recognised that the Germans would be likely to stop using the insecure method of sending the message key twice and determined to find an alternative method of cracking Enigma traffic. He decided to try to take responsibility for the German Naval Enigma, as no one else was looking into it. This was because they considered that the superior operator discipline and method of conveying the chosen key for the day, rendered the task very much more difficult. Turing, however, diagnosed the system, but was not able to decrypt the traffic on a regular basis. The first break into Enigma traffic was performed by him in December 1939, but it was into some German naval signals intercepted in November 1938.

In January 1940, Alan Turing spent several days at Bruno conferring with his Polish mathematician colleagues. He took with him a large number of the "Zygalski sheets" which were based on the Polish-supplied information, but were not working well for the British. It turned out that the wiring of the fourth and fifth Enigma rotors that Rejewski had worked out, had been copied down incorrectly. Correcting this error allowed the Poles to make, on 17 January 1940, the first break into wartime Enigma traffic — from 28 October 1939.²³

On 10 May 1940 the Germans stopped transmitting a repeated message key, thus putting an end to the Poles' original method of cryptological attack.

After the French-German armistice, the Polish Cipher Bureau continued its work in France's southern "Free Zone" and in French Algeria, at constant risk of discovery and imprisonment or worse. When Germany took over Vichy France in November 1942, the Poles once again had to flee. The Cipher Bureau's chiefs, Colonel Gwido Langer and Major Maksymilian Ciężki, and some of the technical staff were captured by the Germans but, despite extensive interrogation, managed to preserve the secret of Enigma decryption. The mathematicians Marian Rejewski and Henryk Zygalski, after a perilous odyssey that took them across France, into a Spanish prison, to Portugal and at last by ship to Gibraltar, finally made it to Britain. (The third mathematician, Jerzy Różycki, had perished in the sinking of a passenger ship while returning in 1942 to southern France from a tour of duty in Algeria.) In Britain, Rejewski and Zygalski were inducted as privates into the Polish Army. Eventually they were promoted to second lieutenant, then lieutenant, and put to work breaking German SS and SD ciphers at a Polish signals facility in Boxmoor; they were not invited to work on Enigma at Bletchley Park.

Apart from the less-than-ideal inherent characteristics of the machine, the way Enigma was used proved its greatest weakness in practice. Errors by German Army and Air Force Enigma-machine operators were common and the Poles had become very experienced in exploiting even very subtle cryptological mistakes that the Germans made. A blatant one had been the printing of a complete set of plaintext-key-ciphertext as a training example in an early Enigma manual, a copy of which the Bureau had obtained. Another was the use of easily-guessed key sequences, which became known as "cillies", apparently after an operator who used CILLIE for keys, presumably the nickname of his wife or girl friend. One person responsible for preparing the settings sheets, re-used some of the columns of wheel orders, ring settings or plugboard connections from previous months.²⁴ This was named "Parkerismus" after Reg Parker who had spotted it.

Laziness in the operator choice of keys gave rise to the Herivel tip. The Herivel tip, suggested by John Herivel, was an effect that relied on operators being lazy and failing to choose random rotor positions for their indicators after changing the rotor ring settings, effectively sending the ring settings almost in the clear.

Sources of Cribs

The word "crib" was coined at Bletchley park for some plaintext that was known, or hoped, to be present at some point in the enciphered message. This was a fundamental part of the British approach to breaking Enigma.

In one instance an operator was asked to send a test message, and simply hit the T key repeatedly and sent the resulting letters. A British analyst received from the intercept stations a long message without a single T in it, and immediately realised what had happened. Some Enigma operators used "form letters" for daily reports, notably weather reports, in which case the same crib might be used every day. In another common operational fault an entire source message (e.g. a weather forecast intended for submarines) would be re-sent after a change of settings which gave additional advantage to the codebreakers.

When a captured and interrogated German revealed that Enigma operators had been instructed to encode numbers by spelling them out, Alan Turing reviewed decrypted messages, and determined that the number “eins” (1) appeared in 90% of messages. He automated the crib process, creating the Eins Catalogue, which assumed that “eins” was encoded at all positions in the plaintext. The catalogue included every possible position of the various rotors, starting positions, and key settings of the Enigma.

British Bombe

One important approach to breaking the ciphers relied on the fact that the reflector (a patented feature of the Enigma machines) guaranteed that no letter could be enciphered as itself. This was combined with knowledge of common German phrases such as "Heil Hitler" or "please respond," which might occur frequently in certain plaintexts; such a successful guess at a plaintext was known at Bletchley as a crib. With a probable plaintext fragment and the knowledge that no letter could be enciphered as itself, a corresponding ciphertext fragment could often be guessed by trying every possible alignment of the crib against the ciphertext, a procedure known as crib dragging. Out of the possible guesses, some would turn out to be true plaintext-ciphertext pairs. This provided a clue to message settings.

The British bombe, designed by Alan Turing and Gordon Welchman, relied on cribs. Assume that a triple loop is found, e.g. abc. That means that, with a crib, plaintext letter a is mapped to cipher b, plain b to c, and plain c to cipher a again within a short distance (ideally plain: abc, cipher: bca). Now the rotor mechanisms of three Enigmas are assembled serially in-line and set to the original rotor positions, with their offset (here 1 step each) accordingly. Then a corresponding physical wire closed loop is obtained. This can be detected with lamps connected to the rotor contacts. The lamp in the wire loop will stay dark. Now the rotor systems are turned synchronously. If only one lamp stays dark because of the one wire loop, the Steckerfeld (Plug Field) may be quickly calculated, and the positions with all lamps lit rejected. This typically happens several times in the 17,576 possible rotor settings.

Naval Enigma

Kriegsmarine procedures were much more secure, and the Navy Enigma variant featured a set of eight rotors from which the three operating ones were selected. This meant that there were 336 possible rotor combinations alone. Bletchley Park made no useful headway into Kriegsmarine Enigma until mid-1940 with the capture of the armed trawler, Polares. The latter yielded enough intact cryptographic material that by June or July 1940, Hut 8 at least knew what content to expect in Kriegsmarine messages, and knew the details of the encipherment and decipherment procedures. However, the 336 possible rotor selections, together with a lack of usable cribs, made the usual cryptanalysis methods almost useless.

Hut 8 therefore developed "Banburismus," a method using Bayesian statistics to derive a bombe menu from the "message settings" rather than the messages themselves. In doing so, they would identify at least the rightmost rotor being used in the cipher that day. If Hut 8 were lucky, they managed to identify the rightmost and middle rotors, leaving only six wheel orders to be run on the bombes.

Later in the war, British codebreakers learned to fully exploit a crucial security flaw associated with German weather reports: they were broadcast from weather ships to Germany in lower-level ciphers, easy to decrypt, then retransmitted to U-boats at sea in Enigma, thus giving Bletchley Park regular cribs. This was crucial in attacking the special four-rotor U-boat Enigma machine introduced in 1942.

Cipher material was captured at sea. The first capture of Enigma material occurred in February 1940, when rotors VI and VII, the wiring of which was at that time unknown, were captured from the crew of U-33. On 7 May 1941, the Royal Navy captured a German weather ship, together with cipher equipment and codes. They did it again shortly afterwards. And two days later U-boat U-110 was captured, complete with Enigma machine, settings book, operating manual and other information. As a result, Naval Enigma was readable directly through the end of June, and from then on Banburismus allowed it to be read fairly continuously until newer, faster Bombes rendered the procedure unnecessary in mid-1943.

In addition to U-110, Naval Enigma machines or settings books were captured from a total of seven U-boats and eight German surface ships, including U-boats U-505 (1944) and U-559 (1942), two German weather-reporting trawlers, and a small vessel (the Krebs) captured during a raid on the Lofoten Islands off Norway. Several other imaginative techniques were dreamed up, including Ian Fleming's suggestion to crash captured German bombers into the sea near German ships, hoping the planes' crews would be rescued by the ships' crews, which would then be taken captive, along with the ships' cryptographic materials, by commandos concealed in the planes.

American Bombe

German suspicions

By 1945, almost all German Enigma traffic (Wehrmacht, Kriegsmarine, Luftwaffe, Abwehr, SD, etc.) could be decrypted within a day or two, yet the Germans remained confident of its security. They considered Enigma traffic sufficiently secure that they openly discussed their plans and movements, handing the Allies huge amounts of information, not all of which was properly used. For example, Rommel's actions at the Kasserine Pass were clearly foreshadowed in decrypted Enigma traffic, but the information was not properly appreciated.

After the war, American TICOM project teams found and detained a considerable number of German cryptographic personnel. Among the things the Americans learned was that German cryptographers, at least, understood very well that Enigma messages might be read; they knew Enigma was not unbreakable. They just found it impossible to imagine anyone going to the immense effort required. ²⁵ When Abwehr personnel who had worked on Fish cryptography and Russian traffic were interned at Rosenheim around May 1945, they were not at all surprised that Enigma had been broken, only that someone had mustered all the resources in time to actually do it. Admiral Dönitz had been advised that that was the least likely of all security problems.

Since World War II

Modern computers can be used to solve Enigma, using a variety of techniques.²⁶ There is even a project to decrypt some remaining messages,²⁷ using distributed computing.

Notes

^ Singh (1999) p.17
^ Singh (1999) pp. 45-51
^ Singh (1999) pp.63-78
^ Singh (1999) p. 116
^ Or 16,900 (= 26 × 25 × 26) for those machines with "double stepping" of the second rotor
^ Sale, Tony, Military Use of the Enigma: The Message Key and Setting Sheets, Codes and Ciphers in the Second World War: The history, science and engineering of cryptanalysis in World War II, http://www.codesandciphers.org.uk/enigma/enigma3.htm, retrieved on 21 October 2008
^ Smith (2006) p. 23
^ Mahon, Patrick (2004), "History of Hut 8 to December 1941", in Copeland, B. Jack, The Essential Turing: Seminal Writings in Computer Logic, Philosophy, Artificial Intelligence and Artificial Life Plus the Secrets of Enigma, Oxford: Oxford University Press, p. 302, ISBN 9780 198250807, http://books.google.co.uk/books?id=x7mMr4twnloC&pg=PA302&lpg=PA302&dq=crib+crash+cipher&source=web&ots=QXjU9n7KXE&sig=b1ekd3e-_yTNMML5tvXuuFxxKuE&hl=en&sa=X&oi=book_result&resnum=3&ct=result
^ Welchman (1984) pp. 301-307
^ Wilcox (2001) p. 2
^ Singh (1999) p.146
^ Wilcox (2001) p. 5
^ Welchman (1984) p. 36 and p. 60
^ Wilcox (2001) p. 6
^ Singh (1999) p. 153
^ Welchman (1984) p. 15
^ Bomby is the plural of bomba
^ Rejewski, Marian (1984), Kozaczuk, W, ed., Enigma: How the German Machine Cipher Was Broken, and How It Was Read by the Allies in World War Two, University Publications of America, London: Arms and Armour Press, p. 290
^ Singh (1999) p. 157
^ Ralph Erskine, "The Poles Reveal their Secrets: Alastair Denniston's Account of the July 1939 Meeting at Pyry", pp. 294-305, Cryptologia 30(4), December 2006
^ Hodges, Andrew (1995), Part 4: The Second World War, Alan Turing: a short biography, http://www.turing.org.uk/bio/part4.html, retrieved on 23 October 2008
^ Welchman (1984) p. 11
^ Smith (2006) p. 27
^ Welchman (1984) p. 167
^ Bamford, J. (2001). Body of Secrets, Doubleday. p.17. ISBN 0-385-49907-8.
^ Sullivan, Geoff; Weierud, Frode (July 2005), "Breaking German Army Ciphers", Cryptologia 24 (3): 193-232, http://www.tandf.co.uk/journals/pdf/papers/ucry_06.pdf, retrieved on 16 October 2008
^ M4 Message Breaking Project, http://www.bytereef.org/m4_project.html, retrieved on 16 October 2008

References

Stephen Budiansky, Battle of Wits: the Complete Story of Codebreaking in World War II, 2002, ISBN 0-7432-1734-9.
Jim DeBrosse and Colin Burke, The Secret in Building 26: The Untold Story of America's Ultra War against the U-boat Enigma Codes, 2004, ISBN 0-375-50807-4.
Kris Gaj, Arkadiusz Orłowski, Facts and Myths of Enigma: Breaking Stereotypes, EUROCRYPT 2003: 106–122. Online version (PDF).
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.
James J. Gillogly, "Ciphertext-only Cryptanalysis of Enigma," Cryptologia, 19 (4), 1995, pp. 405–412. Online version.
David Kahn, Seizing the Enigma: the Race to Break the German U-Boat Codes, 1939-1943, 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. (A substantially revised and augmented translation of W kręgu enigmy, Warsaw, Książka i Wiedza, 1979, supplemented with additional appendices by Marian Rejewski.)
Władysław Kozaczuk, Jerzy Straszak, Enigma: How the Poles Broke the Nazi Code, Hippocrene Books, 2004, ISBN 0-7818-0941-X. (Largely an abridgment of Kozaczuk's 1984 Enigma, minus Rejewski's appendices, here supplanted with appendices by other authors.)
A. Ray Miller, The Cryptographic Mathematics of Enigma, 2001, [1].
Hugh Sebag-Montefiore, Enigma: The Battle for the Code, John Wiley, 1984.
Marian Rejewski, "An Application of the Theory of Permutations in Breaking the Enigma Cipher," Applicationes mathematicae, 16(4), 1980. Online version (PDF).
Alan M. Turing, "Treatise on Enigma" (parts online, PDF): [2]
Gordon Welchman The Hut Six Story: Breaking the Enigma codes, M & M Baldwin, 3rd Edition, 1997, ISBN 0-947712-34-8
Gordon Welchman, From Polish Bomba to British Bombe: the Birth of Ultra, Intelligence and National Security, 1986.
Gilbert Bloch, "Enigma before Ultra: Polish Work and the French Contribution," translated by C.A. Deavours, Cryptologia, July 1987.
Zbigniew Brzezinski, "The Unknown Victors," pp.15–18 in Jan Stanisław Ciechanowski, ed., Marian Rejewski, 1905–1980: Living with the Enigma secret, Bydgoszcz, Bydgoszcz City Council, 2005, ISBN 83-7208-117-4.
Welchman, Gordon (1984). The Hut Six Story: Breaking the Enigma Codes. Harmonsworth, England: Penguin Books.
Wilcox, Jennifer E (2001), About the Enigma, National Security Agency, http://www.nsa.gov/publications/publi00021.pdf, retrieved on 13 October 2008
Singh, Simon (1999), The Code Book: The Science of Secrecy from Ancient Egypt to Quantum Cryptography, London: Fourth Estate, pp. 143-189, ISBN 1-85702-879-1
Smith, Michael (2006), "How it began: Bletchley Park Goes to War", in Copeland, B Jack, Colossus: The Secrets of Bletchley Park's Codebreaking Computers, Oxford: Oxford University Press, ISBN 978-0-19-284055-4

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