Mathematics 116
Spring, 2009
Tu Th 11:10AM-12:30 PM, 3 Evans Hall

Course control number 54488
Current enrollment information

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

Prerequisites

Math 55 is the official prerequisite. In addition, it would be helpful to have had one or more of Math 110, Math 113, Math 115. If you have had a couple of these upper-division courses but have never taken Math 55, that should be fine.

Textbook

An introduction to mathematical cryptography (Springer, Amazon) 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.

Examinations

Please do not plan travel on these dates:

First midterm exam, February 19, 2009 in class (Questions and possible solutions, written by Ribet);
Last midterm exam, April 2, 2009 in class (Questions and possible solutions, written by Ribet);
Final Exam, Tuesday May 19, 2009 8-11 AM (Lucky exam group 13) in 3 Evans (Questions and possible solutions, written by Ribet).

The registrar's 2008-2009 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. If you are contemplating these actions and think that you could benefit from some information and/or advice, contact me immediately after the relevant exam.

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, except that we'll likely have to omit Chapter 6. (See Further notes for the instructor on page xv.) My best guess is that we will have just finished §3.3 (RSA) right before the first midterm and that we will have begun to study elliptic curves before the second midterm. After the second midterm, we will complete our discussion of elliptic curves and then talk about digital signatures and "additional topics."

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

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 due January 29, 2009: 1.6b, 1.7d, 1.8d, 1.9c-d, 1.10c-d, 1.12a, 1.15d, 1.16h, 1.17fg, 1.24, 1.27, 1.28c, 1.29
Problems due February 5, 2009: 1.30a, 1.31, 1.34, 2.4, 2.6, 2.7, 2.8, 2.10
Problems due February 12, 2009: 2.17a, 2.18ab, 2.23 (all parts), 2.24 (all parts), 2.25 (both parts), 2.28a, 2.34ab, 2.35a, 2.37, 2.38, 2.39 with "some" replaced by "two"
Problems due February 24, 2009: 2.40, 3.1ac, 3.4abc, 3.6, 3.8a, 3.9, 3.10, 3.13cde, 3.15, 3.21b, 3.23a, 3.25b
Problems due March 3, 2009: 3.35, 3.38, 3.40, 3.41ab, 3.42, 4.1, 4.2, 4.5
Problems due March 10, 2009
Problems due Thursday, March 19: 4.31, 4.35, 4.37, 4.38, 4.42, 4.46
Problems due Tuesday, April 7: 4.48, 4.51, 4.52, 4.53 plus the following problem.
Problems due April 14: 5.2, 5.5 (a,c), 5.7 (c,d), 5.8, 5.9, 5.11 (c), 5.13, 5.14, 5.18 (a,c)
Problems due April 23: 5.20, 5.22 (a-c), 5.26, 5.28, 5.30, 5.31, 5.32
Problems due May 5: 5.34, 5.35, 5.36, 5.37, 5.38, 5.39, 5.40, 5.41, 7.1, 7.3, 7.4, 7.6, 7.7, 7.11

Grading

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. 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, 2009 Tu Th 11:10AM-12:30 PM, 3 Evans Hall