Teachers As Scholars (Steven J Miller: sjm1
LQWURGXFWLRQ WR FUBSWRJUDSKB (Introduction to
Cryptography) and Benford's Law
Monday, January 24 and 31, 2011
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
- 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:
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):
Law of Digit Bias, or: Why the IRS cares about Number Theory (lecture
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
- 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.
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).
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.