The History and Mathematics of Codes and Code Breaking

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Divided and Conquered

The Allied cryptanalysts were victorious over the German cryptographers due to a variety of reasons; however, one rather simple reason is often overlooked: the allies had a much larger and much more unified base of cryptologists than the Germans.

Germany had a total of around 30,000 people working in the intercepting, decoding, and coding of messages. The European Axis powers had a grand total of 36,000 people working in those endeavors. The Allied powers had a number closer to 60,000 people doing the same jobs, nearly twice that of the Axis powers. Just think about the implications of this: more people leads to more intercepted messages, which leads to more cipher text to work with (a historically beneficial resource in terms of breaking codes), and more people leads to more brain power trying more techniques to break the same code.

In addition to this, Germany did not have a central cryptology base like the Allies did at Bletchley Park. The Germans were spread out among 6 different bases, and would often overlap in each other’s efforts, duplicating each other’s work and thus wasting time and resources. There was some collaboration but not nearly to the degree of the Allies. In Andrew’s blog post, he discusses the importance of collaboration in the field of cryptology so I will not expand upon this as much.

Finally, the Germans never created a bombe-like machine that could decipher messages which can very easily be attributed to the division and smaller size of the German forces. Without this key technology, Germans had to do a lot of the leg work manually which is much much more time-consuming and much less reliable. The bombe and other machines like it (Colossus and Tunny) exponentially increased the cryptographic progress of the Allies, catapulting them far ahead of the Axis powers.


Click here to see my primary source.

The Importance of Logic

A confidence boost from winning the First World War led many Allied countries to lose their motivation for solving Enigma as they lacked the driving factors of fear and hardship that had provoked their initial incentive to win the war. This quickly caused German overconfidence in the security of Enigma, instigated by both the lacking effort of the Allied forces and the strength of the code itself. The Germans’ unshakable faith in their coding system would ultimately lead to their defeat as they mistakenly viewed Enigma as unbreakable.

While many other Allied countries initially gave up in most of their attempts to solve the code, Poland luckily realized the importance of having skilled cryptanalysts. Poland’s decision to hire mathematicians to solve the mechanical cipher of Enigma was one of the most crucial factors in the Allied success. By taking this mathematical approach, the cryptanalysts studied the machine’s operations and were thus able to analyze the scramblers’ and plugboard cablings’ effects.

Though creativity is an essential part of cryptanalysis, the Allied cryptanalysts used mathematics to focus more on the logical aspect of code breaking. By attacking Enigma through the discovery of repetition within the codes, the Allies were able to find patterns that uncovered the plaintext of the German code. In order to break Enigma, having a well-trained team of mathematicians was critical. Solving this highly advanced technology required a similar scientific approach in cracking its message.

Without Poland’s mathematical approach to solving Enigma, the Allied cryptanalysts would arguably never have cracked the code, as logic was the key factor in exposing the messages created by Germany’s cryptographers.

German Cryptography is still Human Cryptography

During WWII, Germans sent out thousands of messages encrypted using the supposedly unbreakable Enigma machine. It was discovered after the war that German intelligence knew that these messages could be captured by the Allies, but they could not think anyone would have the time or resources to possibly decipher them. This strongly held idea that Enigma was unbreakable was perhaps the greatest mistake of Germany.

Another factor, besides German overconfidence, that allowed the Allies to decipher German messages were the patterns discovered when Enigma was used. These patterns were precisely the result of non-randomness that describes human nature. Some keys were easily guessed because the letters on the Enigma keyboard were next to each other. Other keys may have been similarly predictable because they resembled German names, or they were used repeatedly. These were called “cillies.” Ironically, an effort to consciously combat human un-randomness was also a mistake on Germany’s part. By avoiding “obvious” plugboard settings and arranging rotors to avoid repeated positions, the amount of possible settings were drastically reduced.

Human nature in and of itself is never truly random; this is a basic fact we learn in our statistics classes. If you asked a population to randomly choose a number between 1 and 4, would a fourth of the people choose each of the numbers? Polls have shown that, instead, a clear majority would choose the number 3. In the same manner, cipher keys are not always a random garble of letters. They are often derived from meaningful words or phrases that may be pertinent to the message or the receiver/sender of the message.

Rejewski and Turing

One of the main reasons for the success of the Allied cryptanalysts at Bletchley Park over German cryptographers is the acquisition of the previous work of the Polish on the German Enigma. Polish cryptanalyst, Marian Rejewski, led the polish to first break Enigma in 1932, and kept up with breaking any new security the Germans implemented to strengthen Enigma, until in 1939, when the Germans increased the number of plugboard connections from 5 to 8 to 7 to 10, which made cryptanalysis extremely more difficult. This spurred the Polish to disclose all their work on Enigma to the Allies, especially as the likelihood for another war was growing. Thus, when war broke out and the need to break Enigma became of utmost importance, the Allies had a head start on breaking the codes, as they already had acquired intelligence on Enigma.

Another curious and more indirect reason why the Allies were ultimately successful was because Britain never found out that Alan Turing was a homosexual. Turing was the one of the most important men in the war in that he led the cryptanalyst team at Bletchley Park to victory in breaking Enigma. At the time, homosexuality in Britain was illegal and it was very fortunate that the state never found out about Alan Turing’s case during the war, otherwise Turing probably would never had made it to Bletchley.  Needless to say, if Turing had not been working for the Allies during the war, Enigma may never have been broken and the Germans may have won.

Allies versus Germans: they won because they were Allies

It is my opinion that one of the prominent and yet overlooked reasons that the Allied cryptanalysts were able to end up winning against German cryptographers was that they were indeed Allies. Although there were times when they kept information from each other, they were able to share their breakthroughs in a way that Germany could not share with its allies. Every time an advancement in breaking the code was made it was possible for them to share that advancement with each other, and this allowed them to break more codes faster. Germany, on the other hand could not share breakthroughs with codewriting and codebreaking with its allies. This is for a pretty obvious reason.

The Allies were only intent on defeating Germany and its allies, to keep the world balance as it was. Germany and its allies were intent on conquering as much territory as possible. This meant that Germany was afraid to share information with its allies, because there was always the chance that once they defeated the Allies, they would turn on each other. An interesting parallel of this would be that of supervillains. The issue with them joining together to defeat superheroes was and is always that they can’t work together for very long before turning on each other.

The Allies could communicate with each other. Germany could not do so. This, as simple as it is, is one of the key reasons that the cryptanalysts worked so efficiently. The Allies were allies.

The Value of Mathematicians as Cryptanalysts

A variety of factors contributed to the Allied cryptanalysts’ success over German cryptographers, including espionage, determination, and cooperation. One main element that contributed to the Allied success was the employment of mathematicians and scientists in their cryptanalyst units. The Polish breakthroughs in cracking Enigma demonstrated the value of mathematicians as codebreakers. Marian Rejewski, the main Polish cryptographer working on cracking Enigma, was a mathematician. Enigma was a highly complex machine requiring much logical and mathematical thinking in order to break it. In Britain, linguists and classicists had always dominated Room 40. The addition of mathematicians and scientists to the team greatly strengthened the unit and brought in a new perspective on how to break the ciphers. Analyzing the ciphers from a mathematical lens provided valuable new insight that was necessary to break the Enigma code.

Alan Turing is known for identifying Enigma’s greatest weakness, which made it possible to crack the Enigma cipher in tough circumstances. He was a master of math, science, and logic. His advanced skills in these areas helped him think through the different layers of Enigma and figure out how to approach and tackle the haunting task of cracking the code. Turing’s unique background in mathematical machines allowed him to create his bombes. These bombes tested Enigma settings much faster than they could be tested by hand. Without mathematicians like Turing, who could conceptualize and build such machines, it’s possible that Allied cryptanalyst units would never have broken the Enigma machine ciphers.

Nowadays, when I think of a modern cryptographer, the first thing that pops into my mind is a mathematician. However, a cryptographer has to be fluent in a variety of subjects, including mathematics, science, logic, and linguistics. The ability to integrate knowledge from diverse fields when attacking a cipher is what makes an exceptional cryptographer. Although the Germans’ overconfidence in the strength of Enigma played a significant role in the success of the Allied cryptanalyst efforts, many other factors were instrumental in the cryptanalysis as well. The realization that mathematicians could be important additions to cryptanalytic staffs was vital to the Allied cryptanalysts’ successes over German cryptographers.

The Navajo Code Talkers

A major reason for the Allied success during World War I was German overconfidence in the Enigma.  Because of their overconfidence, they were unaware that the British were deciphering their messages.  Although this played a large role, it is not the sole cause for the victory of the Allies.  In order to win the war, the Allies also had to have some defenses of their own.  One of these defenses came in the form of code.  The United States was in need for an impenetrable code so that its communications could be secure.  The answer came in the form of the Navajo, a Native American tribe.

The Navajo language is incredibly complex; it is unique and does not stem from any other language.  Singh quotes Philip Johnston, the mastermind behind using the Navajo language as code, “the Navajo tribal dialect is completely unintelligible to all other tribes and all other people.”  The United States government employed 420 Navajo code talkers.  With these code talkers, the United States had a secure means of communication, which allowed for them to prevent disasters from happening and anticipate potential threats.  After the war, the Japanese even admitted that they had not made a dent in breaking the Navajo code.

Having a secure code is vitally important.  This is evidenced by the German failure to keep a secure code.  Once the British had broken the Enigma, German communications were readily available to the Allies.  This allowed for the Allied forces to gain the upper hand.  On the other hand, with the United States having a secure code, the Allies were able to communicate without fear of German or Japanese decipherment.  The Germans and Japanese may have been able to intercept the messages, but without knowledge of the Navajo language, decipherment was essentially impossible.

A Boost of Motivation

The mindset for the Allies had changed between the First World War and the Second World War. After their success in cracking Germany’s ciphers in the First World War, the Allies felt like that could crack anything Germany tries to encipher. However, once the Germans started using the Enigma machines, the Allies were stumped. This change in attitude might be attributed to the fact that they were not in direct threat at that time so they didn’t have the motivation to try to decipher the messages. That along with the hopelessness that might come with failed attempts would make them lose motivation. Poland, however, was threatened so they had to do everything they could to decipher those messages. Therefore, with the help of Schmidt and Rejewski, they reached a breakthrough in cracking the enigma. If it wasn’t for their breakthroughs, the Allies may not have been able to crack it. Gaining that knowledge may have been the motivation they needed to fully uncover how the Enigma machine works. The Allies were also able to pick up on some keys that Germany’s operators would send. The operators would sometimes pick three consecutive letters from the keyboard which the Allies started picking up on.

Photo credit: "Enigma Machine (Bletchley Park)" by Tim Gage via Flickr CC

Photo credit: “Enigma Machine (Bletchley Park)” by Tim Gage via Flickr CC

Sometimes they would repeat the same keys and therefore the cryptanalysts would be able to predict them. Overall, cracking the Enigma took the efforts and collaboration of many individuals working as a team.

On the Shoulders of Giants

In the beginning of World War II, Great Britain was under less threat from the ever-expanding German forces on the European continent. Poland, on the other hand, was sandwiched between the Soviet Union to the east and rapidly encroaching Nazi armies to the west. Under the pressure of otherwise being forced under Nazi rule, the Polish cipher bureau made incredible headway in analysis of early German Enigma intercepts.

Bletchley Park Bombe

“Bletchley Park Bombe”
Photo by Antoine Taveneaux- Licensed under Public Domain by Wikipedia Commons

When Great Britain’s ships were attacked by German subs, a greater need to decipher Enigma arose. What was perhaps the most important contribution to British cryptanalysts’ efforts was the fact that they were able to build upon the Polish cryptanalysts’ work. Without those insights, the analysts at Bletchley Park might never have developed a full image of how the Enigma machine worked. Or rather, they might not have fully understood the weaknesses of the cipher (as well as its operators). For example, the Polish cipher bureau supplied copies of the military models of the Enigma machines to the British and French, and also provided the operator procedures that were in use at the time. This allowed groups like Bletchley Park’s Hut 6 to focus on finding a way to crack Enigma without the use of fragments such as the six-letter message key repeats at the beginning of every message. Additionally, the cryptological bombe that Alan Turing developed was based on a model designed by Polish cryptoanalyst Marian Rejewski, mechanizing the process of working out daily message keys.

Through collaboration and the ability to build upon the work of earlier cryptanalysts, the British were able to break the Enigma cipher. Not only did this save Allied lives and make victory much easier, but also the decryption shortened World War II in Europe and saved the lives of many in the Axis states who would have otherwise been killed in the longer fighting.

Human Error and Forced Flaws

Photo Credit: “Chiffriermaschine ‘Enigma’ ” by Walther licensed by Wikimedia Commons under Creative Commons

The Enigma Machine was practically impregnable if all of its information was kept secret and all its operators worked without human error. With billions and billions of possible settings, it would have taken cryptanalysts an obscene amount of time to sort through all of the possible keys. Additionally, with an ever changing pattern and shifting scramblers it would be incredibly hard to find a method to deduce the plaintext if all one had was pure ciphertext, even with if the most brilliant minds in Britain working on a solution. Essentially, cracking the Enigma required some sort of “crib”, some insight into how the code was working on a specific day that would take out some of the possible Enigma settings.  Because of this, the cryptanalysts would not have had the success that they did without the help of two things; German cryptographer’s mistakes in using the code and the espionage and tricks of the Allied forces.

A clue into the how the code was being run a certain day was often acquired from the mistakes of the Germans. For example, when German operators were picking keys they would often choose “three consecutive letters from the Enigma keyboard” or even use the same key as they had used previously (Singh, p.164). These mistakes, known as cillies, became vital to Bletchley Park’s decoding of the Enigma machine. Because they knew that some keys were more likely to show up than others, they could try their hunches first and would save valuable time if they were proven correct. Basically, the Enigma machine was still doing its job; it’s just that the operators proved to be too predictable. In addition, the Germans took efforts to make the Enigma machine more secure that often backfired and lessened the impregnability of the cipher. For instance, they decided that a scrambler couldn’t stay in the same position for two days in a row (Singh, p.164). This may seem to make it more random, but it actually excluded many of the possible scrambler arrangements that British cryptanalysts had to weed through.

When all else failed, however, and German mistakes and bright Bletchley park minds didn’t produce a crib, espionage and trickery became key. When they couldn’t find a crib, it seemed, British cryptographers would create one. By manufacturing situations where the German U-Boats would have to send messages with a specific location in the cipher, the British cryptanalysts could gain insight into the way the cipher was working. Because they knew the location of whatever the U-Boats had sighted (be it a convoy or a mine), the British had a bit of plaintext to work with. With this plaintext, they could employ Turing’s loop method and decrypt the scrambler and plug board settings of the day. All in all, using the openings found in the Germans operational mistakes and those created by Allied operations, the cryptanalysts at Bletchley Park could decrypt the Enigma, collecting valuable information that would help them win the war.

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