The History and Mathematics of Codes and Code Breaking

Author: vreelaap

Turning’s Bicycle and Modular Arithmetic


A passage that I found especially intriguing in Neal Stephenson’s Cryptonomicon, is when Waterhouse and Turning are on a bike ride in the English countryside. In these few complex mathematical pages, we are taught about modular arithmetic, which helps us understand how the Germans used the Enigma machine during the war. The part of this passage that I grasped, is when they are describing modular arithmetic through the cycles of Turning’s bike, which has one bent spoke on the back wheel. Turning knows that when the link and the spoke come in contact with each other, the chain will break and the bike will be useless. However, he figures out a way around this malfunction as it only occurs when the wheel and the chain happen to coincide.

Stephenson incorporates math when he proceeds to introduce variables.  He lets l stand for the number of links in the chain, n for the number of spokes, and C for the position of the chain. In order to calculate the value of C, you must do modular arithmetic. Stephenson explains the steps by saying “if the chain has a hundred links (l=100) and the total number of links that have moved by is 135, then the value of C is not 135, but 35” (166). The values of C, each time the wheel spins one full rotation, is n mod l, 2n mod l, etc. Ultimately this tells us that the chain will fall off his bike, when some multiple of n happens to match up with a multiple of l as well. This passage is a clever way to describe modular arithmetic, which we have covered in class.


Settling at 100,000,000,000,000,000

The National Security Agency’s request to have a standard encryption system for businesses to communicate with one and other is a controversial topic, yet definitely justifiable. Horst Feistel developed the Lucifer system, which was known to be one of the strongest encryption products. In fact it was too strong to be adopted by the National Security Agency (NSA) because it had too many possible keys. Once the NSA decided to limit the number of keys to 100,000,000,000,000,000, it was officially chosen to be the NSA’s standard program and renamed the Data Encryption Standard (DES).

The point of encrypting codes is to hide information for the intended recipient’s eyes only. Thus one can imagine that businesses would be angry at the NSA’s decision to use DES, which ensures that they can break any possible code. However, it is justified because “the same message can be encrypted in a myriad of different ways depending on which key is chosen” (249). Had the businesses only been able to use one designated key, complaints would be understandable. However, there are 100,000,000,000,000,000 different keys to choose from. Although one can object that the NSA lowered the number of possible keys to weaken the cipher, you can further justify their decision. The number of possible keys they decided on is secure because “within the civilian community, […] no civilian organization had a computer powerful enough to check every possible key within a reasonable amount of time” (250). Therefore the NSA is the only source that can hack into a company’s information.


We Don’t Know What We Have Until It’s Gone

Chapter 4 of Little Brother circles around the idea of a want for familiarity. After the main character, Marcus Yarrow, is taken hostage, he is stripped of his belongings and dignity. Marcus talks about wanting to be back with his friends and parents, as anyone would in this situation, but what intrigues me is how much we all take such familiarities for granted. As Marcus realizes he is getting onto a boat and leaving his homeland, he becomes sick to his stomach at the thought of never seeing his parents again. However, had Marcus been leaving to go on a vacation without his parents, he most likely wouldn’t have thought twice about parting with his parents. On a similar note, seeing a pizza carton’s familiar logo causes Marcus to be sad and nostalgic of his free life. Yet before taken hostage, the pizza logo was virtually meaningless. This chapter has made me realize that we take so many things for granted and don’t realize what we have until its gone. Marcus even stated that he missed his school, which he couldn’t have hated more in the first few chapters. Aside from parents and friends, we also don’t realize the freedoms we as American citizens have, and all take for granted. At the end of the chapter, Marcus is finally released and couldn’t be happier have his usual clothes back and hear the familiar sounds, which went unnoticed before this event, in his familiar city. Marcus now appreciates these feelings and items that used to be overlooked. 

Motivation for Cracking Ciphers

The Beale Cipher consists of three cipher texts and only one has been cracked throughout the past one hundred years. One of the ciphers allegedly leads to where the treasure of over $20 million is located and the other two texts, one of which has been deciphered, are apparently descriptions of the actual treasure itself. The world’s most clever cryptographers have been trying to decipher the Beale Cipher for years, but it is hard to imagine why they would continue to try. Human beings are naturally curious, thus whether one is trying to decipher the Beale Cipher or a code we receive on our problem set, we strive to figure it out no matter how long it takes. When specifically looking at the motivation for solving the Beale Cipher, the incentive is of course the apparent money reward. By the same token, the glory one would receive if they were to crack the Beale Cipher would make all of their work worth it.

The Complexity of the Great Cipher

Antoine and Bonaventure Rossignol created the Great Cipher of Louis XIV and made it so complex that it took over 200 years to decipher. Generation after generation attempted to crack the cipher, yet no progress was made. The Rossignol’s both died, which terminated the ciphers use, as well as cutting off any potential collaboration with the creators and knowledge of exact details that could have been useful to the hundreds of codebreakers that tried to uncover the mystery. It was not until Commandant Etienne Bazeries came along and spent three years of his life working on deciphering letters of Louis XIV that the code was finally solved. Bazeries knew that it was not a substitution cipher, as there were 587 different numbers instead of the usual 26 different numbers. To Bazeries’ dismay, the cipher was also not a homophonic cipher, which was a possibility he entertained for months.

His final attempt proved to be worth it after all. The main factor that made this cipher so secure was the fact that each number represented a whole syllable, not a pair of letters. Bazeries finally got on a roll, guessing the remaining letters of an unfinished word, which enabled him to recognize other syllables. Another major deceiving factor in the Great Cipher was the traps that the Rossignols inserted; some numbers occasionally deleted previous numbers instead of standing for another syllable. The combination of the traps, the vague numbers, and the inability to collaborate with the Rossignols created an extremely secure and virtually unbreakable cipher.



The Ever Advancing Society

As an educated young woman in the year 2012, frequency analysis is not a hard concept to grasp. Using it as an aid to breaking ciphers, I apply fundamental knowledge to assess the frequency of certain letters and letter combinations in any given ciphertext. Because I have been reading and writing for over ten years, there is no need to be taught what letters or word combinations occur more often than others, but rather I have the ability to approach the ciphertext on my own. Back in the old days, the ubiquity of education was not as vast, thus citizens would need to be taught the theory of frequency analysis. Because most people worked on the farm and had little time to learn, scholars and highly educated people were the only ones exposed to the idea. It is interesting how these scholars would have to be thoroughly taught math, statistics, and linguistics in order to grasp the idea of frequency analysis, whereas amateurs today can use basic, elementary school logic to decipher a simple code. The fact that young people today can decipher a code using frequency analysis, without having been taught this approach (like we did in class on the first day), highlights just how advanced our society has become.

Powered by WordPress & Theme by Anders Norén