The History and Mathematics of Codes and Code Breaking

Author: mclaugrp

Math Behind the Shark

In a section of Cryptonomicon entitled “Cycles,” there are a few pages that go into detail explaining how the three wheels in the enigma machine give it a certain security level and how the adding of the fourth wheel in the system Shark increased the security of the machine. It is explained by comparing the chain of letters created by the enigma machine to a bicycle with a weak link in the chain. The “weak link” in the enigma machine is the first chain that is created to encipher the first letter and when the same chain is used again which occurs 17,576 letters later. When the Germans added another wheel they increased the number of links in the chain to 456,976 and since their messages were hardly ever that long the weak link usually never came into play.

This section of the book added onto my understanding of the enigma machine and how the 4th wheel added so much more security. The increased security was explained through a number of pages with a lot of mathematics on them which helped me see more clearly what factors were actually playing a role in increasing the security of the enigma machine when adding the fourth wheel and creating Shark.







Image: “Chain,” by Pratanti, Flickr (CC)

Safe Enough?

When Horst Feistel developed the Lucifer system for encrypting information on computers, it had an infinite number of keys that could be used to encipher so it would actually be beyond the code breaking abilities of the National Security Agency (NSA). So when the NSA decided to adopt Feistel’s system as the Data Encryption Standard (DES) they wanted to make sure they limited the possible number of keys so they would still be able to break the encrypted data just by using brute-force with their supercomputers, but at the same time civilians would not be able to break the code. They decided to limit the number of keys to roughly 100,000,000,000,000,000. This number of keys would provide privacy and security within the civilian community, but would still allow the NSA to break into messages if they needed to.

I personally believe that the NSA was justified in limiting the number of keys in the Lucifer cipher. I think it is vital for the NSA to be able to read certain messages if they really need to. If they didn’t limit the number of keys every message would be completely private and secure. This might sound great in theory, but would actually be relinquishing our security as a nation. Anything could be sent by anyone to anyone and no one would ever know about it even if the government were

suspicious. So if two known terrorists were communicating, we wouldn’t be able to read what they were saying. However when the keys are limited to a certain number, any of our messages are completely secure and private within the civilian community, but the government would be able to read it if they wanted to. I think that this is a reasonable violation of our privacy. Also, the government is not going to do anything with the information they read if it is innocent, so most people have nothing to worry about. They are only going to care about things that involve national security. So I think it was justified for the NSA to limit the number of possible keys to a number high enough that the correspondences within the civilian community would be secure, but a number low enough that only the NSA could break into the message if they really needed to.
Image: “Castello di Sermoneta” by Andrea Marutti, Flickr (CC)

Privacy: The True Meaning

In Chapter 4 of Little Brother, Doctorow gives a great definition and explanation of what privacy is and how it is sometimes construed in the modern age when he relates keeping information private and going to the bathroom. He explains that there is not anything inherently wrong, weird, or shameful about going to the bathroom, but that everyone would be hesitant about doing it in public with hundreds of people watching. Of course we would all agree with this, so then why is it wrong to keep some things we do on the internet private? The answer that Doctorow wants to get across is that it is not wrong at all, “it’s about your life belonging to you.”

The reason this particular passage stood out to me the most was because I have always agreed that you should be allowed to keep certain things private on the internet without it being considered weird or wrong, but I never could really come up with a really good reason why. When I read this part of the novel, I was frankly kind of amazed. An analogy that seems so far-fetched at first glance, worked perfectly to explain how I feel about privacy on the Internet. That particular passage was so well done and relatable that after reading it I found myself being much more empathetic towards Marcus as the novel progressed.

Image: “Please!,” by Josh Hallett, Flickr (CC)

Motivation Behind Cracking the Beale Ciphers

In my opinion, I think the main reason people are still trying to break the Beale Cipher is based on the 20 million dollars worth in gold that is supposedly buried. Money can make people do crazy things and people often won’t think about the thousands of other people that have tried and failed and the chance that there is no treasure to be found! Along with the fact that so much money is at stake, the second cipher was cracked using the Declaration of Independence so people probably feel like there is hope. If none of the three ciphers were cracked then less people would try probably. Some other people probably want to crack the code out of pure pride. Since so many people have tried and didn’t succeed, then if the cryptography world would give them a lot of respect.

Image “Fishpool gold coins” by Lawrence OP, Flickr (CC)

What’s So “Great” About the Great Cipher?

The Great Cipher was created by the Rossignols in the 17th century and remained unbroken for the next two centuries due to a number of security features that made it nearly unbreakable. When an expert French cryptographer Bazeries got his hands on letters that were enciphered using the Great Cipher, he spent the next three years trying to break the code. Through his efforts we learned just how secure the cipher really was. The pages of the letter he was trying to decipher contained thousands of numbers but only 587 unique ones were used. At first, Bazzaries assumed that the extra numbers were just homophones, meaning that multiple numbers represented the same letter. After months of trying this method, he decided that the Great Cipher was not a homophonic cipher and moved onto the next idea. He tried to break the code as if it was a digraph, meaning that each number corresponded to a pair of letters. He tried to use frequency analysis on pairs of letters but this failed as well. He then tried a different form of the digraph idea in which each number represented a syllable. After he used frequency analysis on the syllables most used in the French language he found that the phrase “les ennemis” appeared many times on each page. When he replaced every number that corresponded with these syllables he was able to complete the partially completed words and solve the message. While he was solving the message, he was stumped many times because the Rossingols had placed traps in the cipher that were meant to trip up any people trying to break the code. For example, some numbers represented single letters instead of a syllable and to make the cipher even more complicated one of the numbers represented neither a letter nor a syllable, but actually deleted the previous number. It is easy to see why the Great Cipher went unsolved for 200 years because it was so revolutionary in the techniques it used to keep out prying eyes seeking the information held within the cipher.

Cryptanalysis Evolving Over Time

In Simon Singh’s book, The Code Book, it is expressed that “Cryptanalysis could not be invented until a civilization had reached a sufficiently sophisticated level of scholarship in several disciplines, including mathematics, statistics, and linguistics.” I think this is a very true statement because cryptanalysis involves a level of mathematics and linguistics that wasn’t readily available to many people in early civilizations. The argument that amateur cryptanalysts intuitively use frequency analysis on substitution ciphers and therefore a level of scholarship isn’t needed to solve ciphers is truly invalid in my mind. One needs to realize how different modern civilization is compared to the other time periods we are comparing it to. Along with the civilization needing to reach certain levels of scholarship in mathematics, statistics, and linguistics, they also needed to have to time, energy, and willpower to sit down and try to solve ciphers. In the time of Queen Elizabeth and Queen Mary, the normal folk were illiterate and were working very hard to just have enough food to survive. If they came across a note that looked like a code, they first of all wouldn’t have enough leisure time to try to break the code if they wanted to and they also wouldn’t be literate so they wouldn’t even know where to begin.

In conclusion, the level of education in the modern era is light years ahead of what it was many hundred years ago. One example I heard in class that made it very clear to me was that a hundred years ago, a scholar that was very specialized and on a career path towards engineering or something like that would be the only person who would even considering studying physics. In today’s world, almost every high school graduate has taken a full course in physics.

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