Math 645: A Survey of Mathematical Problems I
Frank Sottile

• This week, we start our last topic of this term, Chapter 5, which is about codes and cryptography. These are some of the most common and important applications of mathematics in the modern world. Our digital world would not be possible without these technologies.
      Despite the title of this chapter codes and cryptography are distinct topic, which are related in that they operate on digital information. Cryptography (what may popularly be called codes) is as old as civilization, and systems of them are called ciphers. Its intention is to send information securely, so that only the recipient is able to access the information. We use modern versions of this every day, unwittingly. One of my friends from graduate school, Kristin Lauter, was a major force in the development of the cryptography that ships in all Microsoft products.
      Coding (not writing computer code) is more modern. It is set up to solve the problem of how to protect data from degradation in transmission, retrieval, and storage. The US library system is an example of a relatively inefficient way to protect information; many copies of books are distributed into several different libraries, which protects against the loss of any one. In 1946 Claude Shannon proved the fundamental result in formation theory, which may be interpreted that such security of data may be achieved with minimal redundancy, and modern codes strive to reach this bound.
      Codes enable us to read and write data, and send it over the Internet. Each of these operations can and does degrade the data, introducing errors, and codes allow us to recover the original data from the degraded data.
      The first reading (25 by Kenneth H. Rosen) is about simple ciphers, and the second (26 by Joseph Gallian) is about a simple code used on German money (which no longer exists, due to the Euro). Note the portrait of Gauss on the Zehn Deutsche Mark; many countries put cultural and scientific heroes on their money, and not politicians. The third reading is a real treat, it is a reprint of the original article by Rivest, Shamir, and Adleman on their RSA method for cryptography, the second most important (but most popular) article in cryptography. (The most important is by Diffie and Hellman in 1976 that introduced the notion of public=key cryptography.) Those two papers laid the foundation for modern information security. While a treat, this is a real research paper, and it will be a challenge for you to read it and understand it. Dr. Geller has a few comments on this in her Lecture 12, which is part of your reading.

• Read 25 by Kenneth H. Rosen, 26 by Joseph Gallian, and 27 by Rivest, Shamir, and Adleman.
• Chapter 5 of A Survey of Mathematical Problems.
• Dr. Geller's Lecture 12.

1. Exercises 5.1–5.6.
2. Problems 5.1–5.3.