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

LQWURGXFWLRQ WR FUBSWRJUDSKB (Introduction to Cryptography) and Benford's Law

Monday, January 24 and 31, 2011

• Lecture Notes: First Lecture: Monday, January 24, 2011.
• Lecture Notes: Second Lecture: Monday, January 31, 2011.
  • Benford's Law lecture notes
  • Cryptography lecture notes
• List of riddles from first lecture.
• Link to my math riddles homepage.
• quick links:    Cryptography    Benford's Law    Math Riddles

Cryptography (lecture one notes here    lecture two notes here)

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.

Reading list: the following files are from a book in progress, tentatively titled `The Mathematics of Encryption' by myself, Midge Cozzens and Wesley Pegden), for a general audience. Feel free to download these files, but please do not distribute further. Comments on choice of topics and exposition are especially welcome; email me at sjm1 AT williams.edu.

• 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.
• 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 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, correcting and factorizations: The first 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. The second part concerns the important problem of factorization, and discusses some of the ways to try to factor numbers and detect primes. We'll discuss at least the main themes of these chapters, and go into detail as time permits. You should skim the part on error detection and correction before class.
• Diffie-Helman-Merkle key exchange.
• Articles from the NSA on cryptography (this is a link to many subpages). Two especially good and accessible ones deal with the German code Enigma, and Ultra, the allied deciphering of it. I strongly urge you to look at the links here and share these with your students and colleagues. Another nice one is on the Battles of Coral Sea and Midway.
• Civil war message just decoded -- fortunately it wasn't needed! (Another version of the story here.) A nice application of the Vigenere cipher. See also the notes by my colleague here.
• An encrypted message to Thomas Jefferson.
• Here are some books recommended by the class:
  • The Code Book: The Science of Secrecy from Ancient Egypt to Quantum Cryptography: By Simon Singh
  • In Code: A Mathematical Journey: By David Flannery and Sarah Flannery
  • Applied Cryptography: Protocols, Algorithms, and Source Code in C: By Bruce Schneier

For more resources on cryptography, see my webpage from the Winter Study course I taught at Williams in January, 2010. See also the riddles list.

Here are some programs I wrote (in Mathematica):

• Caesar Cipher
• Vigenere Cipher
• Random alphabet permutation
• Miller cipher: Caesar cipher with a twist
• Another Miller cipher: adding and adding
• Schrock polynomial cipher: how secure is this?

Benford's Law of Digit Bias, or: Why the IRS cares about Number Theory (lecture notes here)

How can you tell if a sequence of numbers is 'random' (and, of course, what does random mean here)? We'll discuss a variety of real world problems where the behavior of the leading digits is not what you would expect. For example, the first digit of the Fibonacci numbers or equals 1 about 30% of the time, not 10% or 11% as one might expect. The IRS uses this digit bias to detect fraudulent corporate tax returns. This phenomenon, known as Benford's Law, was first noticed by observing which pages of log tables were most worn from age -- it's a good thing there were no calculators 100
years ago! We'll discuss some of the theory and many of the applications, using this as a springboard to discuss basic probability and difference equations. This seminar is most appropriate for middle school and high school math teachers, but anyone who enjoys numbers and problem solving is welcome.

Introduction to Benford's Law: This is the first chapter of a book I edited on Benford's law, and explains what the phenomenon is and some of the places it occurs. You should download and read this chapter.
• Recurrence relations and Benford's law: This is essentially Chapter 9 of my book `An invitation to modern number theory'. Much of this chapter uses advanced results from Fourier analysis, but much of it should be accessible to a general audience. In particular, some of the general framework of what a recurrence relation is, why they typically obey Benford's law, and the basics of hypothesis testing. This is written at a high level with all the details; in the class we'll talk about some special cases that highlight what's going on. You should look at this to get a sense of what mathematics is used in attacking these problems, but not worry much about the details. You should quickly skim this before class.
• Detecting fraud and errors using Benford's law: This chapter for the book, written by Mark Nigrini, talks about some of the applications to detecting fraud. Due to copyright restrictions, this cannot be posted online. I'll hand this out at the first class, and you should read before the second class.
• Excel spreedsheet program to test for Benford behavior. This is an Excel program I wrote a few years ago to allow you to quickly test the first as well as first two digits against Benford's law.
• Great movie about the Fibonacci numbers in Nature.
• Topics in Mathematical Modeling: Chapter One (available online here) is a nice introduction to some properties of the Fibonacci numbers.
• Mathematica program to solve recurrence relation for double plus one.
• Webpage on Roulette strategies -- they don't mention the dangers of double plus one (they call it the martingale method).

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.