“Sometimes it is the people no one imagines anything of who do the things no one can imagine.”
Some of you may have heard of this quote from the famous movie “the Imitation Game.” It’s a story of the renowned English computer scientist and cryptanalyst Alan Turing who decrypted codes of the German army to help Allies win World War II.
In the movie, the code was used for secure communication. and the study and practice of techniques for secure communication is called cryptography.
This summer, Exeter starts a brand new course— cryptography. It’s my honor to be one of the first 7 students to experience the fun of cryptography. When I was selecting my class in Exeter, I saw this class, which caught my eye immediately, by chance. Due to my deep interest in math, the unable-to-hold-oneself-back feeling pushed me to sign up for this course, instead of for multivariable calculus or statistics.
The truth is that this is a really interesting class indeed, and suitable for everyone, no matter whether you are just interested in math or have got plenty of prizes in math competitions worldwide or regionally like me. In class, we learn cryptography techniques as well as math in a wholly different way compared to any other math class.
Basically, we learned different methods of encrypting plaintext and decrypting codes, such as using the Caesar Cipher, Rectangular Arrays, or Cipher with Modular Arithmetic, etc. Take the Caesar Cipher for example: we first set a number less than 26 but larger than 0 as the key. Then we convert the letters into numbers according to the alphabet. Assume that A=0, so B=1, C=2…Z=25. After that, we can finally encrypt our plaintext by adding the key number to the numbers of letters, which will lead us to a new number. The last step is to reverse the first step— convert the number into letter, and we will get our encrypted text. Sounds easy, right? This is the easiest way to encrypt plaintext.
It seems that this is the end of this kind of method. I can just tell you that it’s partly right. What if I get rid of the limit that the number must be less than 26?
The upshot is that the number is larger than 26, which can’t be represented as a letter. This problem leads us to a new method of encrypting text— the Vigenere Cipher. Modular Arithmetic is of great importance in this method. Since there are 26 letters altogether, we may set up the Modular system for mod 26 in order to be convenient when calculating. Assume that we get a number 35, it’s the same with 9 in mod 26 because in mod 26, the only existing numbers are 0 to 25. Since the result that subtracting 26 from 35 equals 9, then 35 and 9 are equivalent in mod 26.
However, there is another solution to the problem. Why not just use numbers to take the place of letters? This thought then leads us to another new method, whose cipher text is composed of numbers.
From all the example above, you may find the difference between cryptography class and other classes. We gain math knowledge from various cipher texts and encrypting methods. One particular method can lead to another more complex one, which requires more knowledge. Hence, when we are learning ways to encrypt messages, we also have a better understanding of Modular Arithmetic, the property of relative prime, and definition of the inverse function in a more vivid way.
With the Harkness and enlightenment from our teacher, Karen Lassey, we expanded the problem into how to make codes safer and more confusing or how to solve problems when decrypting the code. For instance, in the Vigenere Cipher, we discussed what will happen if we use symbols to represent numbers larger than 26 instead of subtracting 26 from them. Is it more confusing? What shortcoming does it have? What’s more, we also talk about the possibility of applying letter frequencies to create a new cipher. If these ideas will be put into use sometime in the future, it will certainly enhance the safety of the codes.
Actually, cryptography knowledge has been put into use in our daily life. Some of you may know the ISBN numbers, but everyone should have noticed the bar codes and the numbers below on anything you bought. These numbers are called the ISBN number, which are used to identify whether this product is a real one or a fake one. When identifying it, Modular Arithmetic is used. If you have interest in that, you can search for more information on the Internet.
Cryptography is a new-born field with much potential and can be developed much further than it is today. I am convinced that cryptography will continually promote our information security, and Exeter opens a door for us to access this mysterious subject. I hope sincerely that there will be more students taking cryptography class in the next few years.