The History and Mathematics of Codes and Code Breaking

Tag: Alan Turing

A comment on another student's analysis of how the enigma was broken.

( (link to original blog post)

In his blog post titled "An Interdisciplinary Approach," Browkm10 shows how the creativity of the minds in Bletchley park heavily contributed to the success of the team. We talked about in class how breaking a cipher involved a certain degree of logic, creativity, and skill. Browkm10 discusses how all the major players like Turing brought diverse expertise to the table. He talks about how there were chess champions, bridge builders, and machine experts all congregated together working on the same problem. He ultimately argues that it was the combination of creativity and logic that made the defeat of the German enigma possible.

I do think, however, that he/she left out an important aspect that had to take place for the enigma to be broken, which was luck. The cipher was only able to be solved because of a few key mistakes that were made by the Germans. They didn't allow switchboards to have connections to adjacent letters, which lowers the total number of combinations by a huge amount. They also had rules about scambler placement that had the same effect. It was the logic and creativity that made breaking the enigma possible, but there were a good amount of mistakes made by the Germans as well that contributed to the Enigma's demise. I think overall the blogger made some very good points, but I think that this nuance's his/her argument.


Too Much to Lose

Although German overconfidence played a major role in the success of Allied cryptanalysts, there were many other factors at play. One of the most significant reasons for Allied success was that the Allies had much more to lose. Initially, Marian Rejewski cracked Enigma because the threat of a German invasion of Poland was extremely high. Whereas other countries such as France had given up on breaking the Enigma, the Polish had too much to lose should they fail. Rejewski and his team spent a full year creating a book full of all of the potential keys for the Enigma. When it became clear that a German invasion of Poland was inevitable, Rejewski and his team handed over their work to the British, in hopes that they might be able to use it as well.

As the Germans added features to the Enigma to strengthen its encryption, such as additional plug board options, the Allies had to step up their game. Once again, the Allies had too much to lose for them not to invest the time and resources into cryptography. For each message the British failed to decipher in time, thousands of lives could be lost. The message could be about the location of the next air raid, or where the German troops were planning to move. Should the Allies have been able to know this information in advance, they might have been able to evacuate areas or adjust their strategies. Therefore, it was incredibly important to them that they be able to break Enigma. As a result, despite some reluctance on their commanding officer’s part, cryptologists at Bletchley Park were eventually given enough resources for Alan Turing to create his Turing Machine; a machine that was reliably able to crack the daily settings for Enigma.

When the stakes are higher, people work harder. German overconfidence certainly helped the Allies to be more successful with their cryptography, however, without the imminent German threat it is unlikely that people like Marian Rejewski and Alan Turing would have had the dedication or the resources, respectively, to break Enigma. Without cracking Enigma, the war could have turned out very differently.

An Interdisciplinary Approach

While German overconfidence in the enigma did eventually contribute to the cracking of the code and their subsequent downfall,  the use of mathematicians and scientist as opposed to linguists and classicist ultimately made the most difference. Many mathematicians and scientists are interdisciplinary. For example, Alan Turing was both experienced in math and building machines. Being interdisciplinary helped the breaking of the Enigma to be approached in both a creative and a logical sense. As opposed to the linguist and classists, that may just look for the patterns within the code, mathematicians and scientists will find the algorithm within the code and  the applicability behind the code and then find a method to apply it broadly. But while there were many mathematicians and scientists, Bletchley park was also made up of "an authority on porcelain, a curator from Prague Museum, the British chess champion and numerous bridge experts" (Singh, 178). This gives breaking the code many different ways of thinking and approaches. I think this mixture of knowledge filled the need to balance both creativity and logic. This is a balance that is needed to break any code. I also think that their motivations behind breaking the codes also had a huge impact. For the British it was a matter of breaking this codes in order to save more lives and keep their country safe. On the other hand, the original way of finding out more about the Enigma was through Hans-Thilo Schmidt's want for revenge on his brother and damage of his country's security. So while the Bletchley park codebreakers were compelled to figure out Enigma for their own countries' sake, Schmidt was driven by hate for his country and his own brother.

How can we make security more "secure"?

On page 99 of Little Brother by Cory Doctorow, Marcus delineates the flaws of cryptology and how ultimately cracking the Enigma led to the victory against the Nazis in WWII. One of the flaws was secrecy; after Alan Turing cracked the Enigma, any Nazi message could be deciphered because “Turning was smarter than the guy who thought up Enigma” (99). As a result, it sparked the thought that any security system is “vulnerable to someone smarter than you coming up with a way of breaking it” (99). Bruce Schneier also refers to flaws of a security system in his Afterword, explaining that it is useless for you to come up with a system entirely by yourself because there is no way for you to detect flaws in your creation. You are limited to your knowledge. Outsiders with different levels of thinking would help by suggesting different views in which people can think of in order to break the system.

I think that this concept is interesting; you are limited by what you know. And everyone around us knows something that we don’t. Recently I read a passage in Harvard Business Review on how companies and organizations should welcome people in different kinds of fields to evaluate an idea because they won’t think the same way that people in a particular company does; a mathematician thinks differently than a historian does, and the distance between their thinking has the potential to bolster ideas, limit flaws, and suggest new ideas that haven’t been thought of yet. Could this be the way to strengthen our current security systems? What kind of people do we need to evaluate them? How many people do we need (until we pass the point to where the security measure is too widely known and therefore ironically more vulnerable)?

I believe this is one of the fundamental qualities of Cryptology and all security measures: how do we know a system is safe to use? Truth is, we really don’t know, but we can always come closer by cross referencing and past experiences, allowing security to get better and better with each step of the way.


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.

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.

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