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Breaking the Enigma cipher

BOMBE was the name of an electro-mechanical machine, developed during WWII by Alan Turing and Gordon Welchman, whilst working as codebreakers at Bletchley Park. It was used to help breaking the German Enigma codes and was (partly) based on the so-called BOMBA, an earlier machine developed by Polish mathematicians in 1938. From 1943 onwards, an improved version of the 4-wheel BOMBE was built in the US by the US Navy and, separately, by the US Army.
The Polish Bomba
The Poles were the first to break the military variant of the Enigma in 1932. At section BS-4 of Biuro Szyfrów (Polish Cipher Bureau), three young Polish mathematicians, Marian Rejewski, Henryk Zygalski and Jerzy Rózyki, had recovered the wiring of the military Enigma machine.
Marian Rejewski Jerzy Růzycki Henryk Zygalski

Their success was based on pure mathematical analysis, assisted by information from a German spy by the name of Hans-Thilo Schmidt (codename Asche), and a commercial Enigma intercepted in the Polish mail. They then bought a commercial Enigma and used the gathered information to convert it into a military one. This converted machine is commonly known as the Polish Enigma.
From 1930 onwards, the Germans had been using a very simple key management scheme, in which the randomly chosen message key was sent twice in encrypted form at the beginning of each message. This allowed the daily key to be recovered relatively easy with hand methods.

On 1 October 1936, the Germans stopped using six Steckern and started using five to eight of them. This defeated the hand methods and the Poles thought out a catalog attack based on the cycle length of the indicator permutations. Rejewski subsequently developed a machine to assist in building the catalog: the Cyclometer.
Rejewski's Cyklometr. Image supplied by CODEBREAKERS.EU [10]. Click for further information.

The catalog contained the length and number of cycles for all 17,576 positions of the wheels for any possible wheel order. As there were three wheels at the time, there were six possible wheels orders (3 x 2 x 1), giving a total of 105,456 catalog entries. Building the catalog took well over a year, but when it was ready, it allowed the keys to be obtained in about 15 minutes.
On 1 November 1937, the Germans replaced the existing reflector (UKW A) for a new one (UKW B), and the Poles had to build a completely new catalog, which would probably take another year.

Then, on 15 September 1938, the Germans entirely changed the procedure for enciphering the message keys, and the catalog method had become completely useless. This prompted the Poles to come up with new solutions, such as the Zygalski Sheets and the Bomba Kryptologiczna (crypto­logic bomb), often abbreviated to Bomba.

The Bomba is based on the principle that the random 3-letter message key is sent twice at the beginning of each message and that every now and then, a particular plaintext letter, yields the same ciphertext letter three positions further on.

As an example, assume the indicator is AWB TWY. We see that the letter W occurs twice in the second position. The codebreakers at Bletchley Park would later call such occurencies females. As both letter groups (AWB and TWY) originate from the same plaintext (e.g. ZXS), we know that the letter X is encrypted into W twice with an interval of 3 steps. This is a unique property that can only occur with a limited number of settings.

According to Rejewski, if enough females were found, the Bomba could be used to recover the current Enigma settings in less than two hours.
Artist's impression of the Bomba with only one wheel set installed. Image supplied by CODEBREAKERS.EU [10].

Although all Bomby were destroyed by the Poles in 1939, just before the Germans invaded Poland, Rejewski made a sketch based on his memories many years later. This sketch appeared in Brian Johnson's book The Secret War in 1978 [1]. An improved version of this sketch appeared in a report of 1979, written by Marian Rejewski himself [2]. This report was provided as an appendix to Władysław Kozaczuk's book W kręgu Enigmy [11] and was translated into English in 1984 [12].
The image on the right 1 shows the improved sketch provided by Rejewski in 1979. The beautiful illustration above was made by the people behind the CODEBREAKERS.EU website and are based on Rejewski's sketch. It gives a good idea of what the Bomba may have looked like.

As the Germans had three different Enigma wheels at the time, there were six possible wheels orders to be considered. This was done by running six bomby in parallel. Each Bomba had 6 full Enigma rotor-sets 2 at its top (1), connected in pairs. Each pair was used to solve one (of the three) females.

Although the exact operation of the Bomba is still unknown, many have tried to explain its principle by reconstructing a theoretical model. A plausible attempt was made by David Link in Cryptologia in 2009 [3].

For the recovery of the Ringstellung, the plug board configuration (Steckerbrett) does not have to be taken into account as it doesn't move during the encipherment. All that matters is that the same input letter produces the same result twice, three steps apart. In [3] Link describes a simple method to solve the Steckern once the wheel order and the Ringstellung is known.
Rejewski's original sketch of the Bomba

  1. Many thanks to the people of CODEBREAKERS.EU for supplying the beautiful artwork [10].
  2. Only one full rotor set is shown in the image. Click here to see the machine with all six rotor sets installed.

Original drawing of the Bomba by Rejewski Artist's impression of the Bomba with only one wheel set installed. Image supplied by CODEBREAKERS.EU [10]. Artist's impression of the Bomba with all wheels installed. Image supplied by CODEBREAKERS.EU [10]. Artist's impression of the Bomba with top lid in place. Image supplied by CODEBREAKERS.EU [10].

The British Bombe
Turing-Welchman Bombe

Based on the information presented by the Poles, British mathematician Alan Turing developed a machine that was capable of recovering the key settings even if the Germans would drop the double encryption of the message key at the beginning of each message. The machine was called Bombe (later: Turing-Welchman Bombe) and was built by the British Tabulating Machine Company (BTM) in Letchworth, Hertfordshire (UK) under supervision of Harold (Doc) Keen [4].
The name was derived from Bomba, a similar machine developed by the Poles shortly before the outbreak of WWII. The Bomba exploited the fact that the same message indicator was sent twice at the start of each message, a major flaw in the German cryptographic procedures.

Although the concept of the Bomba was known to Bletchley Park, Turing wanted to try a different approach. When the Germans discovered the weakness they gave up the doubly enciphered indicator on 10 May 1940 [9], the day Germany attacked France, rendering the Bomba useless.
Close-up of the rotors of a Bombe machine

Turing designed the British Bombe in 1939. Compared to the Polish Bomba, it used a completely different approach. It was based on the assumption that a known (or guessed) plaintext, a so-called crib, is present at a certain position in the message. The Bombes were built by the British Tabulating Company (BTM, later ICL) at Letchworth (UK) under the supervision of Harold 'Doc' Keen [1]. The first machine, called 'Victory', was delivered at Bletchley Park on 18 March 1940.

The Bombe was further enhanced with the so-called diagonal board, an invention of fellow codebreaker Gordon Welchman, that greatly reduced the number of steps needed for the codebreaking effort. A second Bombe, with Welchman's diagonal board present, was installed on 8 August 1940. It was named 'Agnus Dei', later shortened to 'Agnes' or 'Aggie'. The first machine (Victory) was later modified with a diagonal board as well.

During the course of the war, over 200 Turing-Welchmand Bombes were built. To avoid the risk of losing them in case of a bomb attack, they were spread between Bletchley Park and its so-called Outstations in Wavendon, Adstock, Gayhurst, Eastcote and Stanmore, where they were operated by WRNS, RAF-technicians and civillian personnel [3].
Enigma M4
Like the German Wehrmacht (Army) and Luftwaffe (Airforce), the Kriegsmarine (Navy) used a three-wheel Enigma machine with Steckerbrett (plug-board). The Navy called it the M3. Unlike the Wehrmacht and the Luftwaffe however, the Navy used additional codebooks for shortening their messages. The shorter their radio broadcasts, the smaller the risk of Direction Finding (DF).

Because Admiral Karl Dönitz, head of the U-Boat section of the German Navy had his doubts, three extra wheels were introduced (VI, VII and VIII), exclusively for the Kriegsmarine. But that wasn't all he changed. On 2 February 1942, completely out of the blue, the German Navy introduced the Enigma M4, a four-wheel machine, exclusively for U-Boat communication. At the same time they changed their codebooks, leaving the codebreakers at Bletchley Park in the dark.
The Enigma M4 had an extra rotor, called the Zusatzwalze (extra wheel), inserted between the 3rd wheel and the Umkehrwalze (reflector). As the Bombe was built specifically for breaking three-wheel Enigma traffic, it was not suitable for attacking the new M4 machine.

Apparently, Dönitz, had managed to fulfill the logistic nightmare of replacing all existing M3 machines by the new M4. But as the code was changed overnight, and the rest of the Navy was still using the 3-wheel machine, Turing figured that the machines were somehow compatible.

He turned out to be right. When the 4th wheel was set at 'A', the machine would behave like the old M3. This was done, for example, when communicating with the Naval Weather Service. Furthermore, the 4th wheel never moved during encypherment, making it effectivelty a selector between 26 different reflectors (UKW). Turing soon managed to recover the full wiring of the extra wheel, but was in desperate need for new codebooks in order to read the U-Boat traffic.

In the event it took until 30 October 1942, nearly 9 months after the introduction of the M4, before new codebooks were captured. In the meantime, some 3-wheel Bombes had been adapted for attacking 4-wheel Enigma traffic and orders were given to Doc Keen at BTM for the development of an enhanced Bombe.
The American Bombe
Although the British 4-wheel Bombe worked as expected, it faced many problems. At this stage of the war, Great Britain had shortages of nearly everyhing. As a result, BTM was unable to deliver enough machines in time and the machines that were delivered often had contact problems.

Since their involvement in the war in 1942, the Americans had been pushing the Brits to share their knowledge about the Bombe and allow them to copy its design. Finally, in late 1942, when the British 4-wheel Bombe was facing problems and the daily losses in the Battle of the Atlantic were accumulating, the Brits finally gave in and allowed the US to build its own Bombe.
The US-Bombe was built by the National Cash Registers (NCR) in Dayton (USA), where it was developed by Joe Desch. Initially, the US Navy wanted him to build a fully electronic machine, but Desch found this to be impractible, as it would require the machine to have more than 70,000 electronic valves (tubes).

Towards the end of 1942, Desch proposed a less elegant but far more realistic approch: an electro-mechanical machine, just like the British Bombe, but much faster and more reliable. The US Navy immediately approved the project.
Photograph copyright NSA [5]

Desch set out to build the machine and by mid-1943, he had the first prototypes running. Although the initial design faced reliability problems, he managed to improve it considerably. In the end, the US Bombe turned out to be very reliable, fast and effective. By December 1943, 120 machines were installed. For the remainder of the war, the US took care of breaking the majority of German Naval Enigma traffic and in particular the messages of the dreaded German U-Boats.

By the end of the war, the US Bombe would have saved thousands of lives, but had destroyed that of Joe Desch, who had suffered a nervous breakdown and would never really recover from that. Deborah Anderson, Desch's daughter, has dedicated a website to the work of her father and his collegues at NCR [4]. After the war, all US Bombes were destroyed, except for one unit which is now on permanent display at the National Cryptologic Museum (NCM) at Fort Meade (USA) [5].
Rebuilding the British Bombe
In the UK, all British Bombes were destroyed or dismantled once the war was over. As far as we know, not a single machine has survived. This has triggered a group of enthusiasts - led by John Harper - to start a rebuilding project in the mid-1990s. The aim of the project was to recreate a fully functional replica of a war-time Bombe machine, which was completed in 2007 [2].
It is now on public display at the Bletchley Park Museum and is part of the permanent exhibition in B-Block. The Bombe is being demonstrated on a regular basis and is currently being used to do real WWII codebreaking work. A far more detailed description of the Bombe Rebuild Project is available on John Harper's website [2].

The image on the right was taken at the 2009 Enigma Reunion at Bletchley Park, where the Bombe was demonstrated by John Harper. It shows the Bombe in action when trying to break a real war-time Enigma message.
A view of the rebuilt Bombe at Bletchley Park

Please note that different versions of the Bombe exist. Furthermore, a number of attachments and 'special versions' were built for specific tasks and experiments. The version reconstructed by the Bombe Rebuild team is the later 3-wheel 36-Enigma version, with high-speed Siemens-type sense relays. During the war, 69 of such machines were built. It is featured in the video below.

This video shows the reconstructed Bombe in action. It was made during the Enigma Reunion at Bletchley Park in 2009, when project manager John Harper was giving live demonstrations. Some of the moving parts inside the Bombe are clearly visible. The video also shows the Checking Machine and a British Typex cipher machine that was converted to act as an Enigma machine.
It is sometimes thought that Enigma was broken with help from Colossus, the first valve-based electronic digital computer, that was developed at Bletchley Park during WWII. It is even claimed in some books and TV documentaries.

This claim is not correct however, as Colossus was used to break the far more advanced Lorenz SZ-40/42 cipher machine, that was used by Hitler's High Command.

 More about Colossus
Click to read about Colossus

  1. Brian Johnson, The Secret War
    1978, BBC.

  2. Marian Rejewski,
    Mathematical foundations of the German Enigma cipher machine solutions (Polish)

    Rejewski's report of 1979.

  3. David Link, Resurrecting Bomba Kryptologiczna: Archaeology of Algorithmic Artefacts, I
    Cryptologia, Volume 33, Issue 2, April 2009, pp. 166-182.
    ISSN: 0161-1194

  4. John Keen, Harold (Doc) Keen
    ISBN: 0-947712-42-9

  5. John Harper, The Bombe Rebuild Project
    Website, showing the progress and the various stages of the project.

  6. Milton Keynes Heritage Association, Outstations from the Park

  7. Dayton Codebreakers, US Navy Cryptanalytic Bombe
    Website by Deborah Anderson.

  8. National Cryptologic Museum (NCM)
    Based in Fort Meade (USA) near the NSA building.

  9. Kris Gaj and Arkadiusz Orlowski, Facts and myths of Enigma: breaking stereotypes
    George Mason University, Fairfax, VA 22030, USA.
    Institute of Physics, Polish Academy of Sciences Warszawa, Poland.
    Chapter 7. Eurocrypt 2003. Retrieved January 2012.

  10. International Cryptology Game, Artist's inpressions of Bomba and Cyclometer
    Website CODEBREAKERS.EU. Received June 2016.

  11. Władysław Kozaczuk, W kręgu Enigmy
    1979.  Translated into English in 1984 [12].

  12. Władysław Kozaczuk, Enigma: How the German Machine Cipher Was Broken,
    and How It Was Read by the Allies in World War Two. 1
    Translated from [10] and edited by Christopher Kasparek.
    30 June 1984. ISBN 978-0-31327-007-9 (first edition).
    US edition: ISBN 978-0-89093-547-7 (2nd edition).

  1. Not te be confused with Enigma - How the Poles broke The Nazi Codes, which is basically a subset of the 1984 book in which Rejewski's contributions have been replaced by (less important) material by others.

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© Crypto Museum. Created: Friday 23 November 2012. Last changed: Saturday, 10 September 2016 - 10:15 CET.
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