Mathematics 116
Spring, 2010
Tu Th 11:10AM-12:30 PM, 3109 Etcheverry Hall

Course control number 54482
Current enrollment information

Professor Kenneth A. Ribet
email:
Telephone: (510) 642-0648
Fax: (510) 642-8204
Office hours (885 Evans Hall)

photo of Campanile repair in 2009

Prerequisites

Math 55 is the official prerequisite. However, the essential mathematical content of this course overlaps significantly with the three courses Math 110, Math 113 and Math 115. Experience with these courses is helpful, though not necessary.

The exercises in this course involve calculations that cannot be performed by hand. At the beginning of the course, all that is needed is a calculator. By the end, you will need to use Sage or Magma to do the homework. Since Sage is free, while Magma is a commercial product, I urge you to download Sage and try it out as soon as possible. You can also work with Sage online instead of downloading it and using it locally.

Textbook

An introduction to mathematical cryptography by Hoffstein, Pipher and Silverman.

Although this book describes itself as "self-contained," it includes compact summaries of material from and abstract and linear algebra and from number theory. If you haven't had courses in these subjects, be prepared for moments when you will need to digest a lot of material in a short amount of time. As we go through the course, look ahead so that you can get a head start on problematic passages.

Be sure to consult the authors' errata list if you think that you've spotted an error. The authors will be grateful to readers like us if we send them additional corrections.

The book is available for purchase directly from Springer or from Amazon. However, if you have an IP address that is associated with the campus and navigate to http://www.springerlink.com/content/978-0-387-77993-5, you should see things like

Institutional Login 
Recognized as: 
University of California at Berkeley (414-47-457) 
California Digital Library Springer (798-02-082)

and Buy a Print Copy of this Book for $24.95 Including Shipping. Further, you can download the book, chapter by chapter, as a series of eight .pdf files from the springerlink site. Accordingly, you can have the book available to you on your laptop, desktop or other digital device. If you are not physically on campus but need access to electronic resources for which UC is a subscriber, see the UC Berkeley Library Proxy Server help page for some pointers.

Examinations

Please do not plan travel on these dates:

First midterm exam, February 18, 2010 in class (Questions and skeletal solutions from last year's class and the analogue from this year's class), mean of scores 23.53, standard deviation 3.85;
Next midterm exam, April 1, 2010 in class (Questions and skeletal solutions from last year's class and the analogue from this year's class); cumulative distribution, showing the 33 scores, ranging from 4 to 25;
Final Exam, Thursday May 13, 2009 8-11 AM in 3109 Etcheverry Hall (Questions and possible solutions from last year's class and the analogue from this year's class); mean = 30.3; standard deviation = 9.6.

The College's 2009-2010 student calendar lists drop and grade-change deadlines. You can still drop the course the day after the first midterm and still change your grading option to P/NP the day after the second midterm.

Lectures

The catalog description (which was written by me and/or Craig Evans) is very terse: "Construction and analysis of simple cryptosystems, public key cryptography, RSA, signature schemes, key distribution, hash functions, elliptic curves, and applications." The book covers these topics and more. I plan to follow the textbook, ending somewhere in Chapter 6 (the chapter on lattice-based cryptography).

To follow the lecture-by-lecture pace of the course, please consult the Math 116 iCal calendar.

As I stress above, the book can be viewed as self-contained only because it includes quick summaries of a number of topics that are best viewed as inputs to a study of cryptography. Among these topics are

Basic linear algebra as in Math 54 and Math 110;
Elementary number theory as presented in Math 55 and Math 115;
Not-so-elementary number theory (quadratic residues, quadratic reciprocity), which is usually discussed in Math 115;
Group theory as studied in Math 113;
Commutative ring theory (including ideals and quotient rings) as studied in Math 113;
The basic theory of elliptic curves (not typically studied in our undergraduate courses).

You can do yourself a big favor by checking out these sections (§§1.2-1.4, §2.5, §2.10, §3.1, §3.9, §§5.1-5.2) ahead of time to see whether they are likely to be difficult or easy for you. I will of course discuss them in class, but my treatment will be a bit fast for people who have never thought about the relevant subjects in their lives.

Homework

You may find the authors' Snippets from Selected Exercises helpful if you want to paste strings into a computer application.

Problems from Chapter 1 due January 28, 2010: 1.5, 1.7bc, 1.8bd, 1.9ad, 1.10ad, 1.11bd, 1.13, 1.16 (all parts). For some of these problems, it will be helpful to have an open Sage notebook available for calcuations.
Problems from Chapter 1 due February 4, 2010: 1.17fg, 1.18, 1.21, 1.23ac, 1.27, 1.28c, 1.30c, 1.31, 1.33, 1.34
Problems due February 11, 2010: 2.4bc, 2.6, 2.8 (all parts), 2.10 (all parts), 2.17 (all parts), 2.19, 2.23 (all parts)
Just a few problems due on February 18, 2010: 2.25, 2.28ab, 2.34a
Not that many problems due on February 25, 2010: 2.38, 3.1ac, 3.2, 3.5a, 3.5b(i), 3.6; also use the Pohlig-Hellman algorithm to find x such that 5^x is congruent to 67 mod 103.
Problems due March 4, 2010: 3.8b, 3.9ab, 3.10, 3.13b (iii and iv), 3.14ace, 3.15, 3.18
Problems due March 11, 2010: 3.21ab, 3.22bf, 3.23a, 3.24a, 3.25c, 3.39, 3.40
Problems due March 18, 2010: 3.41bc, 4.2abd, 4.3de, 4.5 (all parts), 4.21, 4.22, 4.24, 4.27, 4.29
Problems due March 32 (a.k.a. April Fool's Day), 2010: 4.32, 4.34a, 4.35, 4.37, 4.38, 4.39
Problems due April 8, 2010: 4.42, 4.43, 4.47
Problems due April 15, 2010 (Use sage as much as you want, but try to do a few computations by hand to get a sense of what sage is doing): 5.1, 5.2 (including bonus if you can), 5.3, 5.4 (all parts, using sage), 5.5 (using the sage command list), 5.6, 5.7 (where the trace of Frobenius is what I called a_p).
Problems due April 22, 2010: 5.9, 5.10cd (use sage as appropriate, but try to follow the problem's instructions), 5.11cd, 5.13, 5.15
Last assignment, due May 4, 2010: 5.18b, 5.20 (using sage), 5.30 (using sage), 5.34, 5.39, 5.40

Grading

The scheme will be similar to that of last year's course, where each student had two midterm exam grades between 0 and 30; a final exam grade between 0 and 50; and 11 homework scores, each between 0 and 12. For each student, we computed a composite homework score between 0 and 114 by adding together the 9 highest homework grades and 1/2 of the second lowest homework grade. We then calculated a composite course grade between 0 and 100 by adding together the average of the midterm exam grades, the final exam grade and 20/114 times the composite homework score. The final letter grades respected the ranking by composite course grade. In 2009, there were 29 students who took the final exam. Letter grades were distributed as follows: 11 As, 15 Bs, 3 Cs.

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Mathematics 116 Spring, 2010 Tu Th 11:10AM-12:30 PM, 3109 Etcheverry Hall