Winter Study 2013:
Math 10: LQWURGXFWLRQ WR FUBSWRJUDSKB
Professor Steven J Miller (sjm1@williams.edu)
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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 years. The course will be a mix of history and theory.
Prerequisites: Math 102 or its equivalent
Meeting time: main classes are Tuesday, Wednesday and Thursday from 10am to noon in Chem 202; we have Tuesday and Thursday afternoons from 1pm to 4pm in Chem 123 as needed.
Meetings / Syllabus: After the first week it is all tentative.
Week 1: Dec 31 to Jan 4: First meeting is Thursday, January 3rd from 10am to noon in Chem 202, followed by a lunch at a dining hall for anyone interested in trying to help me get to know everyone (we have 67 students so far), followed by a second meeting on Thursday January 3rd from 1:30 to 3pm in Chem 123. The classes will be a combination of an organizational meeting, discussion of the general concepts and themes of the course, and a lecture on some classical cryptography.
Week 2: Jan 7 to Jan 11: There will be no lectures by me this week as I will be in San Diego for the Joint Mathematical Meetings; I am hoping for a volunteer from class to give a lecture on LaTeX to anyone who has not used it. You can also view my video LaTeX tutorial here. You will be responsible for reading some chapters and being prepared to move fast the next week. If you are interested in a project (i.e., reading and presenting an aspect of cryptography to the class either alone or in a group, please let me know as this is a good week to work on it). I am strongly considering recording a lecture or two for you to listen to. If someone is able to meet with me to record it, that would be appreciated (and, of course, you would then be able to hear the lecture and ask questions).
Week 3: Jan 14 to Jan 18: There will be classes Tuesday, Wednesday and Thursday from 10am to noon in Chem 202. Tentatively Wednesday's class will be for mathematical review. There will be individual meetings with groups on Tuesday and Thursday afternoon, but these may need to be scheduled around job talks by candidates for the math position.
Week 4: Jan 21 to Jan
25: There will be classes Tuesday, Wednesday and Thursday from 10am to noon
in Chem 202. Rest TBD, but again potential scheduling issues with math job
candidates.
.
Evaluation: Writing up solutions to cryptography problems.
Note: Please feel free to swing by my office or mention before, in or after class any questions or concerns you have about the course. If you have any suggestions for improvements, ranging from method of presentation to choice of examples, just let me know. If you would prefer to make these suggestions anonymously, you can send email from mathephs@gmail.com (the password is the first seven Fibonacci numbers, 11235813).
Readings: to be distributed in class and by email. Click here for additional comments.
LaTeX links: my LaTeX handout and links to sites to download LaTeX.
Decrypting links:
This webpage is under construction -- more information will be added, but here are some fun cryptography puzzles and reading.
Consider an army with 10 generals. One wants a security
system where any three of them can determine the code to launch nuclear
missiles, but no two of them can. It is possible to devise such a system by
using a quadratic polynomial, such as a x2 + bx + c -- to launch
the missiles, one must input (a,b,c). One cannot just tell each general one of
a, b or c (as then it is possible that some subset of three generals won't
know a, b and c); however, if a general knows two of (a,b,c), then a set of
two generals can launch the missiles! What information should be given to the
generals so that any three can find (a,b,c) but no two can? What about the
general situation with N generals and any M can launch (but no set of M-1)
can?
You have 7 generals and a safe with many
locks. You assign the generals keys in such a way that EVERY set of four
generals has enough keys b/w them to open ALL the locks; however, NO set of
three generals is able to open ALL the locks. How many locks do you need, and
list how many keys the first general gets, the second, .... Is there more than
one way that works?
Somehow a man and a woman end up on two different desert islands with the man wanting to propose to the women by sending an engagement ring. Each has a lock and a key. Neither has a ship; however, there are pirates in the area who have a box. They're nice but untrustworthy pirates; they'll transport anything anywhere for free, but if you place something in an unlocked box, they'll take whatever is inside (if the box is locked, they'll transport it). Is there any way for the man to safely send the engagement ring?
Some readings
Notes on abstract algebra and group theory and RSA cryptography (from my
book An Invitation to
Modern Number Theory).
Markov
chains and cryptography: Persi Diaconis, The Markov Chain Monte Carlo
Revolution. (2008). Bull. Amer. Math. Soc. Nov. 2008.
Two centuries on,
a cryptologist cracks a presidential code: Rachel Emma Silverman, Wall
Street Journal, July 2, 2009.
20 years of attacks on RSA: Dan Boneh, Notices of the AMS,
February 1999, 203-213.
Group Theory in
Cryptography: Blackburn, Cid and Mullan.
Braid
Group Cryptography: David Garber
Elliptic curve Cryptography:
Expander graphs based on GRH with an application to elliptic curve cryptography: Jao, Miller (not me!) and Venkatesan
Do
all elliptic curves have the same difficulty of discrete log?: Jao, Miller (not me!) and Venkatesan
Quantum Cryptography:
Quantum Cryptography: From theory to practice: Ma (PHD dissertation, University of Toronto)
General articles from the NSA/CSS historical publications.