The History and Mathematics of Codes and Code Breaking

Author: leehe

Turning Wheels

Cryptonomicon creates a fictional setting in which Neal Stephenson recreates the chaos of World War II cryptography. A passage of particular interest involved the relation of Turning's bicycle wheel to the cipher machine. Waterhouse eyes the bent spoke and weak link in the chain of Turning's bicycle and his mind immediately goes to the mathematical implications of the weak parts. By describing the math involved to figure out when the chain will entirely fall off - which only happens once the weak link of the chain and bent spoke come into contact with each other - and applying it to the mathematics involved in the rotors of an Enigma machine. Just as Turning's bicycle wheel has a certain period of rotation until the chain will fall off and the bicycle will be useless, Stephenson explains that the rotors also have a period. With three rotors, the period, or the number of times until the nth letter is enciphered with the same letter as the first letter of the message, is 17,775;

This passage not only represents the complicated mathematics involved in solving the Enigma, but also the ingenuity on part of the Germans in adding another rotor to their cipher machine. Because the period of the machine increased by a factor of twenty-five, and messages sent are unlikely to reach a length of 456,976 characters, the Germans greatly increased the security of their cipher through the introduction of the fourth and fifth rotors, a concept we have previously discussed, but the mathematics presented in this passage helped me to further understand the exact implications of the addition. yet, with a fourth rotor added, this period increases to 456,976 letters.

Image: Yellow Bike, Flickr

The NSA Standard: Invasion or Protection?

Computers, by their simple invention, launched the complexity of cryptography to levels of security that before had seemed unattainable. Ciphers could not only be computed quicker, with more efficiency, and with less chance of human error, but also the amount of cryptanalysis required to decipher encrypted text multiplied. With such advancements, the government saw its ability to monitor communications effectively slip from its hands as mathematicians such as Horst Feistel created new cryptography systems that utilized the new technology. The National Security Association, in November of 1976, created a Data Encryption Standard, or DES, that would allow businesses and people to communicate through secure processes that involved up to 100,000,000,000,000,000 keys, yet t

The adoption of the DES by the NSA was a reasonable standard, seeing as the number of keys were limited to 56-bit, a number that could only be broken by the technology available to the NSA (Singh, 250). Businesses are still able to communicate with optimal security, depending on which encryption system they use, yet, the NSA can intercept and decipher messages they perceive to be dangerous to the country. While businesses can express concern that DES could allow rival companies to be able to hack into their data and use it to their personal advantage, the NSA has determined that the DES is strong enough to protect against such actions (Singh 250). Businesses and citizens are asked a small price to pay in return for their nation's security. The NSA uses the data they've collected to ensure the safety of citizens, and the enforcement of the DES is a small price to pay for what the NSA gives back. he NSA could still decipher it. The transparency required by the standard does not correlate to the government's need to monitor all forms of communication, but rather the necessity of an efficient system of monitoring communications that could result in the harm of the United States and its citizens.Another concern that may arise when considering the justification of the NSA's decision is whether or not limiting the number of keys invades the privacy of citizens. While such a controversial issue cannot be simply proven wrong or right, considering the issue at hand, if a person or company has the necessity of using an encryption system that not even NSA could break, then there actions may prove to be illegal, in which case the NSA has every right to know about and intercept any information communicated via ciphertext.

Image: 30 seconds of my life by Jeff Carson, Flickr

The Power of Publicity


In Cory Doctorow's Little Brother, the author examines the boundaries of invasion of privacy in today's society. As the main character, Marcus, and his friends fight against the  U.S. Department of Homeland Security's intense surveillance of all citizens following an astronomical terrorist attack, they must establish methods for communicating without their messages being interrupted by the DHS, whose head members are scrambling to accumulate evidence that Winston took part in planning the attack. 

In chapter six of Doctorow's social criticism, Marcus explains that he will need to encrypt his messages to avoid the prying eyes of the government. In his brief discussion of cryptography and its effectiveness, Marcus makes a startling affirmation. "You have to publish a cipher to know that it works," he claims. While this idea initially seems to violate the idea of cryptography, encoding messages to keep the content safe from being revealed to anyone but the intended receiver, after some thought Marcus's bold statement reveals his true wisdom. He explains that while he could create his own cipher, he would never know if it was secure from others because he had created it himself without first testing its security. Contrary to "anyone" who can create their own cipher system that to them is unbreakable, Marcus suggests first publicizing said cipher system before use. This method would release one's code into cyber space or print, encouraging others to attempt to crack it. Marcus concludes his argument by simply stating that in today's society, you do not simply create your own cipher and assume it is secure; rather, he emphasizes using "stuff" that has been around forever, but has never been successfully cracked.

I initially found Marcus's assertion that publicizing one's cipher was the ultimate way to ensure security to be naive, but with further examination found it to be most insightful. Initially publishing the cipher you created to see if others could indeed break it seems silly. You would not be able to utilize the cipher if it is released and then cracked, and simply utilizing the cipher without checking its security could be more efficient; however, if you publish your cipher for everyone to see, and encourage others to break it, you are utilizing the most valuable source for ensuring its security. Ultimately, having an insecure cipher is more detrimental than losing speediness while searching for an effective cipher.

Image: In The News by paurian, Flickr (CC)

Beale Cipher Continues to Confound Cryptographers

The Beale Cipher has, for many years, stumped the best and brightest cryptographers in their quest to not only decipher the text, but also discover the treasure behind it. Despite years of unsuccessful attempts to decipher the complex cryptography, many cryptanalysts continue to analyze the cipher Beale created. The fruitless efforts of many analysts must have a much deeper cause than a simple search for treasure.

The enigma of the Beale Cipher drives cryptanalysts to further pursue its deciphering. The motivation comes from the mystery that lies behind its message and its key. A sort of reverse psychology plays a role in its mystery. The cipher has been deemed unattainable to any that have tried it; yet, the inherit inability to solve it motivates other cryptanalysts to try and break it. Just as children who are told they should not touch the stove do it anyways, cryptanalysts regard the difficulty of the cipher not as a warning, but as a challenge.

In an attempt to define the motivation behind cryptanalysts' quest, one must also consider our ever changing world. Each day, new technology emerges, developments in research are made, and new masterpieces are created. With this constantly developing society comes the social drive to outdo others' achievements. While no one has yet solved the Beale cipher, cryptanalysts see the challenge as an opportunity to outdo their peers, using the technological advancements of today to drive their discovery.

Image "Bound to Make the Connection" by Jackson, Flickr (CC)

The Great Cipher Eludes Great Minds

The Great Cipher was elusive to even the greatest scholars for more than two centuries, creating a whole span of encrypted letters containing enigmatic answers to some of the biggest speculations in history. When the breakthrough in the pattern of the cipher came in 1983 at the hands of Bazeries, the reasons the cipher was exponentially difficult to crack were revealed.

Not only did Bazeries discover that there were 587 different numbers in total, but he also learned after painstaking exploration that the numbers were not homophones. The simple fact that the Great Cipher did not follow the common practice of substituting one or multiples numbers for a single letter further complicated the ability for cryptanalysts to crack it. Furthermore, the Great Cipher was additionally not combinations of double letters indicated by numbers, but instead contained numbers that represented the syllables in the French language.

The Rossingols were not secure in simply allowing numbers to represent syllables. To further complicate the cipher, the number of digits in the numbers representing each syllable did not correspond to the number of letters in each syllable. For example, in the first word Bazeries decrypted, "les ennemis," while the three digit number 124 represents the syllable "les", three digit numbers also represent "ne" and "s", 125 and 345, respectively. The Rossingols additionally created numbers that represented not a syllable, but deletion of the previously stated syllable.

Together with the death of the Rossingols before the secrets of the cipher could be revealed, these factors created an entirely secure cipher. One so secure it would take the human race an additional two hundred years of discovery to crack.



Cryptanalysis and Critical Thinking

Certainly, the level of scholarship experienced today is far more advanced than that of the past. Each day, each hour, each second advances are made that further expand the breadth of human knowledge; however, this knowledge stands on multiple foundations of  understood ideas,  preconceived notions, and intuitive reasoning. Cryptanalysis today can be accomplished by those of relatively common education because of the foundations set by previous eras. Thus, as Singh describes, civilizations in the past required more time, effort, and education to establish the principles of frequency analysis simply because they had not acquired the foundation of knowledge that is common today. In the era in which frequency analysis was invented, an "educated person" of the society would have the knowledge to read, write, and do arithmetic, and those skills were a great accomplishment for that era. Yet today eight-year-olds throughout the world have already acquired those skills, along with ways to collaborate with others, communicate effectively, and think critically. This advancement was only possible with a strong foundation of knowledge.

Critical thinking skills also allow ordinary cryptanalysts to intuitively use frequency analysis without training. Today's constantly shifting society forces each person in it to think critically every day. From the rate that technology is advancing to daily medicinal miracles to constant national turmoils, each day provides an opportunity to create something new, something more effective, something no one else could conceive. In this modern society, critical thinking is vital to survival. Conversely, in the years in which frequency analysis was invented, civilizations expanded at a much slower rate, and thus critical thinking at the rate at which it occurs today was much less prevalent in earlier populations.

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