Cryptography

The History and Mathematics of Codes and Code Breaking

Month: August 2015

Evolution of Knowledge

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On August 30, 2015

In Student Posts

Isn’t it strange to think that modern high schoolers undoubtedly know more about mathematics and various disciplines of sciences than ancient or even not so ancient scientists who devoted their entire lives to certain subjects? I mean, if you think about it, it’s not too ludicrous. They definitely have a greater grasp on mathematics than, say, Pythagoras, who’s crown jewel of a discovery is currently being taught to 6th graders around the world. Take Isaac Newton, for example, a man who appears in textbook after textbook as the late-renaissance wonder man who invented integral and differentiable calculus. He discovered the basics for physics, including the laws of motion, gravity, and optics. However, they are exactly that, the basics. Now, do not think I am bashing Newton by any means. Nor am I saying that high schoolers know all that he knew or are more intelligent than him. He is possibly the greatest thinker in the history of man; however, modern education has advanced so much so that today’s teenagers now take the knowledge that past scientists and mathematicians spent their lives discovering for granted. More and more advanced knowledge and problem solving skills are being exposed to a younger and younger audience in today’s education system. That is not to say that past “ground breaking” discoveries were by any means easy. It was no more easy than a scientist today discovering the secrets to the quantum world.

Because of this, it is no surprise that seeming “amateurs” use frequency analysis and other cryptanalysis strategies that took centuries to develop. Let’s take the credentials of a possible “amateur” cryptanalyst into consideration. He/She probably has some sort of upper level (comparative to a few hundred years ago) mathematic training including calculus and statistics and probably multiple years of taking English courses, all of this learned from high school. According to my previous assertion, wouldn’t this count as a “sufficiently sophisticated level of scholarship in several disciplines, including mathematics, statistics, and linguistics” (Singh 15)? I think so.

The Infinite Struggle Between Makers and Breakers

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On August 30, 2015

In Student Posts

The continuous tug-of-war between cryptographers and cryptanalysts has triggered the expansion and prominence of cryptography in the world today. Just as quickly as new encryption methods are developed, new decryption strategies are implemented, creating a state of equilibrium in cryptography.

The art of secret writing dates back to Herodotus (Singh 3) and has had various uses in history, including military communication and murder plots. In more recent years, the explosion of technology has propelled the expansion and necessity of cryptography to the point where it has an effect, either direct or indirect, on every individual. Cryptography has a range of important functions, such as keeping bank account information private and ensuring that government information is kept secret.

Cryptanalysis was not invented until the Islamic golden age under the Abbasid caliphate (Singh 14). According to Singh, cryptanalysis could not be invented until a civilization had reached a sufficiently sophisticated level of scholarship in several disciplines, including mathematics, statistics, and linguistics (15). The Muslim civilization provided an ideal cradle for cryptanalysis because the Muslims were educated in a variety of fields and pursued knowledge in all of its forms (Singh 15).

This breadth of knowledge associated with the Muslim civilization is valued in most modern society today. Most individuals are at least somewhat proficient in an assortment of disciplines. Primary education sets the field by educating children about a variety of different subjects in order to equip them with a large intellectual toolset. Many secondary education institutions stress a liberal education in which students obtain an educational background in the humanities as well as the math and science fields.

Because of this, it is no longer necessary to undergo formal education in cryptanalysis in order to use elementary strategies such as frequency analysis. With the broad education that modern citizens receive, combined with the availability of abundant resources, especially those on the Internet, many of the older, simpler codes can be decrypted by the layperson. However, modern encryptions are much more complex than those of the past and may require serious study.

During the past few centuries, cryptography has exploded due to the expansion of technology. Cryptography affects numerous fields ranging from government information to banking, and brings in knowledge from a variety of areas including mathematics, logic, linguistics, and statistics. Modern education allows us to engage in the world of cryptography even as the complexity and number of cryptographic techniques increase exponentially.

Standing on the Shoulders of Giants

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On August 30, 2015

In Student Posts

It is curious to think of how the world was just one hundred years ago. So many things that are now obvious were unknown or shrouded in mystery. In 1915, barely anything was known about heredity and the cellular functions that were involved in it. Now, we have mapped the entire human genome and can even change DNA. In Newton’s time, gravity was a concept that hadn’t been explored. Now, it is an obvious fact, one that forms the most basic aspect of sciences that have now advanced to ever-increasing complexity. What is gravity when compared to the Higgs boson?

In this same way, the discoveries made by cryptanalysts of centuries past have now become obvious to us. This is by no fault of theirs—without civilization’s ability to analyze statistics and linguistics and apply mathematical concepts, decrypting encrypted messages could never have been attempted.

The thing is, civilization didn’t stop there. It continued to grow and make new discoveries while standing on the foundation laid by its predecessors. Discoveries of algebraic concepts that excited prominent scholars hundreds of years ago are now taught in middle school classes to unappreciative twelve-year-olds. They are no longer new and complicated and exciting, but old news, taken for granted, never thought of unless they are used for the springboard into some novel inquiry.

Another aspect to consider is the advance of technology and information. Doing statistics by hand is a painstaking process that can now easily be bypassed by calculators and software. In addition, so many more people have access to information now than they used to, whether that be in a classroom or on the Internet. Especially when considering the Internet, where any question can be answered easily in a matter of seconds, it is not all that surprising that amateur cryptanalysts can “wing it.”

This is not to diminish the strategies implemented by older cryptanalysts. Rather, it is to show how far we have been able to come since their time because of the nature of their discoveries. As Isaac Newton said, “If I have seen further it is only by standing on the shoulders of giants.”

The Beginning of an Era of Secrecy

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On August 29, 2015

In Student Posts

Imagine for a moment that everything you’ve ever hidden is completely public. Everyone has access to your private emails and your bank account information, among other things. This would be life without cryptography.

Cryptography is hiding the meaning of a message, and it is typically used in most forms of modern communication. In The Code Book, the author, Simon Singh notes that secret writing has been a part of human civilization since the fifth century (Singh 4), but it was widely accepted that the most typically used cipher of ancient times, the substitution cipher (Singh 13), was impossible to crack until the 8th century when the Abbasid caliphate’s place as a center of learning allowed it to become the homeplace of cryptanalysis (Singh 16).

Singh defines cryptanalysis as “the science of unscrambling a message without knowledge of the key” (Singh 15). As Singh states on page 15, cryptanalysis was only possible in the beginning due to the Muslim civilization achieving “a level of scholarship in several disciplines, including mathematics, statistics, and linguistics”.

The important debate, though, is whether or not this level of scholarship is still necessary in today’s society. I am of the opinion that, although our society’s era of secrecy necessitated well-trained cryptanalysts, this is no longer the case. As our class demonstrated on only the second day, breaking of the more common codes is fairly simple for most modern humans. The difference, I believe, lies in the fact that in ancient times, education and widespread knowledge had not progressed to the point it has now reached. The internet, as a portal to almost all human knowledge, has made it simple for anyone to pursue any knowledge or expertise that they desire.

Without modern technology, and modern education, I am of the opinion that intense training would still be required to become a cryptanalyst. However, due to our civilization’s widespread resources, it has become much easier for individuals to discover and crack codes and ciphers on their own.  Cryptology began as a secretive science, but has become an integral part of modern society, and as such we are all cryptologists in some form or another.

Don’t Overlook the History of Seemingly Simple Things

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On August 29, 2015

In Student Posts

Cryptography and cryptanalysis are two fields whose progress is intertwined; both make advances to either get an advantage over the other or to compensate for a breakthrough the other has made. As societies have progressed, the need for more complex methods of both encryption and decryption has risen along with the complexity of society.

Messages have been hidden and encrypted from prying eyes since the fifth century B.C. (Singh 4), but it wasn’t until around 800 A.D. in the flourishing Arab empire that cryptanalysis was invented (Singh 17). Singh notes, “Cryptanalysis could not be invented until a civilization had reached a sufficiently sophisticated level of scholarship in several disciplines, including mathematics, statistics, and linguistics” (Singh 15). But ever since, the battle between cryptographers and cryptanalysts has employed the growing knowledge and technology of civilization, to the point where the decryption methods discovered by the greatest of Arab thinkers is now almost common sense for any elementary school child.

Why is that? Well consider for example how mathematical technology has advanced along with society. The abacus appeared in China as early as 500 B.C., followed by the invention of Arabic numerals in 1202, and just think of all the different models of TI-calculators that can be found at Target now (source). For a young child, counting with Arabic numerals is pretty simple, but the use of more complicated and advanced calculators takes more work and learning. The same goes for code-breaking: basic substitution ciphers can be easily figured out, but more complex codes and ciphers will take more time and effort.

This shows how far society has come in a couple thousand years; it shows how human knowledge builds upon itself to reach even higher. The breakthroughs by Arab cryptanalysts over a thousand years ago sparked entire industries and professions, and who knows, maybe a discovery in 2015 could change the entire future of cryptography and cryptanalysis.

The Evolution of Ciphers and the Human Mind

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On August 29, 2015

In Student Posts

Over two thousand years ago, Julius Caesar’s methods of secret keeping were deemed groundbreaking.  His use of the substitution shift cipher was effective and secure.  Today, however, many would find his seemingly indestructible cipher to be elementary.

Like most human inventions, codes and ciphers have constantly evolved over the years to become more complicated and much more difficult to break.  Because of this advancement, what were once the world’s best ciphers and codes thousands of years ago have become simple puzzles that high schoolers, even middle schoolers, can deconstruct.

According to the author of The Code Book, Simon Singh, “[c]ryptanalysis could not be invented until a civilization had reached a sufficiently sophisticated level of scholarship” (Singh, 15).  The Muslim civilization achieved this heightened academic prowess due to its emphasis on being well-rounded, insightful humans.  Citizens studied a wide range of subjects, “including mathematics, statistics and linguistics” (Singh, 15).  Europeans, conversely, were stuck in the Dark Ages, unable to pursue the high scholastic level of the Islamic civilization.  While the Arabs were creating new ciphers and breaking old ones, Europe was far behind.

The Arabs had a significant advantage over the Europeans due to their overall knowledge of various subjects.  Similarly, humans today have an advantage over the Muslim civilization.  The ciphers that the Arabs were creating and breaking were much simpler than the ciphers are today, and after studying history and learning of ciphers and codes from old times, people today are able to easily decrypt old ciphers and codes.  Presently, individuals do not have to be trained in cryptanalysis because subjects such as mathematics and statistics are available to the average citizen.  Society today focuses almost entirely on “secular subjects” (Singh, 16), which is what led to the success of the Muslims.  People today also have greater opportunities to learn and have learned from history, so we are able to combat difficult problems, specifically ciphers, on our own, using past methodology, logic, creativity, and luck.

The evolution of the cipher is directly connected to the expansion of the human mind.  Substitution ciphers that were used by Julius Caesar are now commonly recognized and easy to decipher.  People today have a much more extensive knowledge of ciphers and codes, making the ciphers and codes easier to figure out.  As humans continue to advance, so will ciphers and codes and the means to breaking them.

 

Singh Chapter 1 – Reading Questions

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On August 28, 2015

In Reading Qs

The-Code-Book-Singh-Simon-9780385495325As I mentioned in class, you’ll need to read the first chapter of The Code Book for Monday. In case you’d like a little guidance for your reading or would like to prepare for Monday’s discussion, here are a few questions about Chapter 1 you might consider. I’m not expecting you to answer these questions (on the blog or in writing), I’m just providing them as a resource.

  1. On page 41, Singh writes, “The cipher of Mary Queen of Scots clearly demonstrates that a weak encryption can be worse than no encryption at all.”  What does Singh mean by this and what does it imply for those who would attempt to keep their communications secret through cryptography?
  2. Most of the examples of cryptography in Chapter 1 were associated with well-resourced people—monarchs, military leaders, etc.  Is that because those are the only examples that have survived or is that because cryptography and cryptography development is dependent on exceptional resources?  If the latter, do you think that has changed over time?  What implications does that have for today’s uses of cryptography?
  3. Given that Singh was presumably trying to write an interesting and engaging book, why do you think he chose these examples for Chapter 1 instead of other potential examples of classical cryptography?

Blog Assignment #1

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On August 28, 2015

In Blog Assignments

Ghost WriterFor your first blog assignment, write a post between 200 and 400 words that responds to the following question:

On page 15 of The Code Book, author Simon Singh writes, “Cryptanalysis could not be invented until a civilization had reached a sufficiently sophisticated level of scholarship in several disciplines, including mathematics, statistics, and linguistics.”  If such a level of scholarship was required for the development of the frequency analysis approach to solving substitution ciphers, what do you make of the fact that amateur cryptanalysts today often use that approach “on their own,” so to speak, without being trained in it?

Please give your post a descriptive title, and use the “Student Posts” category for your post. Also, give your post at least three tags, where each tag is a word or very short phrase (no more than three words) that describe the post’s content. You’re encouraged to use tags already in the system if they apply to your post.

Your post is due by 9:00 a.m. on Monday, August 31st. If you have any questions about sharing your first post here on the blog, don’t hesitate to ask.

Update: Here are some basic instructions for posting to WordPress that you might find useful. Also, via xkcd, here’s the secret to using any kind of computer technology.

Image: “Ghost Writer,” by me, Flickr (CC)

Welcome to Class!

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On August 26, 2015

In Resources

I’m excited to teach “Cryptography: The History and Mathematics of Codes and Ciphers” again this fall. It’s my favorite course to teach, and I hope you find it interesting, too.

Here’s a copy of the Fall 2015 Syllabus.  Please read this before class on Friday, when we’ll talk about various aspects of the course and I’ll take your questions on the syllabus.

I thought I would share a couple of links related to Edward Snowden and the NSA. During class, I shared this graphic from ProPublica summarizing the various NSA programs that we’ve learned about from Snowden’s link. I also recommend listening to this interview with one of the reporters who put together the chart, which aired on ProPublica’s podcast.

I’ll also remind you that you should read the first chapter in our textbook, The Code Book by Simon Singh, before class on Monday. We’ll talk about other upcoming assignments during class on Friday.

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