The History and Mathematics of Codes and Code Breaking

Tag: enigma (Page 1 of 3)

There's Always Someone Smarter

A passage from Cory Doctorow's Little Brother that caught my attention was "[t]he problem had been that Turing was smarter than the guy who thought up Enigma.  Any time you had a cipher, you were vulnerable to someone smarter than you coming up with a way of breaking it (99)."  It first caught my attention because we just recently discussed World War II and Bletchley Park, where Alan Turing broke Enigma.  Second, when Marcus, the main character of the novel, said "you were vulnerable to someone smarter than you coming up with a way of breaking it," I was reminded of the several in-class discussions about always assuming your cipher is breakable.

Throughout history, there have been several instances where cipher makers have been overconfident in the security of their ciphers.  For example, in the 16th century, Mary Queen of Scots employed a substitution cipher, using numbers and symbols.  Unfortunately for her, she placed too much confidence in both her cipher and her contacts, and by the use of frequency analysis, her cipher was broken, and her plot to escape imprisonment and murder Queen Elizabeth I was unveiled.  In 1587, Mary Queen of Scots was executed.

Another example of cipher overconfidence came from the Germans during WWII.  Their Enigma was incredibly secure, so naturally the Germans assumed that it was unbreakable.  However, the British had a team of highly intelligent mathematicians on their side, including Alan Turing, who discovered a flaw in Enigma, and was therefore able to break it.  He created a machine called a "bombe" that was able to break the daily German Enigma key.  The Germans were unknowingly sending intercepted and decipherable messages.  Their overconfidence in their cipher led to their ultimate downfall.

Overconfidence in one's security plays a large role in Little Brother.  With the Department of Homeland Security monitoring nearly everything from transportation routes to Internet usage, Marcus and his friends were in serious need of ciphers and codes to protect their privacy.  Marcus learns early on that with every bright idea, there exists another that is better.  Most likely, there is always someone who's smarter.

The Allies Work Better Under Pressure

It is no secret that Allied code breakers bested German code makers during World War II which contributed enormously to an Allied victory in the war. Germany's overconfidence in the strength of its Enigma cipher definitely contributed to the Allies' code breaking success, but another main contribution was the pressure that Germany forced against the Allies. The Allies were on their heels trying to defend against Germany, which led countries to band together and individual cryptographers to band together to fight a common enemy. The necessity for the Allies to break Enigma in order to thwart the Axis' attacks brought Poland, England, France, and America together which gave them the resources to crack Enigma and Purple (Japan's encryption method).

Without the pressure the Axis powers were putting the Allies under, they would not have felt the urgency to break Enigma and Purple. The Allies won this war on intelligence because they were on the defense and needed to break Enigma and Purple in order to turn the tables against the Axis, while the Axis got complacent and confident about their machines because they were able to advance through Europe and the Pacific without their code being decrypted. Since the Allies were under such pressure, they had to find a way to gain the advantage. Therefore, countries such as Poland and England teamed up and individuals such as the mathematicians at Bletchley Park teamed up to crack the Enigma and Purple ciphers. Without the pressure that the Axis' exerted on the Allies, the Allies would not have been so desperate to find any way possible to crack Germany and Japan's seemingly unbreakable ciphers.

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Two Steps Forward and One Step Back

While German overconfidence in the strength of the enigma machine was partially responsible for the downfall of the cipher, many other reasons also influenced this ultimate collapse of German enciphering. I think that another main reason that the enigma was able to be broken lies in the fact that the enigma itself was simply a machine. The industrial mechanization of the early-mid 1900’s transitioned the world from simple methods to more efficient and technologically advanced means of production and thinking. These new technologies greatly impacted the way that war was waged; planes and radios and bombs all allowed for higher casualty rates while new cryptographic methods allowed for more methodical enciphering.

Even though this mechanization of enciphering sped up the process, the complexity of the machine was almost outweighed by the simplicity of its engineering. Because it was ‘just’ a machine, the enigma machine was able to be almost reverse engineered by Alan Turing. The industrial shift in society was not just reflected in product manufacturing, but also in the ways that people thought. In one of his many papers, Turing proposed the idea of an automated calculator. While this was well ahead of the technologies available to him at the time, this shows the logical thought process which was now being used to approach breaking ciphers.

In addition, new technologies made it easier for messages to be intercepted. From the telegraph to the radio to modern communication over the internet, lines of communication are becoming increasingly more accessible to spectators. By no means am I saying that technological advances have hindered enciphering, I just think that it is important to consider how the mechanization of society influenced the thought processes and methods of decoding in and the ease with which these encoded messages could be accessed by outside forces.


Let's Go De-fense! *clap, clap, clap-clap-clap*

Certainly the Germans' overconfidence in the power of Enigma led to their loss in the battle of cryptography in World War II, and my classmates have brought up many other great reasons: how the Allies worked together, a few genius individuals working for the Allies, the American use of Navajo code talkers, human error on Germany's part, and Poland's (specifically Marian Rejewski) contributions to cracking Enigma. But stepping back from that, I think in the grand scheme of things it comes down to who was playing offense and who was playing defense. In general, Germany was on the offensive: the Blitzkrieg bombing of Britain, the invasions into France, and their U-boats in the Atlantic. This made the Allies often on defense, not entirely sure where and when Germany would attack next. Because of this, I think they found it more imperative to crack Enigma; if the Allies knew when the next bomb would fall, or where German troops were camped, or where the U-boats were headed, they wouldn't be caught by surprise and could be far better prepared to fight back. Therefore they were willing to hire thousands of codebreakers to work at places like Bletchley Park, and invest money in a seemingly crazy machine to break Enigma. Until Enigma was completely broken, Germany did have the upper hand, and they weren't as worried about deciphering Allied messages as long as they kept winning battles and advancing.

When the Allies (thanks to Alan Turing and his machine) were finally able to decipher any Enigma message everyday, British officers recognized the advantage they now had, one that would only be kept if Germany continued to think they were still on offense. So the Allies were very careful to not let Germany know of their success, and only here does Germany's overconfidence in Enigma come into play. Up until this point they had every right to be confident in the secrecy of many of their communications, and it showed as they swept across Europe. But England could never do a perfect job covering up what they knew, and Germany's overconfidence in Enigma led them to ignore that. In the football game of World War II, the Allies defense had intercepted the football and were running for the touchdown, while Germany's offense still thought they were advancing towards their field-goal range and their cryptographic defense was off taking a water break in the locker room.

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A Uniquely American Code

One of the foremost advantages the Allies had during the Second World War was the United States’ Navajo Code Talkers. Because Native American tribes developed language and culture separately from Europe and Asia, there was no basis for the German cryptographers to begin to decrypt their codes. An extra layer of encryption was that the Navajo code corresponded to words or letters in the English language, rather than their own meanings, which made decryption more than simply understanding the Navajo language (which the Germans and Japanese were unable to do, anyway).

Though German forces were overly confident in the Enigma cipher and its complexity and impermeability, it did not mean they were unable to gain ground with cracking the Allies’ ciphers. Codes and cipher machines such as Type X and SIGABA may have been more effective than Enigma because Allied cryptographers were more careful than German ones. However, there was always a risk that the German cryptanalysts had begun to crack the codes, and decrypting the messages sent by those machines were also very slow. Implementing the Navajo Code Talkers made it basically impossible for the Germans to crack the code, and also expedited the process of sending and deciphering messages that greatly contributed to the Allies’ victory.

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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.

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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.

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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.

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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.

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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.


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