Teachers As Scholars (Steven J Miller: sjm1 AT williams.edu)

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Wednesday, March 20th and Wednesday, March 27th, 2019

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Cryptography

The ability to encode information so that only certain recipients can read it (or, conversely, to read information you are not supposed to have!) contains some of the most exciting applications of pure and applied mathematics. Since at least the time of Julius Ceasar (the title to this course is encoded with the cipher he made famous), codes and ciphers have been used to protect important information. We'll discuss various cryptosystems used over the centuries, mixing history and theory. In the course of our studies we'll discuss results from number theory, group theory, graph theory and combinatorics. This seminar is most appropriate for middle school and high school math teachers, but anyone who enjoys numbers and problem solving is welcome.

DAY ONE: March 20, 2019:

DAY TWO: March 27, 2019:

Reading list: the following files are from an earlier version of a book I wrote, `The Mathematics of Encryption', with  Midge Cozzens, for a general audience. Feel free to download these files, but please do not distribute further. If you want to use variants of these in your schools, just let me know. Comments on choice of topics and exposition are especially welcome; email me at sjm1 AT williams.edu. I've also posted some links to related material for those who want to read more.

• Classical cryptography: This is the core chapter. It begins with one of the oldest ciphers, and continues through subsequent improvements. You should download and read all of this before class.
• Enigma and Ultra: The Germans believed their Enigma code was secure; fortunately for us the Allies were able to crack it. In this short chapter we'll discuss some of the mathematics behind Enigma, and see why the Germans reasonably felt that it would be impossible to break.
• Public key encryption: In this chapter we discuss some modern cryptosystems. These are believed to provide a high level of security through the use of difficult math problems. The danger, of course, is just because no one has found a simple way to do a problem doesn't mean that there isn't a simple way. We discuss two systems, one based on graph theory (which we won't cover due to time constraints) and the other (RSA) based on primes. You should download and read this chapter before class.
• Mod p arithmetic, group theory and cryptography: The public key encryption notes need some knowledge of primes and group theory. This chapter (from my book `An invitation to modern number theory', joint with Ramin Takloo-Bighash), goes through much of the number theory, though at a very high level. We probably won't discuss much of this, but I've included this for completeness. This is mostly for reference -- don't worry about reading this before class.
• Errror detecting and correcting: This chapter here deals with transmitting information in such a way that, not only can we detect when we've made certain errors, but we can also correct them! The mathematics is motivated through some riddles. These riddles can be used to excite and interest students, and naturally lead to the more advanced material. You should skim, but only skim, the part on error detection and correction before class.
• You've seen these ideas all your life with UPC codes.
• Here are some notes I have written with Jim Kupetz, which are being used to develop computational modules for high school courses.
• Some cryptography links (no need to read these before the class):

Math Riddles
I also maintain a math riddles page. Please feel free to share these riddles with your colleagues and your students, and let me know if there is anything I can do to make the site more useful for you and your classes. The goal is to add a student / teacher's corner sometime in the spring to facilitate using these in classrooms. If you're interested in helping, or want updates on the progress, let me know.

Notes from 2012: lecture notes taken by David Strasburger,   RSA implementation by Tom Chiari